On Quaternions and Octonions - The Eye · 2020. 1. 17. · rotation can be speciﬁed by a single...

On Quaternions and Octonions

10

A K PetersNatick, Massachusetts

On Quaternions and Octonions:

Their Geometry, Arithmetic, and Symmetry

John H. Conway

Derek A. Smith

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A K Peters, Ltd.63 South AvenueNatick, MA 01760www.akpeters.com

Copyright © 2003 by A K Peters, Ltd.

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Library of Congress Cataloging-in-Publication Data

Conway, John Horton. On quaternions and octonions : their geometry, arithmetic, and symmetry / John H.Conway, Derek A. Smith. p. cm. ISBN 1-56881-134-9 1. Quaternions. 2. Cayley numbers. I. Smith, Derek Alan, 1970- II. Title.

QA196 .C66 2002 512’.5–dc21

2002035555

Printed in Canada07 06 05 04 03 10 9 8 7 6 5 4 3 2 1

This book is dedicated to Lilian Smith and Gareth Conway,without whom we would have finished this book much sooner.

Contents

Preface xi

I The Complex Numbers 1

1 Introduction 31.1 The Algebra R of Real Numbers . . . . . . . . . . . . . . 31.2 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . 51.3 The Orthogonal Groups . . . . . . . . . . . . . . . . . . . 61.4 The History of Quaternions and Octonions . . . . . . . . . 6

2 Complex Numbers and 2-Dimensional Geometry 112.1 Rotations and Reflections . . . . . . . . . . . . . . . . . . 112.2 Finite Subgroups of GO2 and SO2 . . . . . . . . . . . . . 142.3 The Gaussian Integers . . . . . . . . . . . . . . . . . . . . 152.4 The Kleinian Integers . . . . . . . . . . . . . . . . . . . . 182.5 The 2-Dimensional Space Groups . . . . . . . . . . . . . . 18

II The Quaternions 21

3 Quaternions and 3-Dimensional Groups 233.1 The Quaternions and 3-Dimensional Rotations . . . . . . 233.2 Some Spherical Geometry . . . . . . . . . . . . . . . . . . 26

vii

viii Contents

3.3 The Enumeration of Rotation Groups . . . . . . . . . . . 293.4 Discussion of the Groups . . . . . . . . . . . . . . . . . . . 303.5 The Finite Groups of Quaternions . . . . . . . . . . . . . 333.6 Chiral and Achiral, Diploid and Haploid . . . . . . . . . . 333.7 The Projective or Elliptic Groups . . . . . . . . . . . . . . 343.8 The Projective Groups Tell Us All . . . . . . . . . . . . . 353.9 Geometric Description of the Groups . . . . . . . . . . . . 36

Appendix: v → vqv Is a Simple Rotation . . . . . . . . . . 40

4 Quaternions and 4-Dimensional Groups 414.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Two 2-to-1 Maps . . . . . . . . . . . . . . . . . . . . . . . 424.3 Naming the Groups . . . . . . . . . . . . . . . . . . . . . . 434.4 Coxeter’s Notations for the Polyhedral Groups . . . . . . 454.5 Previous Enumerations . . . . . . . . . . . . . . . . . . . . 484.6 A Note on Chirality . . . . . . . . . . . . . . . . . . . . . 49

Appendix: Completeness of the Tables . . . . . . . . . . . 50

5 The Hurwitz Integral Quaternions 555.1 The Hurwitz Integral Quaternions . . . . . . . . . . . . . 555.2 Primes and Units . . . . . . . . . . . . . . . . . . . . . . . 565.3 Quaternionic Factorization of Ordinary Primes . . . . . . 585.4 The Metacommutation Problem . . . . . . . . . . . . . . . 615.5 Factoring the Lipschitz Integers . . . . . . . . . . . . . . . 61

III The Octonions 65

6 The Composition Algebras 676.1 The Multiplication Laws . . . . . . . . . . . . . . . . . . . 686.2 The Conjugation Laws . . . . . . . . . . . . . . . . . . . . 686.3 The Doubling Laws . . . . . . . . . . . . . . . . . . . . . . 696.4 Completing Hurwitz’s Theorem . . . . . . . . . . . . . . . 706.5 Other Properties of the Algebras . . . . . . . . . . . . . . 726.6 The Maps Lx, Rx, and Bx . . . . . . . . . . . . . . . . . . . 736.7 Coordinates for the Quaternions and Octonions . . . . . . 756.8 Symmetries of the Octonions: Diassociativity . . . . . . . 766.9 The Algebras over Other Fields . . . . . . . . . . . . . . . 766.10 The 1-, 2-, 4-, and 8-Square Identities . . . . . . . . . . . 776.11 Higher Square Identities: Pfister Theory . . . . . . . . . . 78

Appendix: What Fixes a Quaternion Subalgebra? . . . . . 80

Contents ix

7 Moufang Loops 837.1 Inverse Loops . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.2 Isotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.3 Monotopies and Their Companions . . . . . . . . . . . . . 86

7.4 Different Forms of the Moufang Laws . . . . . . . . . . . . 88

8 Octonions and 8-Dimensional Geometry 898.1 Isotopies and SO8 . . . . . . . . . . . . . . . . . . . . . . 89

8.2 Orthogonal Isotopies and the Spin Group . . . . . . . . . 91

8.3 Triality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8.4 Seven Rights Can Make a Left . . . . . . . . . . . . . . . . 92

8.5 Other Multiplication Theorems . . . . . . . . . . . . . . . 94

8.6 Three 7-Dimensional Groups in an 8-Dimensional One . . 95

8.7 On Companions . . . . . . . . . . . . . . . . . . . . . . . . 97

9 The Octavian Integers O 999.1 Defining Integrality . . . . . . . . . . . . . . . . . . . . . . 99

9.2 Toward the Octavian Integers . . . . . . . . . . . . . . . . 100

9.3 The E8 Lattice of Korkine, Zolotarev, and Gosset . . . . . 105

9.4 Division with Remainder, and Ideals . . . . . . . . . . . . 109

9.5 Factorization in O8 . . . . . . . . . . . . . . . . . . . . . . 111

9.6 The Number of Prime Factorizations . . . . . . . . . . . . 114

9.7 “Meta-Problems” for Octavian Factorization . . . . . . . . 116

10 Automorphisms and Subrings of O 11910.1 The 240 Octavian Units . . . . . . . . . . . . . . . . . . . 119

10.2 Two Kinds of Orthogonality . . . . . . . . . . . . . . . . . 120

10.3 The Automorphism Group of O . . . . . . . . . . . . . . . 121

10.4 The Octavian Unit Rings . . . . . . . . . . . . . . . . . . 125

10.5 Stabilizing the Unit Subrings . . . . . . . . . . . . . . . . 128

Appendix: Proof of Theorem 5 . . . . . . . . . . . . . . . 131

11 Reading O Mod 2 13311.1 Why Read Mod 2? . . . . . . . . . . . . . . . . . . . . . . 133

11.2 The E8 Lattice, Mod 2 . . . . . . . . . . . . . . . . . . . . 135

11.3 What Fixes λ ? . . . . . . . . . . . . . . . . . . . . . . . 138

11.4 The Remaining Subrings Modulo 2 . . . . . . . . . . . . . 140

x Contents

12 The Octonion Projective Plane OP 2 14312.1 The Exceptional Lie Groups and

Freudenthal’s “Magic Square” . . . . . . . . . . . . . . . . 14312.2 The Octonion Projective Plane . . . . . . . . . . . . . . . 14412.3 Coordinates for OP 2 . . . . . . . . . . . . . . . . . . . . . 145

Bibliography 149

Index 153

Preface

This is a book on the geometry and arithmetic of the quaternion and oc-tonion algebras. These algebras are intimately connected with special fea-tures of the geometry of the appropriate Euclidean spaces, which makesthem a useful tool for understanding symmetry groups in low dimensions.For example, there is a special relationship between 3- and 4-dimensionalgroups that is clearly revealed by the quaternions because a 3-dimensionalrotation can be specified by a single quaternion, and a 4-dimensional oneby a pair of quaternions. The details are subtle, because certain maps are2-to-1 instead of 1-to-1.

Many people are familiar with quaternions, so in the first part of ourbook we take their properties for granted and use them to enumerate thefinite groups in 3 and 4 dimensions (following a similar treatment of 2-dimensional groups using complex numbers). We close the first part with adiscussion of what geometry says about the arithmetic of Hurwitz’s integralquaternions, and in particular establish the unique factorization theorem.

A major theme of the second part of our book is the remarkable “trialitysymmetry” that arises in connection with the octonions. However, theproperties of the octonions are not so familiar, so we start by proving thecelebrated theorem of Hurwitz that R, C, H, and O are the only algebrasof their kind, since this also yields the best way to construct them. Usingthe methods of the proof, we show that these algebras form Moufang loops,and we study Moufang loops in a way that exhibits the triality, applyingthis to the particular case of the 8-dimensional orthogonal group.

The study of the arithmetic of the integral octonions defined by Dickson,Bruck, and Coxeter has not progressed very far. In the final chapters ofour book, we improve this situation in several ways. We use a method dueto Rehm to create a new factorization theory for the integral octonions.

xi

xii Preface

We also describe the action of their automorphism group on importantsubrings, finding maximal subgroups to be stabilizers of certain of these.In one case, we are led to read the integral octonions mod 2.We close with a very brief chapter on the celebrated octonion projec-

tive plane.Very complicated arguments have been needed to deduce properties

of the algebras over arbitrary fields from minimal hypotheses such as thealternative law. We have preferred to keep our arguments simple by startingfrom the composition property and restricting to the classical algebras overthe real field.The programmers of video games make heavy use of quaternions, as do

the controllers of spacecraft, since in both these disciplines it is necessary tocompose rotations with minimal computation. We have eschewed writingof these and other practical applications, which are amply treated in therecent book of Kuipers [29].We thank John Baez, Warren Smith, Daniel Allcock, Warren Johnson,

and Mohammed Abouzaid for their useful comments on various things inthe book. Derek thanks the Academic Research Committee at LafayetteCollege for its financial support through a summer research fellowship.Alice and Klaus Peters have been the most helpful publishers one couldimagine. We thank them, Jonathan Peters, Heather Holcombe, DarrenWotherspoon, Ariel Jaffe, and Susannah Peters for their work on this book.Finally, we thank our wives Barbara and Diana for their patience.

John Conway and Derek Smith

November 2002

I

The Complex Numbers andTheir Applications to 1- and2-Dimensional Geometry

We suppose you understand the real numbers! The complexnumbers are formal expressions x0 + x1i (with x0, x1 real),combined by

(x0 + x1i) + (y0 + y1i) = (x0 + y0) + (x1 + y1)i

(x0 + x1i)(y0 + y1i) = (x0y0 − x1y1) + (x0y1 + x1y0)i;that is to say, they constitute the algebra over the realsgenerated by a basic unit i that satisfies

i2 = −1.

1

1

1

1

Introduction1.1 The Algebra R of Real Numbers

After the work of the ancient Greek and later geometers, it is customaryto parametrize the Euclidean line by the algebra R of real numbers. Inthis parametrization, the distance between a and b is the absolute value oftheir difference, |a − b|, which can also be written as (a− b)2. It is animportant property of R1 that |xy| = |x||y|.The isometries (i.e., the maps that preserve distance) of R1 are the

translations x→ k + x and reflections x→ k − x.

So the 1-dimensional general orthogonal group GO1 consists of the twoisometries x→ ±x that fix the origin.The real numbers R contain the rational numbers Q, and in particular,

the (rational) integers Z that are the subject of number theory. In partic-ular, they satisfy the unique factorization theorem (the essentials of whichwere also discovered by Euclid) whose traditional statement is that eachpositive integer is a product of (positive) prime numbers in a way that isunique up to order.For the purposes of this book, it is better to delete the positivity require-

ment when the statement becomes that each positive or negative integeris a product of positive or negative primes in a way that is unique up toorder and sign change.We briefly sketch the traditional proof, which makes essential use of the

fact that any number n of Z can be divided by any nonzero number d toleave a remainder r strictly smaller than the divisor. (In fact, we can make|r| ≤ 1

2 |d|; see Figure 1.1.)3

4 1

1

1

1 Introduction

0 d 2d 3d 4d

12 |d| 1

2 |d|

Figure 1.1. Maximal remainder for Z.

An ideal of Z is a subset I with the following properties:• 0 ∈ I• The sum of any two elements in I is in I• All of the multiples of any member of I by arbitrary integers areagain in I

It is a principal ideal only if it consists of all multiples of some integer g,called the generator.Then we have

Lemma 1. Any ideal of Z is principal.

Proof. 0 is principal. If I = 0, let d be a non-zero element ofI with smallest absolute value. We show that I must consist of just themultiples of d. For if I contains any other integer, say n, we can write

n = qd+ r, where 0 ≤ r < |d|.But since n and d are in the ideal, so is r (of smaller absolute value thand), a contradiction.We use this to prove

Lemma 2. If p is a prime number, then p divides a product if and onlyif it divides one of the factors.

Proof. It will suffice to suppose that p divides ab. Then the ideal ofnumbers of the form mp+ na must be a principal ideal whose generator gmust be a divisor of p, and so can be chosen to be p or 1. But if g = p,then p divides a, while g = 1 implies 1 = mp + na for some m,n, and sob = mpb+ nab, which is a multiple of p.These lemmas combine to prove the theorem, for if

p1p2 . . . and q1q2 . . .

are two prime factorizations of the same number, we can deduce that q1must divide one of the pi, and so be that pi–say p1–up to sign. Aftercancelling p1, we find that q2 must equal p2 (say) up to sign, etc.

1.2. Higher Dimensions 5

1.2 Higher Dimensions

The results with which we have just opened Chapter 1 concern 1-dimen-sional space R1 and its sublattice Z. It is the purpose of this book togeneralize these results to certain higher dimensions.

Chapter 2 handles the 2-dimensional case, where we discuss plane geom-etry in terms of the algebra C of complex numbers, and say somethingabout its two most famous arithmetics Z[i] of Gaussian integers and Z[ω]of Eisenstein integers.

It is a remarkable fact that the analogous discussion of 3-dimensionalgeometry requires a 4-dimensional algebra, that of quaternions. This madethem hard to discover, as is seen in the famous story Baez quotes later inthis chapter.

In Chapter 3 we define the quaternions H and use them to enumeratethe finite groups of 3-dimensional isometries. In Chapter 4, we providea similar service for 4-dimensional space, which is of course the naturalsetting for the quaternions.

The correspondence between chapters and dimensions is broken in thenext few chapters. Chapter 5 discuss what it means for a quaternion tobe “integral.” We find that Hurwitz’s system of integers has a form ofunique factorization, and so must be preferred to the more naive system ofLipschitz.

What other algebras have properties like those of the quaternions? Itturns out that the most important property [xy] = [x][y] is the one thatdefines “composition algebras.” In Chapter 6 we prove Hurwitz’s famousresult that the only composition algebras are the famous ones in dimen-sions 1, 2, 4, and 8. It turns out that multiplication in the 8-dimensionalcomposition algebra O of octonions is not associative, but it satisfies theMoufang laws, an intriguing substitute for associativity. In Chapter 7, weexplain how the Moufang laws may be regarded as a symmetry condition.

Chapter 8, analogous to Chapter 4, discusses octonions and 8-dimen-sional geometry. We discuss Cartan’s wonderful “triality” of PSO8, anduse it to prove our “7-multiplication theorem.”

Chapter 9, analogous to Chapter 5, explores the correct definitionof integrality for octonions and then erects the correct factorizationtheory for it. The next two chapters study the automorphisms of theoctavian integers in great detail. Chapter 10 explores the units, andChapter 11 shows how more information can be obtained by workingmodulo 2.

Finally, Chapter 12 uses the octonions to construct a most intriguingprojective plane.

6 1

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1

1 Introduction

1.3 The Orthogonal Groups

The General Orthogonal group GOn is the set of all isometries of n-dimensional Euclidean space Rn that fix the origin. The following lemmaimplies that GOn is generated by reflections.

Lemma 3. Every element α of GOn that fixes a k-dimensional subspacecan be written as a product of at most n− k reflections.Proof. Take a vector v not fixed by α, say v → w. Then reflection in

v − w restores w to v (see Figure 1.2) while fixing any vector u fixed by α(for v − w is orthogonal to u since we must have [u, v] = [u,w]).

w vv−w

Figure 1.2. Reflection in v − w.

We shall suppose that the reader is familiar with the notion of deter-minant. Since the determinant of a reflection is −1, the determinant ofany element of GOn is ±1, and the elements of determinant +1 form asubgroup of index 2, the Special Orthogonal group SOn. The productof any two reflections (say in r and s) is a simple rotation – it rotatesthe plane r, s through some angle and fixes the n− 2 dimensional spaceorthogonal to that plane. By grouping reflections in pairs, we see that SOnis generated by these simple rotations.Finally, if in GOn and SOn we ignore the difference between α and −α,

we obtain PGOn and PSOn, the Projective General and ProjectiveSpecial Orthogonal groups. The relationship between the four groupsGOn, SOn, PGOn, and PSOn depends on the parity of the dimension andis discussed in Chapters 3 and 4.It is a consequence of the existence of complex numbers that SO2 and

PSO2 are commutative; of the existence of quaternions that PSO4 ∼=PSO3×PSO3; and of the existence of octonions that PSO8 has a “triality”automorphism of order 3.

1.4 The History of Quaternions and Octonions

When we planned this book, we intended to put our own description ofthe history of our subject into this chapter. Time has prevented us from

1.4. The History of Quaternions and Octonions 7

doing so, but fortunately we have been saved by John Baez [5]. With hispermission, we quote from his paper, “The Octonions”:

Most mathematicians have heard the story of how Hamiltoninvented the quaternions. In 1835, at the age of 30, he haddiscovered how to treat complex numbers as pairs of real num-bers. Fascinated by the relation between C and 2-dimensionalgeometry, he tried for many years to invent a bigger algebra thatwould play a similar role in 3-dimensional geometry. In modernlanguage, it seems he was looking for a 3-dimensional normeddivision algebra. His quest built to its climax in October 1843.He later wrote to his son, “Every morning in the early part ofthe above-cited month, on my coming down to breakfast, your(then) little brother William Edwin, and yourself, used to askme: ‘Well, Papa, can you multiply triplets?’ Whereto I wasalways obliged to reply, with a sad shake of the head: ‘No, Ican only add and subtract them’.” The problem, of course, wasthat there exists no 3-dimensional normed division algebra. Hereally needed a 4-dimensional algebra.

Finally, on the 16th of October, 1843, while walking with hiswife along the Royal Canal to a meeting of the Royal IrishAcademy in Dublin, he made his momentous discovery. “Thatis to say, I then and there felt the galvanic circuit of thoughtclose; and the sparks which fell from it were the fundamentalequations between i, j, k; exactly such as I have used them eversince.” And in a famous act of mathematical vandalism, hecarved these equations into the stone of the Brougham Bridge:

i2 = j2 = k2 = ijk = −1.

One reason this story is so well-known is that Hamilton spentthe rest of his life obsessed with the quaternions and their ap-plications to geometry [21], [24]. And for a while, quaternionswere fashionable. They were made a mandatory examinationtopic in Dublin, and in some American universities they werethe only advanced mathematics taught. Much of what we nowdo with scalars and vectors in R3 was then done using realand imaginary quaternions. A school of “quaternionists” de-veloped, which was led after Hamilton’s death by Peter Taitof Edinburgh and Benjamin Peirce of Harvard. Tait wrote 8books on the quaternions, emphasizing their applications to

8 1

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1

1 Introduction

physics. When Gibbs invented the modern notation for thedot product and cross product, Tait condemned it as a “her-maphrodite monstrosity.” A war of polemics ensued, with lu-minaries such as Kelvin and Heaviside writing some devastatinginvective against quaternions. Ultimately the quaternions lost,and acquired a slight taint of disgrace from which they havenever fully recovered [12].

Less well-known is the discovery of the octonions by Hamilton’sfriend from college, John T. Graves. It was Graves’ interest inalgebra that got Hamilton thinking about complex numbers andtriplets in the first place. The very day after his fateful walk,Hamilton sent an 8-page letter describing the quaternions toGraves. Graves replied on October 26th, complimenting Hamil-ton on the boldness of the idea, but adding, “There is still some-thing in the system which gravels me. I have not yet any clearviews as to the extent to which we are at liberty arbitrarily tocreate imaginaries, and to endow them with supernatural prop-erties.” And he asked: “If with your alchemy you can makethree pounds of gold, why should you stop there?”

Graves then set to work on some gold of his own! On Decem-ber 26th, he wrote to Hamilton describing a new 8-dimensionalalgebra, which he called the “octaves.” He showed that theywere a normed division algebra, and used this to express theproduct of two sums of eight perfect squares as another sum ofeight perfect squares: the “eight squares theorem” [23].

In January 1844, Graves sent three letters to Hamilton expand-ing on his discovery. He considered the idea of a general theoryof “2m-ions,” and tried to construct a 16-dimensional normeddivision algebra, but he “met with an unexpected hitch” andcame to doubt that this was possible. Hamilton offered to pub-licize Graves’ discovery, but being busy with work on quater-nions, he kept putting it off. In July he wrote to Graves pointingout that the octonions were nonassociative: “A·BC = AB ·C =ABC if A,B,C be quaternions, but not so, generally, with youroctaves.” In fact, Hamilton first invented the term “associative”at about this time, so the octonions may have played a role inclarifying the importance of this concept.

Meanwhile the young Arthur Cayley, fresh out of Cambridge,had been thinking about the quaternions ever since Hamiltonannounced their existence. He seemed to be seeking relation-ships between the quaternions and hyperelliptic functions. In

1.4. The History of Quaternions and Octonions 9

March of 1845, he published a paper in the Philosophical Mag-azine entitled “On Jacobi’s Elliptic Functions, in Reply to theRev. B. Bronwin; and on Quaternions” [9]. The bulk of thispaper was an attempt to rebut an article pointing out mistakesin Cayley’s work on elliptic functions. Apparently as an after-thought, he tacked on a brief description of the octonions. Infact, this paper was so full of errors that it was omitted fromhis collected works–except for the part about octonions [10].

Upset at being beaten to publication, Graves attached a post-script to a paper of his own which was to appear in the fol-lowing issue of the same journal, saying that he had known ofthe octonions ever since Christmas, 1843. On June 14th, 1847,Hamilton contributed a short note to the Transactions of theRoyal Irish Academy, vouching for Graves’ priority. But it wastoo late: the octonions became known as “Cayley numbers.”Still worse, Graves later found that his eight squares theoremhad already been discovered by C. F. Degen in 1818 [13], [14].

Why have the octonions languished in such obscurity comparedto the quaternions? Besides their rather inglorious birth, onereason is that they lacked a tireless defender such as Hamilton.But surely the reason for this is that they lacked any clear ap-plication to geometry and physics. The unit quaternions formthe group SU2, which is the double cover of the rotation groupSO3. This makes them nicely suited to the study of rotationsand angular momentum, particularly in the context of quan-tum mechanics. These days we regard this phenomenon as aspecial case of the theory of Clifford algebras. Most of us nolonger attribute to the quaternions the cosmic significance thatHamilton claimed for them, but they fit nicely into our under-standing of the scheme of things. The octonions, on the otherhand, do not.

Our attempts to understand the “cosmic significance” of the octonionshave led to this book!To Baez’s brief history we add that although Hamilton really does seem

to have been the first to construct the quaternions as an algebra, they have aprior history, starting with Euler’s discovery of the 4-square identity (whichreally defines them) in 1748. Also, O. Rodrigues, in work culminating ina brilliant paper [38] of 1840, parametrized the general rotation by fourparameters (often incorrectly called the Euler-Rodrigues parameters) thatare in fact the coordinates of the corresponding quaternion. This makeshim a precursor of Hamilton, since the multiplication rule he gives for

10 1

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1

1 Introduction

rotations is the same as Hamilton’s 1843 formula for the product of twoquaternions. For a fuller description of Rodrigues’ work, see Rotations,Quaternions, and Double Groups by S. Altmann [3].The most important later contributions relevant to this book are Hur-

witz’s characterization of composition algebras [26]; Dickson’s work (see[15]), including his doubling rule and his discussion of “integrality”; andCoxeter’s geometric [11] and Rankin’s arithmetic [36] investigations of theintegral octonions O. Here and elsewhere in the book we give very fewreferences and historical remarks. We recommend that interested readersconsult

http://www.akpeters.com/QANDO ,

which contains additional references and a link to the online version ofBaez’s paper.

2

Complex Numbers and2-Dimensional GeometryWe suppose that our readers are familiar with the algebra of complex num-bers. In this chapter we focus on their relation to the geometry of theEuclidean plane, as a prelude to the corresponding applications of quater-nions and octonions to higher-dimensional Euclidean spaces. We also dis-cuss the arithmetic of the two most interesting subrings of the complexnumbers–the Gaussian and Eisenstein integers.

2.1 Rotations and Reflections

The geometrical properties of the complex numbers follow from the factthat they form a composition algebra for the Euclidean norm

N(x+ iy) = x2 + y2,

which means that

N(z1z2) = N(z1)N(z2).

This entails that multiplication by a fixed (nonzero) number z0 multipliesall lengths by N(z0); that is to say, it is a Euclidean similarity (seeFigure 2.1).In the important case when N(z0) = 1, z → z0z is a Euclidean con-

gruence, or isometry. Moreover, any such “unit” z0 can be continuouslyreached from 1 by a path that traverses only units (see Figure 2.2), sothis congruence can be obtained by a continuous motion from the identity.Such congruences are called rotations.

11

12 2 Complex Numbers and 2-Dimensional Geometry

The standard formula

x = x cos θ − y sin θy = x sin θ + y cos θ

for a rotation follows immediately from the multiplication

(cos θ + i sin θ)(x+ iy) = (x cos θ − y sin θ) + i(x sin θ + y cos θ)

and the definitions of the trigonometric functions (see Figure 2.3).However, reflections are another kind of Euclidean congruence, exem-

plified by

x = xy = −y

which corresponds to complex conjugation

x+ iy = x− iy.

We have

Theorem 1. If u is a complex unit, then the map z → uz is a rotation,while z → uz is a reflection.

The matrices of these two are

cos θ sin θ− sin θ cos θ

andcos θ sin θsin θ − cos θ

whose determinants, 1 and −1, show that they belong respectively to SO2and GO2\SO2.Moreover, every element (aij) of the 2-dimensional orthogonal group

GO2 is of one of these two types. For the first orthogonality conditiona211+a

212 = 1 implies a11 = cos θ, a12 = sin θ for some θ, and the remaining

conditions imply a21 = ∓ sin θ, a22 = ± cos θ.We have identified the 2-dimensional orthogonal groups:

Theorem 2. SO2 consists of all multiplications z → uz by complexunits, while GO2 consists of these together with the maps z → uz.

This provides a topological identification of SO2 as the circle of realangles θ, considered mod 2π. GO2 has two components, each of which is acircle.

2.1. Rotations and Reflections 13

z0

1

i

z

iz0

z20

zz0

Figure 2.1. The Euclidean similarity z → zz0.

1

z0

Figure 2.2. Euclidean congruences are connected to the identity.

1−1i

z

zi

zi2 = −z

Figure 2.3. The Euclidean rotation z → zi.


2.2 Finite Subgroups of GO2 and SO2

Let us consider the smallest positive θ for which the rotation through θbelongs to a given finite subgroup G of SO2. Then θ has the form

2πm , for

if m is the smallest positive integer for which mθ ≥ 2π, we must in facthave mθ = 2π, since otherwise if mθ = 2π + θ , we have 0 < θ < θ.Also, G is generated by the rotation through 2π

m , since if it containeda rotation through any other angle θ, we could write θ as a multiple of 2πmplus a positive “remainder angle” φ strictly less than 2π

m , a contradiction.This is the rotational or chiral point group m• in the “orbifold notation”(see Appendix).So, we have proved that

Theorem 3. The general finite subgroup of SO2 is m•, consisting ofrotations through multiples of 2πm .

The subgroups of SO2 are called chiral (“handed”) because they pre-serve the left- and right-handedness of 2-dimensional objects (see Fig-ure 2.4). Correspondingly, the subgroups of GO2 not contained in SO2 areachiral (“unhanded”) since they can reverse handedness (see Figure 2.5).By adjoining the reflection in any line through the origin, we obtain the

reflectional or achiral point group, denoted ∗m• in the orbifold notation.It is now easy to see that

Theorem 4. The general finite subgroup of GO2 is ∗m•

(since its chiral subgroup of index 2 must be m• for some m).

!

Figure 2.4. 2-dimensional chirality.

?

Figure 2.5. 2-dimensional achirality.

2.3. The Gaussian Integers 15

2.3 The Gaussian Integers

Gauss defined a notion for complex numbers analogous to integrality forreal numbers. Namely, the complex number x+ iy is a Gaussian integerjust if its real and imaginary parts are ordinary integers. The Gaussianintegers form a ring, since the sum, difference, or product of two Gaussianintegers is a third.Geometrically, the Gaussian integers form a square lattice; see Fig-

ure 2.6, in which two points closest to each other differ by one of the fourGaussian units, 1, −1, i, −i.Perhaps the most interesting property of the Gaussian integers is their

unique factorization theorem:

Theorem 5. Each nonzero, nonunit Gaussian integer has a factoriza-tion π1π2 . . .πk into Gaussian primes. From any one such factorization,we can get to any other by

• reordering the primes• unit-migration: replace πs,πs+1 by πsu, uπs+1

Here a Gaussian prime is a Gaussian integer whose norm is an ordinaryprime, so that in all its 2-term factorizations, just one factor is a unit.

−4−2i −3−2i −2−2i −1−2i 1−2i 2−2i 3−2i 4−2i

−4−i −3−i −2−i −1−i 1−i 2−i 3−i 4−i

−4 −3 −2 −1 1 2 3 4

−4+i −3+i −2+i −1+i 1+i 2+i 3+i 4+i

−4+2i −3+2i −2+2i −1+2i 1+2i 2+2i 3+2i 4+2i

−2i

−i

0

i

2i

Figure 2.6. The Gaussian integers.


When we proved the unique factorization theorem for ordinary integersin Chapter 1, we only used the fact that you could divide one numberby another and get a smaller remainder. In fact, if you allow negativeremainders, you can get within 1

2 |d| of a multiple of the number d you’redividing by (see Figure 1.1, where |d| is the distance from 0 to d).Figure 2.7 shows how every Gaussian integer is within |d|/√2 of a mul-

tiple of a Gaussian integer d. Since 1/√2 is less than 1, the remainder is

smaller in size than the divisor and our proof of unique factorization willwork as well for the Gaussian integers.The units of the Gaussian integers are the fourth roots of unity. Gauss’s

disciple, Eisenstein, gave an alternative system involving the third and sixthroots of unity. The Eisenstein integers are the numbers a + bω, whereω = (−1 + i√3)/2 is one of the roots of x3 = 1, the other two being 1and ω2 = (−1 − i√3)/2. The Eisenstein integers form a triangular lattice(see Figure 2.8). Figure 2.9 is similar to Figure 2.7 and shows that everyEisenstein integer is within a distance 1√

3|d| of some multiple of a given

Eisenstein integer, so that these integers, too, have unique factorizationinto primes.

0

d

2d

3d

4d(1 + i)d

(2 + i)d

(3 + i)d

(1 + 2i)d

(2 + 2i)d

id

|d|√2

Figure 2.7. Maximal remainder for the Gaussian integers.

2.3. The Gaussian Integers 17

−4−2ω −3−2ω −2−2ω −1−2ω −2ω 1−2ω 2−2ω

−3−ω −2−ω −1−ω=ω2

−ω 1−ω 2−ω

−3 −2 −1 0 1 2 3

−2+ω −1+ω ω 1+ω 2+ω 3+ω

−2+2ω −1+2ω 2ω 1+2ω 2+2ω 3+2ω 4+2ω

Figure 2.8. The Eisenstein integers.

0

d

2d

3d(1 + ω)d

(2 + ω)d

(3 + ω)d

(2 + 2ω)d

|d|√3

Figure 2.9. Maximal remainder for the Eisenstein integers.


2.4 The Kleinian Integers

It is a famous theorem, proved by Heegner and reestablished independentlyby A. Baker and H. M. Stark, that there are only nine imaginary quadraticfields with unique factorization, namely those with discriminants

3, 4, 7, 8, 11, 19, 43, 67, 163.

The next simplest ring to the Eisenstein (discriminant 3) and Gaussian(discriminant 4) ones has been called “the Kleinian ring” , which consists

of all numbers of the form a + bλ (a, b ∈ Z), where λ = −1+√−72 (see

Figure 2.10). The most important features are that this ring has just the

two units ±1, and that 2 factorizes as λµ, where µ = −1−√−72 .

−3−λ −2−λ −1−λ −λ 1−λ 2−λ

−3 −2 −1 0 1 2 3

−2+λ −1+λ λ 1+λ 2+λ 3+λ

−2+2λ −1+2λ 2λ 1+2λ 2+2λ 3+2λ 4+2λ

Figure 2.10. The Kleinian ring.

2.5 The 2-Dimensional Space Groups

We have already discussed the 2-dimensional point groups in the body ofthis chapter. Here we briefly describe the 17 2-dimensional space groups,since their theory is closely related to that of the 3-dimensional point groupsof the next chapter.

2.5. The 2-Dimensional Space Groups 19

2222

333442 632

∗∗ ×× ∗× ∗2222 22×

22∗ 2∗22∗442 4∗2

∗333 3∗3 ∗632Figure 2.11. We here reproduce G. Polya’s 1924 illustration representing the 17different plane symmetry groups, replacing his names with the orbifold names.


In the orbifold notation, the 17 groups are

∗632, 632∗442, 4∗2, 442∗333, 3∗3, 333∗2222, 2∗22, 22∗, 22×, 2222∗∗, ∗×,××,

where

• A digit A ≥ 2 not following a ∗ indicates a type of point with localsymmetry A•.

• A string of digits a1 ≥ 2, a2 ≥ 2, . . . , ak ≥ 2 after a ∗ indicates a typeof kaleidoscope, i.e., a connected system of mirror lines on which thereare distinct types of point with local symmetries ∗a1•, ∗a2•, . . . , ∗ak•.

• × indicates a “miracle,” i.e., the presence of an orientation-reversingpath that meets no mirror.

• indicates a “wonder,” i.e., a repetition of motif not yet accountedfor.

(We do not give more precise definitions here.)

Each of these features has a cost:

Feature CostA (A− 1)/A∗ 1

ai after ∗ (ai − 1)/2ai× 1 2

and the “Magic Theorem” asserts that the above 17 groups are preciselythose whose total cost is 2 (see Figure 2.11).The proof of the Magic Theorem relates these names to the features of

the orbifold, the plane considered modulo the group. In this interpreta-tion, ∗a1a2 . . . ak denotes a boundary with k corners having angles π

a1, πa2,

. . . , πak, while A (not after a ∗) is a cone point of angle 2π

A , × denotes acrosscap, and denotes a handle.

II

The Quaternions andTheir Applications to 3- and4-Dimensional Geometry

The quaternions are formal expressions x0+x1i+x2j+x3k(with x0, x1, x2, x3 real), combined by

(x0 + x1i+ x2j + x3k) + (y0 + y1i+ y2j + y3k)

= (x0 + y0) + (x1 + y1)i+ (x2 + y2)j + (x3 + y3)k

(x0 + x1i+ x2j + x3k)(y0 + y1i+ y2j + y3k)=

(x0y0 − x1y1 − x2y2 − x3y3)+(x0y1 + x1y0 + x2y3 − x3y2)i+(x0y2 − x1y3 + x2y0 + x3y1)j+(x0y3 + x1y2 − x2y1 + x3y0)k;

that is to say, they constitute the algebra over the realsgenerated by basic units i, j, k that satisfy Hamilton’s cel-ebrated equations

ij = k jk = i ki = j

ji = −k kj = −i ik = −j

i2 = j2 = k2 = −1.

3

3

3

3

Quaternions and3-Dimensional GroupsIn the previous chapter, we enumerated the finite subgroups of SO2 andGO2 and related them to complex numbers. Our aim in this chapter isto enumerate the finite subgroups of SO3 and GO3 and relate them toquaternions.

3.1 The Quaternions and 3-Dimensional Rotations

What’s the relation between quaternions and 3-dimensional geometry? Ul-timately, this comes from the fact that the norm of a product of quaternionsis the product of their norms. This implies

N(q1vq2) = N(q1)N(v)N(q2),

showing that the map v → q1vq2 is a similarity of Euclidean 4-space thatmultiplies lengths by N(q1)N(q2), and so is a congruence ifN(q1)N(q2) =1. If this operation fixes v = 1, it will also fix the 3-dimensional spaceperpendicular to 1, whose typical elements are the vectors of the formxi+ yj + zk (i.e., with no “real part”), which we call (3-dimensional) vec-tors. Since the condition for this is q1q2 = 1, we obtain

Theorem 1. The map [q] : v → q−1vq is a congruence of Euclidean3-space.

In fact, this congruence is a simple rotation, that is to say, it fixesa vector (in n dimensions, a simple rotation is one that fixes an (n − 2)-

23

24 3

3

3

3 Quaternions and 3-Dimensional Groups

space). To be more precise, we shall show in the appendix to this chapterthat if q = r(cos θ+u sin θ), where u is a unit 3-vector, then [q] is a rotationthrough the angle 2θ about u.Noting that every simple rotation takes this form (by choice of u and

θ), we obtain an immediate proof of

Theorem 2. (Euler’s Theorem on Rotations) The product of anytwo simple rotations is another.

For, the product of simple rotations

x→ q−11 xq1 and x→ q−12 xq2

is x→ (q1q2)−1x(q1q2), another simple rotation. We shall give a geometric

proof of Euler’s theorem in the next section.Since we showed in Chapter 1 that the special orthogonal group is

generated by simple rotations, we have

Theorem 3. Every element of SO3 has the form x→ q−1xq for somequaternion q, and is a simple rotation.

At first sight, this correspondence between quaternions and rotationsis “∞-to-1” since all the non-zero scalar multiples of a given quaternionplainly yield the same rotation. We reduce this multiplicity by demandingthat q be a unit quaternion; this makes it 2-to-1 since two opposite unitquaternions q and −q yield the same rotation.Theorem 4. The map that takes q to the map [q] : x → q−1xq is a

2-to-1 homomorphism from the group of unit quaternions to SO3.

In the standard language, the set of unit quaternions is a “double cover”of SO3 called the spin group Spin3. The spin groups exist in all dimen-sions, but to give their general definition would lead us too far astray.We shall meet just three more instances, Spin4, Spin7, and Spin8, in laterchapters.One of the subtleties of this subject is the presence of several distinct

2-to-1 maps: our notation makes these explicit by using square brackets(when necessary). Thus, we have seen that two unit quaternions +q, −qdetermine the same 3-dimensional isometry [q] : x → qxq. Again, twon-dimensional isometries +g, −g in GOn determine the same projectiveisometry [g] in PGOn. We’ll use [[q]] rather than [[q]] for the result ofsuccessively applying two such maps to q.The rest of this chapter describes the finite subgroups of GO3 (see

Figure 3.1) and the way they are represented by quaternions. The nextfew sections first enumerate the finite subgroups of SO3. We start ournext section with some useful spherical geometry.

3.1. The Quaternions and 3-Dimensional Rotations 25

Holo-icosahedral group∗532 = ±I

Holo-octahedral group∗432 = ±O

Holo-tetrahedral group

∗332 = TO

Holo-pyramidal group

∗nn

Holo-prismatic group∗22n = ±D(n)/DD(2n)

Holo-antiprismatic group2∗n = DD(2n)/±D(n)

Chiro-icosahedral group532 = +I

Chiro-octahedral group432 = +O

Chiro-tetrahedral group332 = +T

Chiro-pyramidal group

nn

Chiro-prismatic group

22n = D(n)Chiro-antiprismatic group

Pyritohedral group3∗2 = ±T

Pro-prismatic groupn∗ = ±C(n)/CC(2n)

Pro-antiprismatic groupn× = CC(2n)/± C(n)

Figure 3.1. 3-dimensional groups from polyhedra (see also Section 3.9).

26 3

3

3


3.2 Some Spherical Geometry

We begin with

Theorem 5. Any finite group of congruences of Euclidean n-space fixesa point.

For if P1, P2, . . . , PN are the images of any point P1, then1N (P1 + P2 +· · ·+ PN) is plainly fixed.

So, from now on, we will suppose that the finite group we are interestedin fixes the origin, and we study it in terms of its action on the unit sphere.

Theorem 6. The product of simple rotations about the vertices of aspherical triangle, through twice the angles of that triangle (with the signconvention as in Figure 3.2), is the identity.

For if P , Q, R are reflections in the sides of that triangle, then the threerotations are PQ, QR, RP , (where PQ means “P followed by Q”) and

PQ ·QR · RP = PQ2R2P = PP = 1.

In particular, since we can “complete the triangle” for any two rotationsA (through 2α) and B (through 2β) as in Figure 3.3, we obtain a geometricproof of Euler’s theorem on rotations.

P

Q

R

α

β

γ

A

B

C

2α

2β

2γ

Figure 3.2. PQ is the indicated rotation about A through angle 2α, QR is theindicated rotation about B through angle 2β, and RP is the indicated rotation aboutC through angle 2γ.

3.2. Some Spherical Geometry 27

A

B

C

2α

2β

2γ

Figure 3.3. The product of rotations through 2α about A and 2β about B is onethrough 2γ about C, completing the triangle.

So a group generated by simple rotations consists entirely of such ro-tations. These rotation groups are the most important groups in whatfollows, and we’ll see later that each of them can be generated by tworotations.A reflection group is a group generated by reflections.

Theorem 7. A group generated by two rotations R1, R2 is containedto index 2 in a reflection group.

We obtain the desired reflection group from the rotation group by adjoiningM , the reflection in the plane containing the axes of R1 and R2. This isa reflection group, since we can write R1 = MM1, R2 = MM2 for twofurther reflections M1 = MR1, M2 = MR2. Any element of it can berepresented by a word in R1, R2, M , in which we can use the relationsR1M = MR1, R2M = MR2 to bring all occurrences of M to the front,and then use M2 = 1 to put it into the form w or Mw, where w is a wordin R1, R2 only.

Theorem 8. Any finite reflection group is generated by the reflectionsin the sides of a convex spherical polygon whose angles are of the form π

n .

For consider the way that the fixed planes of all of the reflections in thegroup cut the unit sphere. Plainly they divide the sphere into a number ofconvex polygons. Also, when two such polygons abut along a common edge,they are reflected into each other by the reflection across that edge. Thisshows that all the polygons have the same shape, and that the reflectionscorresponding to one generate those corresponding to its neighbor, and soby repetition, generate the whole group.

28 3

3

3


In the orbifold notation, the group generated by reflections in sides of apolygon of angles π

a ,πb , . . . ,

πz is called ∗ab . . . z. We allow digits of 1 to be

freely inserted or deleted in orbifold notations, since this simplifies generalstatements.

Theorem 9. (The Spherical Excess Theorems)

A+B + C = π +4A+B + C +D = 2π +2

A+B + C +D + E = 3π +...

where 4(2, , . . .) denotes the area of a triangle (quadrilateral, pentagon,. . . ) with angles A,B,C (D, E, . . . ).

Figure 3.4 reduces the proof for higher polygons to that of the triangle,which is proved by Figures 3.5 and 3.6.

π +41

π +42

π +4n

Figure 3.4. Triangulating a higher polygon and its angle sum.

θ

Area 2θθ

Figure 3.5. The area of a lune is obviously proportional to its angle θ, and mustbe 2θ since the total area of the sphere is 4π.

3.3. The Enumeration of Rotation Groups 29

A

B

CA

B

C

Area 2A

Area 2B

Area 2C

Figure 3.6. The sphere (of total area 4π) is divided into eight triangles, of whichthe four shaded ones have area 2π (since opposite triangles have the same area).The total area of the three shaded lunes is 2A+2B+2C, which must be 2π+2since the common triangle of area is covered three times.

3.3 The Enumeration of Rotation Groups

We first find the corresponding enumeration of reflection groups, namely:

Theorem 10. Every finite reflection group is one of

∗532 ∗432 ∗332 ∗22n ∗nn (n ≥ 1).

For the angle sum for a k-gon is kπ2 , but by the previous theorem it must

be greater than (k − 2)π, implying k ≤ 3.Now, if k = 3, the largest angle must be π

2 (otherwise the angle sumwould be at most π

3 +π3 +

π3 = π), the next largest must be π

2 orπ3

(otherwise the angle sum would be at most π2 +

π4 +

π4 = π), and finally

if those were π2 and

π3 , the remaining angle must be at least

π5 (otherwise

the angle sum would be at most π2 +

π3 +

π6 = π). This gives the cases

∗22n, ∗233, ∗234, ∗235.If k = 2, the polygon is a lune formed by two great semicircles, whose

two angles are equal, and if they are πn , the group is ∗nn (which when n = 1

includes the case when the “polygon” is a hemisphere).

30 3

3

3


The cornerstone of this chapter is

Theorem 11. The finite rotation groups are

532 432 332 22n nn (n ≥ 1).

Firstly, a preceding argument shows that this is true for those groupsthat can be generated by at most two rotations.Can there be a group G that needs more than two generating rotations?

No, because any two rotations of G generate one of the above, which re-stricts their distances, and a further rotation center would be prohibitivelynear to one of the existing ones (see Figure 3.7).Therefore, we have found infinitely many groups, but we find it more

convenient to speak of them as “five groups,” two of which depend on aparameter.

3.4 Discussion of the Groups

The first three groups are collectively called the polyhedral rotationgroups, since they consist of the rotations that fix the appropriate regularpolyhedra, while the remaining two are the axial rotation groups, sincethey fix an axis.Abstractly, they are isomorphic to various well-known permutation

groups G(n) on n letters, as seen in Figure 3.8. The individual cases are

532 = I = I60 ∼= A(5), the icosahedral group432 = O = O24 ∼= S(4), the octahedral group332 = T = T12 ∼= A(4), the tetrahedral group22n = D = D2n ∼= D(n), the dihedral groupnn = C = Cn ∼= C(n), the cyclic group

These are the traditional, rather confusing names. We shall later pre-fix them by “chiro-” to distinguish these rotation groups from the “holo-polyhedral” groups, which are the full groups of symmetries of the appro-priate polyhedra.There is some confusion in the literature since group theorists usually

write D2n for the dihedral group of order 2n, while geometers prefer namesthat display the parameter n. As geometers who are also group theorists,we sometimes write D2n and sometimes D(n), so doing both!

3.4. Discussion of the Groups 31

3

3

2 5

d3

d3

d5

α

3

3

2 4

d3

d3

d4

β

3

3

2 3

d3

d3

d3

γ

n

n

2 2

dn

dn

d2

δ

Figure 3.7. We’ll write n for a center of n-fold rotation and dn for the shortestdistance from one of these to any other rotation center. The smallest possible valuesfor d5 and d3 are seen in 532, soG can’t contain 532 since any putative new rotationcenter must be within d5 or d3 of 2 (see α). Otherwise, the smallest d4 and d3 arein 432, which is precluded since any new center would be within d4 or d3 of 2 (seeβ). The next smallest value of d3 is in 332, which is precluded for a similar reason(see γ). In all remaining cases, if the largest order rotation is n-fold (n ≥ 2), thenit and any other rotation can only generate a 22n and so dn =

π2 and d2 =

πn ,

which again precludes a further rotation (see δ).

32 3

3

3


12

34 45

53

51 23

31

2441

25

1 2

3 4 1

2 3

. . .n 1 2 3

. . .

n 12

The opposite faceto ij is ji

The Icosahedral group I consists ofthe rotations that fix an icosahedron.If the faces are labelled by pairs ofnumbers from 1, . . . , 5 as shown, itinduces the alternating group A(5) of

degree 5.

Opposite faceshave same label

Octahedral group Oinduces S(4)

Back facelabelled 4

Tetrahedral group Tinduces A(4)

Pyramid

Cyclic group C=Cninduces C(n)

The Dihedral group D=Dn consistsof the rotations that fix a prism, whoserectangular faces are numbered1, . . . , n. It is isomorphic to

D(n) = (12 . . . n), (1 n)(2 n−1) . . . .

Figure 3.8. The polyhedral and axial groups are isomorphic to various permutationgroups D(n) on n symbols.

3.5. The Finite Groups of Quaternions 33

3.5 The Finite Groups of Quaternions

We pause to enumerate the finite groups of quaternions. Under the 2-to-1map that takes a unit quaternion q to the rotation [q] : x → qxq, a finitegroup Q of quaternions must be taken to a group [Q] = [q] | q ∈ Q, say,among C, D, T , O, I. The order of Q will be 2 or 1 times that of [Q],according to whether −1 is or is not in Q.The cases when −1 ∈ Q we call 2Cn, 2D2n, 2T, 2O, 2I, with 2G meaning

q | [q] ∈ G. If −1 /∈ Q, then G can contain no order 2 rotation g, since if[q] = g then q2 = −1 must be in Q. This leaves only the cases Cn (n odd),which is the image of an order n group Q (eg. e2πi/n ), to which we givethe name 1Cn. We summarize:

Theorem 12. The finite groups of quaternions are

2I 2O 2T 2D2n 2Cn 1Cn (n odd).

Generators for these groups are given below:

2I = iI ,ω iI =i+σj+τk

2 , σ =√5−12 , τ =

√5+12

2O = iO,ω where iO =j+k√2, ω = −1+i+j+k2

2T = iT ,ω iT = i

2D2n = en, j en = eπi/n

2Cn = en1Cn = en/2

3.6 Chiral and Achiral, Diploid and Haploid

The word “chiral” (meaning “handed”) was introduced to science by LordKelvin before 1896 to denote objects that cannot be moved into coincidencewith their mirror images. For the opposite concept, the word “achiral”(“unhanded”) or “amphichiral” (“either-handed”) has been used.We shall use these terms not only for such objects, but also for their

symmetry groups. Thus, a symmetry group is chiral if all its operationscan be achieved by rigid motions and achiral otherwise.1

For a finite physical object, all symmetries fix the center of gravity, and,when we take this to be the origin, are represented by orthogonal matriceswhose determinants are necessarily ±1. The group is chiral just if every de-terminant is +1, and achiral if it contains some element of determinant −1.

1We adhere to this standard usage, despite the inconsistency that each 3-dimensional“chiral group” coincides with its own mirror image. We expand on this in Chapter 4.

34 3

3

3


The dual distinction does not seem hitherto have been named. Weshall do so by borrowing a pair of terms from the biologists. We call agroup diploid (“formed of couples”) if it contains the “central symmetry”−1 : x → −x, because then its matrices come in pairs ±g. The othergroups are called haploid (“formed of singletons”).In general, these distinctions yield four classes of groups: chiral diploid,

chiral haploid, achiral diploid, and achiral haploid. However, here one typeis missing. For since we are working in an odd number of dimensions,the central symmetry −1 has determinant −1, and so a diploid group isnecessarily achiral. We therefore get only three classes, namely:• The diploid groups (which are necessarily achiral)• The chiral groups (which are necessarily haploid), and• The hybrid groups (namely those that are achiral but haploid).In the next few pages, we use these notions to enumerate types of

groups.

3.7 The Projective or Elliptic Groups

To work projectively is to regard elements as identical if they are scalarmultiples of each other. For us, the only relevant scalars are ±1, so workingprojectively replaces +g and −g by a single element [g].Recall that GOn, the n-dimensional general orthogonal group, consists

of all n × n orthogonal matrices, or equivalently all congruences of n-dimensional Euclidean space that fix the origin, while SOn, the n-dimensionalspecial orthogonal group, consists of all elements of GOn of determinant1. From these we obtain PGOn and PSOn, the n-dimensional projectivegeneral and special orthogonal groups, by working projectively.However, for odd n, we have det(−1) = −1, which entails that PGOn =

PSOn and also PSOn ∼= SOn. This is because just one of g,−g has de-terminant 1, so that their projective image is the projective image of this“positive” one, while considering SOn projectively involves no identifica-tion. Applied to the finite subgroups, this gives

Theorem 13. The finite subgroups of PGO3 = PSO3 are

[I60] [O24] [T12] [D2n] [Cn].

The typical operation [g] of each is represented by a pair of opposite ma-trices: one we call +g of determinant 1, and one, −g, of determinant −1.Knowing the projective groups enables us to enumerate all the subgroupsof GO3, as follows.

3.8. The Projective Groups Tell Us All 35

3.8 The Projective Groups Tell Us All

Chiral groups: Any such consists just of the elements +g for [g] in someprojective group [G]: We call this +G. The cases are:

+C, the cyclic rotation group+D, the dihedral rotation group+T, the tetrahedral rotation group+O, the octahedral rotation group+I, the icosahedral rotation group

In quaternionic terms, +G consists of the maps x→ q−1xq, q ∈ 2G.Diploid groups: Any such must consist of pairs of elements +g,−g

for [g] ∈ [G]. We therefore call it ±G. The cases are:±C, the diplo-cyclic group±D, the diplo-dihedral group±T, the diplo-tetrahedral group±O, the diplo-octahedral group±I, the diplo-icosahedral group

The group ±G consists of the maps x→ ±q−1xq, q ∈ 2G.The hybrid groups are more subtle: There is a projective group [G]

obtained by ignoring signs, and for each element of [G] we must makea choice of sign. Namely, the sign will be + for a certain subgroup [H]consisting of half of the elements of [G] and − for the remaining ones.So a hybrid group determines and is determined by a pair [H], [G] of

projective groups, namely, its projective image [G] and the “half group”[H] of index 2 in [G], at which the sign is +. In formal situations we willuse the name “+H − G”, which indicates that the group consists of theelements

+h for [h] in [H ] and −g for [g] in the rest of [G].Informally we often abbreviate this to HG. The cases are:

+Cn − C2n = CC2n = CC(2n), the cyclo-cyclic group+Cn −D2n = CD2n = CD(n), the cyclo-dihedral group

+D2n −D4n = DD4n = DD(2n), the dihedro-dihedral group+T12 −O24 = TO, the tetra-octahedral group

The group +H−G consists of the maps x→ +q−1xq, q ∈ 2H , x→ −q−1xq,q ∈ 2G\2H .From the notation we’ve been using, it’s easy to recover the abstract

group structure, so we call it the algebraic notation. The rule is that+G and HG are isomorphic to G, and ±G to C2 ×G.

36 3

3

3


3.9 Geometric Description of the Groups

Figure 3.1 described all of the groups in terms of familiar polyhedra. In theupper portion we have the full (or “holo-”) and rotational (or“chiro-”) groups for the icosahedron (±I = ∗532 and +I = 532), octa-hedron (±O = ∗432 and +O = 432), and tetrahedron (TO = ∗332 and+T = 332), as well as the diplo-tetrahedral group (±T = 3 ∗ 2), moreusually called the pyritohedral group since crystals of iron pyrites oftenhave this symmetry. In the lower portion we have the “holo-” and “chiro-”groups of prisms (∗22n and 22n), antiprisms (2∗n and 2nn), and pyramids(∗nn and nn), as well as the pro-prismatic (n∗) and pro-antiprismatic (n×)groups, obtained from the prismatic and antiprismatic groups by preservingthe sense of rotation about the principal axis.Table 3.1 presents a complete dictionary between these names, the orb-

ifold symbols, and our algebraic notations, while Table 3.2 arranges thegroups according to their orders. There is an irritating “switch” betweenthe geometric and algebraic names, which arises because geometers preferto extend rotations by reflections rather than the algebraically simpler cen-tral inversion −1. Our dictionary therefore contains “even/odd” entries,such as ±D(n)/DD(2n) for ∗22n, meaning that this group is ±D(n) whenn is even but DD(2n) when n is odd.There are many ways in which one of the groups can have small index

in another–all of the index 2 containments appear in Figures 3.9 and 3.10.

The chiro-icosahedral group 532 = +IThe holo-icosahedral group ∗532 = ±IThe chiro-octahedral group 432 = +OThe holo-octahedral group ∗432 = ±OThe chiro-tetrahedral group 332 = +TThe holo-tetrahedral group 3∗2 = TOThe pyritohedral group ∗332 = ±T

even / oddThe (n-gonal) chiro-prismatic group 22n = +D(n)The (n-gonal) holo-prismatic group ∗22n = ±D(n) / DD(2n)

The (n-gonal) holo-antiprismatic group 2∗n = DD(2n) / ±D(n)The (n-gonal) chiro-pyramidal group nn = +C(n)The (n-gonal) holo-pyramidal group ∗nn = CD(n)

The (n-gonal) pro-prismatic group n∗ = ±C(n) / CC(2n)The (n-gonal) pro-antiprismatic group n× = CC(2n) / ± C(n)

Table 3.1. A dictionary of the groups.

3.9. Geometric Description of the Groups 37

+T = 332

±T = 3 ∗ 2

±I = ∗532

+I = 532+O = 432

±O = ∗432

TO = ∗332

5

5

Figure 3.9. Small index containments between the polyhedral groups. (The indexis 2 when not mentioned.)

nn=+C(n)

2nn=+D(n) ∗nn=+CD(n)=

=n∗ ±C(n)

CC(2n) n×n odd n odd

even

even

∗22n ±D(n)

DD(2n) 2 ∗ n

Figure 3.10. Index 2 containments between the axial groups. Such containmentsbetween groups with the same parameter n are indicated by the heavy lines betweenboxes—each group in the upper box contains one group in the lower (the corre-sponding group, if this is meaningful). All index 2 containments between groupswith different parameters n and m = n

2 are indicated by the lighter lines beside theboxes. Thus, ∗22n contains both ∗22m and 2∗m (each twice because there are twolines).

38 3

3

3


Table 3.2. The groups classified by order. All the groups of a given order appearon one line, preceded by their number, with the number of isomorphism types as asubscript. The “rivers” separate groups of distinct structures (as indicated below thetable). The lines for orders 8n, 8n + 1, . . . , 8n + 7 that do not appear in the tableare like those for orders 16, 17, . . . , 23.

3.9. Geometric Description of the Groups 39

Table 3.2. (continued)

40 3

3

3


Appendix: v → vqv Is a Simple Rotation

We first prove this in the case when the axis is i, so q = cos θ+i sin θ. Thenif v = xi+ yj + zk, we have

qvq = q(xi)q + q(yj + zk)q = xi+ q2(yj + zk),

since iq = qi, while jq = qj, kq = qk. By writing this as

xi+ (cos(2θ)− i sin(2θ))(y + zi)j

we see that the 1-space spanned by i is fixed, while the 2-space spanned byj and k is rotated through 2θ.The way we derive the system of quaternions in Chapter 6 actually

shows that any unit imaginary quaternion may be called i, any perpen-dicular one j, and their product k. We now verify (without referring toChapter 6) that the quaternions do indeed have enough symmetries to takeany imaginary unit vector u = li + mj + nk to i. For a rotation of theabove type (fixing i) can be used to make the coefficient of k vanish, afterwhich a similar rotation (now fixing k) will make that of j vanish, too.

44 44 4

Quaternions and4-Dimensional Groups4.1 Introduction

The arguments in Chapter 3 used the fact that a 3-dimensional orthogonalmap has the form

x→ qxq or − qxq = q xqaccordingly as it is chiral or achiral. The first task of this chapter is toestablish that the general orthogonal map in 4 dimensions has the form

x→ lxr or lxr,

where l and r are two unit quaternions.In Chapter 6, we shall explain why in any composition algebra, the map

x→ −qxq(where q is any unit quaternion) is the reflection that fixes the hyperplaneperpendicular to q (which, henceforth, we shall call “the reflection in q”).However, we don’t need to quote this, since we can directly verify it in thequaternionic case. Namely, under this map we have

uq → −qq uq = −uqfrom which it follows that q is negated and iq, jq, kq fixed.As we showed in Chapter 1, every symmetry of a Euclidean object is a

product of reflections. So the product of n reflections takes

41

42 44 44 4 Quaternions and 4-Dimensional Groups

x→ ±qnqn−1 . . . q1(x or x)q1q2 . . . qnand therefore has one of the two forms

[l, r] : x→ lxr (if n is even)

and

∗[l, r] : x→ lxr (if n is odd)

where l and r are unit quaternions (the ∗ in the second of these being analternative name for quaternionic conjugation).

4.2 Two 2-to-1 Maps

This generalizes the 3-dimensional results, since it is easy to see that thesemaps fix 1 just if l = r = q (say), showing that the maps of Chapter 3 are

[q] = [q, q] and ∗[q] = ∗[q, q].

In that chapter, we found the map

[q] : x→ qxq

from unit quaternions to elements of SO3 to be 2-to-1 since the only equal-ity was [−q] = [q]. Our new map from ordered pairs of unit quaternions1

to elements of SO4 has a similar property, since here the only equality is

[−l,−r] = [l, r].

(A map [l, r] that is the identity must fix 1, so from the 3-dimensionalresult, the only non-trivial one is [−1,−1].)If we work projectively, there is slightly more fusion, since the four ex-

pressions [±l,±r] all name the same projective map, in view of the equali-ties

[−l,−r] = [l, r] [−l, r] = [l,−r] = −[l, r].

In conformity with our convention that [x] denotes a function satisfying[−x] = [x] (only), we shall use [[l, r]] (meaning [[l, r]]) for this projectivemap.

1These form a copy of Spin4, the 4-dimensional spin group.

4.3. Naming the Groups 43

4.3 Naming the Groups

Postponing some details for the moment, our name for the typical 4-dimensional chiral group will have the form

1

f[[L×R]] ± 1

f[L×R] or +

1

f[L×R]

in the

projective diploid or haploid

cases, where L and R are two 3-dimensional chiral groups. Such namesare apposite because, for instance, ± 1

f [L × R] correctly suggests that thisgroup consists of a proportion 1

fof the appropriate elements ±[l, r], with

both choices of sign. In particular, the orders of these three groups arerespectively

|L| × |R|f

2|L| × |R|

for

|L| × |R|f

.

1f [[L × R]] consists of the elements [[l, r]] for which l ∈ 2L, r ∈ 2R, and

lα = rβ , where α and β are two homomorphisms from these onto the samefinite group F of order f . The corresponding diploid group ± 1

f [L × R]consists of the pairs of elements ±[l, r], satisfying the same conditions,while a haploid group (if one exists) makes a choice of one sign from eachpair.If any of the above has index 2 in an achiral group, then L and R must

be the same type of group, say Q, and our name for the larger group hasthe general form

1

f[[Q×Q]] · 2 ± 1

f[Q×Q] · 2 or +

1

f[Q ×Q] · 2,

and has order

2|Q|2f

4|Q|2f

or 2|Q|2f,

respectively.Usually, the homomorphisms α and β are sufficiently determined by L

and R and the number f . When they are not, we distinguish cases by some“ornamentation” attached to L or R. Namely, we write a numerical para-meter as a parenthesized superscript, while a bar identifies the less obvious


of two cases, and we allow ourselves to omit 1f or (s) or (t) when f or s or t

is 1. Similarly, in the achiral cases, different choices for the element ∗[a, b]that extends the corresponding chiral group are indicated when necessaryby ornamenting the final ‘2’. The individual ornamentations are sufficientlyexplained by the generating quaternions in Tables 4.1, 4.2, and 4.3, whichenumerate the groups.

Group Generators

±[I×O] [iI , 1], [ω, 1], [1, iO], [1,ω];

±[I×T ] [iI , 1], [ω, 1], [1, i], [1,ω];

±[I×D2n] [iI , 1], [ω, 1], [1, en], [1, j];

±[I×Cn] [iI , 1], [ω, 1], [1, en];

±[O×T ] [iO, 1], [ω, 1], [1, i], [1,ω];

±[O×D2n] [iO, 1], [ω, 1], [1, en], [1, j];

±12 [O×D2n] [i, 1], [ω, 1], [1, en];[iO, j]

±12 [O×D4n] [i, 1], [ω, 1], [1, en], [1, j];[iO, e2n]

±16 [O×D6n] [i, 1], [j, 1], [1, en];[iO, j], [ω, e3n]

±[O×Cn] [iO, 1], [ω, 1], [1, en];

±12 [O×C2n] [i, 1], [ω, 1], [1, en];[iO, e2n]

±[T ×D2n] [i, 1], [ω, 1], [1, en], [1, j];

±[T ×Cn] [i, 1], [ω, 1], [1, en];

± 13 [T ×C3n] [i, 1], [1, en];[ω, e3n]

± 12 [D2m×D4n] [em, 1], [1, en], [1, j];[j, e2n]

±[D2m×Cn] [em, 1], [j, 1], [1, en];

± 12 [D2m×C2n] [em, 1], [1, en];[j, e2n]

+ 12 [D2m×C2n] − , − ; +

± 12 [D4m×C2n] [em, 1], [j, 1], [1, en];[e2m, e2n]

Table 4.1. Chiral groups, I. These are most of the “metachiral” groups—see section4.6—some others appear in the last few lines of Table 4.2.

The enumeration is guided by the (easy) theory of subgroups G of adirect product A×B of two groups, which was found in 1889 by Goursat [20]in this very connection. Namely, the elements l ∈ A, r ∈ B for whichthere exists an element (l, r) ∈ G form subgroups L and R of A and B,respectively, which we call the “left and right subgroups,” and the elementsl, r for which (l, 1) or (1, r) is in G form normal subgroups L0 and R0 ofthese, the “left and right kernels.”

4.4. Coxeter’s Notations for the Polyhedral Groups 45

Now, if l0 ∈ L0, r0 ∈ R0, then(l, r) ∈ G⇔ (ll0, rr0) ∈ G,

so the condition that (l, r) ∈ G really depends only on l and r modulo L0and R0. In fact, this condition sets up an isomorphism between the leftand right quotient groups L/L0 and R/R0, which must therefore be copiesof the same abstract group F . If F has order f , G will have index f inL×R.Since the typical element, [[l, r]] of the 4-dimensional projective group

PSO4 is independent of the signs of l, r, it is determined by the pair [l], [r]of corresponding elements of PSO3, showing that PSO4 ∼= PSO3×PSO3.By Goursat’s theory, its typical finite subgroup therefore has the form

[[l, r]] | [l] ∈ L, [r] ∈ R, [l]α = [r]βwhere L, R are two finite subgroups of PSO3 and α, β homomorphismsfrom these onto the same abstract group F . It does no harm to regard α,β as defined on the corresponding quaternion groups 2L, 2R, with −1 intheir kernels. This shows that our form “ 1f [[L × R]]” does indeed cover allprojective chiral groups, and the proofs for the other cases are similar (inthe haploid cases one must consider the subgroup of 2L×2R for which [l, r]is in the group).We have just seen how quaternions can be used to give a short and

elegant complete enumeration of 4-dimensional groups. However, if we arepresented with a group defined geometrically in some other way, it can behard to determine its quaternionic name, and vice versa. The next sectionaddresses an important case.

4.4 Coxeter’s Notations for the Polyhedral Groups

Coxeter’s notations (adapted from Schlafli) for regular polytopes and as-sociated groups are widely used. In this section, we extend his systemslightly so as to obtain a complete set of notations for the “polyhedral”groups, which are also given in Tables 4.2 and 4.3.Coxeter uses [p, q, . . . , r, s] for the symmetry group of the n-dimensional

polytope p, q, . . . , r, s–this is generated by reflections R1, . . . , Rn, cor-responding to the nodes of the n-node diagram

R1 R2 R3 Rn−1 Rn• • • · · · • •

p q s


Group Generators Coxeter Name

±[I×I] [iI , 1], [ω, 1], [1, iI ], [1,ω]; [3, 3, 5]+

± 160 [I×I] ;[ω,ω], [iI , iI ] 2.[3, 5]+

+ 160[I×I] ; + , + [3, 5]+

± 160 [I×I] ;[ω,ω], [iI , iI ] 2.[3, 3, 3]+

+ 160 [I×I] ; + , + [3, 3, 3]+

±[O×O] [iO, 1], [ω, 1], [1, iO], [1,ω]; [3, 4, 3]+ : 2

± 12 [O×O] [i, 1], [ω, 1], [1, i], [1,ω];[iO, iO] [3, 4, 3]+

± 16[O×O] [i, 1], [j, 1], [1, i], [1, j];[ω,ω], [iO, iO] [3, 3, 4]+

± 124 [O×O] ;[ω,ω], [iO, iO] 2.[3, 4]+

+ 124[O×O] ; + , + [3, 4]+

+ 124 [O×O] ; + , − [2, 3, 3]+

±[T×T ] [i, 1], [ω, 1], [1, i], [1,ω]; [+3, 4, 3+]

±13[T×T ] [i, 1], [j, 1], [1, i], [1, j];[ω,ω] [+3, 3, 4+]

∼= ±13 [T×T ] [i, 1], [j, 1], [1, i], [1, j];[ω,ω] ”

± 112[T×T ] ;[ω,ω], [i, i] 2.[3, 3]+

∼= ± 112 [T×T ] ;[ω,ω], [i,−i] ”

+ 112 [T×T ] ; + , + [3, 3]+

∼= + 112[T×T ] ; + , + ”

±[D2m×D2n] [em, 1], [j, 1], [1, en], [1, j];

±12[D4m×D4n] [em, 1], [j, 1], [1, en], [1, j];[e2m, e2n]

±14 [D4m×D4n] [em, 1], [1, en];[e2m, j], [j, e2n] Conditions

+14 [D4m×D4n] − , − ; + , + m,n odd

± 12f [D2mf×D(s)

2nf ] [em, 1], [1, en];[emf , esnf ], [j, j] (s, f) = 1

+ 12f [D2mf×D(s)

2nf ] − , − ; + , + m,n odd, (s, 2f) = 1

± 1f [Cmf×C(s)nf ] [em, 1], [1, en];[emf , e

snf ] (s, f) = 1

+ 1f[Cmf×C(s)nf ] − , − ; + m,n odd, (s, 2f) = 1

Table 4.2. Chiral groups, II. These groups are mostly “orthochiral,” with a few“parachiral” groups in the last few lines. The generators should be taken with bothsigns except in the haploid cases, for which we just indicate the proper choice ofsign. The “Coxeter names” are explained in Section 4.4.

4.4. Coxeter’s Notations for the Polyhedral Groups 47

Table 4.3. Achiral groups.


In terms of these, it has a presentation

1 = R21 = (R1R2)p = R22 = (R2R3)

q = · · · = R2n−1 = (Rn−1Rn)s = R2n= (RiRj)

2 (j > i+ 1).

This has an obvious subgroup [p, q, . . . , r, s]+ of index 2, consisting of theeven length words in R1, . . . , Rn, i.e., those which “mention R1, . . . , Rnevenly.”When just one of the numbers p, q, . . . , r, s is even, say that between Rk

and Rk+1, there are two further subgroups, namely

[+p, q, . . . , r, s] and [p, q, . . . , r, s+]

consisting of words that mention respectively

R1, . . . , Rk and Rk+1, . . . , Rn

evenly, whose intersection is the index 4 subgroup [+p, q, . . . , r, s+] of wordsthat mention both of these sets evenly. We’ve slightly modified Coxeter’s no-tation–he writes [p+, . . .] for our [+p, . . .] and uses only some special cases.To obtain elegant names for all of the “polyhedral” groups in dimension

4, we supplement Coxeter’s notation by writing G for the “opposite” groupto G, obtained by replacing the element g of G by +g or −g accordinglyas det = +1 or −1. Finally, an initial “2.” indicates doubling the group byadjoining negatives, while a final “:2” or “·2” indicates doubling in someother way, either split or non-split.

4.5 Previous EnumerationsThe enumeration of these groups has a long history, starting with Goursat’senumeration of the elliptic groups in 1889. In 1931, Threlfall and Seifert [40]enumerated all the groups, and discovered some of their implications fortopology. A standard reference in English has been Du Val’s [41] elegantlittle book Homographies, Quaternions, and Rotations.However, we shall point out that Goursat and Du Val omit certain

groups, and that there is another important sense in which all these enu-merations are incomplete. Every group does appear in Seifert and Threlfall’slist, but appears an uncertain number of times, since the equivalences be-tween the groups that depend on parameters are not completely listed. Theonly previous place in the literature when the problem was addressed is inthe determination of the crystallographic groups by Hurley [25], who distin-guished them with an ad hoc list of invariants. Brown, Bulow, Neubuser,Wondratschek and Zassenhaus [7] continue the use of Hurley’s invariantsin their enumeration of 4-dimensional space groups.

4.6. A Note on Chirality 49

That the problem is not entirely trivial is neatly illustrated by the factthat all permissible values of h and k in the group Du Val calls

(D 12nr/Cn;D 1

2nr/Cn)

∗s,h,k−

lead to geometrically isomorphic groups! In our system, with different

parameters, this group becomes ± 12f [D2nf ×D(s)

2nf ] · 2.

4.6 A Note on Chirality

The subject of chirality is more subtle than it seems. We have alreadymentioned that when we call the symmetry group of an object “chiral,”what we really mean is that the object is chiral. In three dimensions, thefinite subgroups of the orthogonal group, even the “chiral” ones, are all thesame as their mirror images.However, among the 219 crystallographic space-groups in 3 dimensions

are 11 that differ from their mirror images. (For this reason some authorsgive the total number as 230.) Such groups we call metachiral. Metachi-rality is common among subgroups of the 4-dimensional orthogonal group,since any of the groups ± 1

f [L × R] or + 1f [L × R] for which L = R is

obviously metachiral.However, there is another interesting phenomenon that first shows it-

self in this dimension. In lower dimensions, any chiral group that is notmetachiral is the chiral part of some achiral group. We call such groups“orthodoxly chiral,” or orthochiral. Another way to say this is that thecorresponding chiral object can be superimposed on a copy of its mirrorimage so as to yield an achiral one.In four dimensions, this is not always true, and the groups for which it

is not we call parachiral. For example, the parachiral group ±17 [C7×C(2)7 ]

is contained to index 14 in the achiral group ±[C7 × C7] · 2, but is not thechiral part of any achiral group since its mirror image is ±1

7 [C7 × C(4)7 ].Yet another distinction is made by some chemists–that between “shoe

chirality” and “potato chirality.” We expect neither shoes nor potatoes tobe achiral, but yet we discriminate left shoes from right shoes, but not “leftpotatoes” from “right potatoes!” Why is this?The answer is that a potato can be continuously deformed into its mir-

ror image while preserving its property of being a chiral potato at all times.Topologists call such a deformation an “isotopy” within the class of chi-ral potatoes, and so we shall call the shoe kind of chirality isochirality(meaning, “chirality” even modulo isotopy).In two dimensions, the class of chiral (i.e., scalene) triangles is isochiral,

since the sides either increase counter-clockwise (for “left” triangles) or


clockwise (for “right” ones). In three dimensions, chiral tetrahedra aren’tisochiral, since all chiral tetrahedra are isotopic inside the class of them.

Appendix: Completeness of the TablesWe establish this by showing that they handle all possible pairs of homo-morphisms from L, R onto the same abstract finite F . But since

[l, r][l1,r1] = [ll1 , rr1 ],

we can replace L, R by independent conjugates of themselves, so we needonly work up to this notion of “geometric equivalence”.

The Completeness of Tables 4.1 and 4.2

Now it is easy to classify normal subgroups of the 3-dimensional chiralgroups: There is just one for each of the orders

1, 4, 12 for T1, 4, 12, 24 for O1, 60 for Im (where m|n) for Cn or Dn,

except that when n is even, D2n has two normal subgroups Cn and Dn oforder n. So the kernel of α or β is determined by the index f , except forthe two cases for D2n when n is even and f = 2:

12 [. . .D2n . . .] indicates that ker = Cn12 [. . . D2n . . .] indicates that ker = Dn.

However, there are different cases that have homomorphisms with thesame kernel because F has an automorphism not induced by any elementof GO3 (and so necessarily outer). It is easily checked that these are:

Our notation Sufficient description of automorphism160 [. . . I . . .] extends A5 to S5+ 124 [. . . O . . .] negates elements of 2S4\2A4 2

14 [. . . D . . .] moves the “cyclic part” 3

1f [. . . D

(s)nf . . .]

1f [. . . C

(s)nf . . .]

acts as sth powermap on cyclic part

There are three groups in Table 4.2 for which we give alternative nota-tions involving T . This peculiar repetition simplifies Table 4.3.

2 This entry can be ignored in the projective and diploid cases.3 Goursat, and following him, Du Val, omitted 1

4[D4m×D4n], which arises because

F ∼= D4 has automorphisms freely permuting its non-identity elements, unlike a largerD2k, whose “cyclic part” Ck is uniquely determined.

Appendix: Completeness of the Tables 51

The Completeness of Table 4.3

Here we obtain G from the “half-group” H corresponding to some isomor-phism L/L0 ∼= R/R0 by adjoining an extending element ∗[a, b], which mustnormalize H . We shall show that (at some cost) the extending element maybe reduced to the form ∗[1, c], and also that (at no cost) c can be multipliedby any element of R0, or altered by any inner automorphism of R, whilefinally c must be in the part of R that is fixed (mod R0) by the isomorphism(since (∗[1, c])2 = [c, c] must be in H).For, conjugation by [1, a] replaces ∗[a, b] by

(∗[a, b])[1,a] = [1, a] ∗ [a, b][1, a] = ∗[aa, ba] = ∗[1, c], say,

at the cost of replacing [l, r] by [l, ara], which changes the isomorphismto a geometrically equivalent one. If r0 ∈ R0, ∗[1, cr0] defines the samegroup as does ∗[l1, cr1] for any [l1, r1] ∈ H , and this reduces to ∗[1, cr1l1]on conjugation by [1, l1], which replaces the r in [l, r] by l1rl1, its imageunder an arbitrary inner automorphism of R.These considerations almost always suffice to restrict the extending el-

ement to

∗[1,±1] = ∗ or − ∗,

notated respectively by ·23 or ·21 (the subscript being the dimension of thenegated space). The exceptions are the “D ×D” and “C × C” cases, forwhich Table 4.3 lists every c, and just two more cases, denoted

±12[O ×O] · 2 and ± 1

4[D4n ×D4n] · 2

in which we can take c = iO and e2n, respectively.As we remarked, the reduction to the form ∗[1, c] comes at the cost of

replacing the isomorphism by a geometrically equivalent one, and in the“T ×T” case, this sometimes replaces the identity isomorphism by the onewe indicate by T , namely

ω → ω and i→ i = −i.

The Last Eight Lines of Table 4.3

For ±[D ×D] · 2, we start from the fact that the extending element ∗[a, b]may be reduced (mod H) and must normalize H , and therefore also E,


since E is a characteristic subgroup of H (except in the easy cases whenf ≤ 2, which we exclude). This puts a and b in eR(1 or j), and so (since[j, j] ∈ H) we can take ∗[a, b] = ∗[eλ, eµ] (leading to ±[D ×D] · 2(α,β)) or∗[a, b] = ∗[eλ, eµj] (leading to ±[D × D] · 2–see footnote 4.) In the firstcase, we must have

[j, j]∗[eλ,eµ] = [j, j][e

λ,eµ] = [je2λ, je2µ] ∈ H,

which forces λ = α2 and µ =

αs+βfs , where α,β ∈ Z. The fact that the

square of this is in H imposes the condition αg + βf ≡ 0 (mod 2).G is unaltered when we increase α or β by 2 since [e, es], [1, ef ] ∈ H .

For a similar reason, s is initially only defined (mod f), but the equation

∗[eα2 , eαs+βf2 ] = ∗[eα2 , eα(s+f)+(β−α)f2 ]

shows that

s,α,β ≈ s+ f,α,β − α ,

so from now on it is better to regard s as defined mod 2f . Since [es, s] =

[e, es]∗[a,b] and [e, es2

] are both in H , we must have s2 = fg + 1 for someinteger g ∈ Z.To discuss equalities, we must consider all possibilities for an element

that transforms this group s,α,β to a similar one s ,α ,β . The trans-forming element can also be reduced mod H and after taking account of ∗and [1, j] (which takes s,α,β to itself or −s,α,−β ), can be supposedto normalize H and therefore have the form [e

a2 , e

as+bf2 ], with a, b ∈ Z. We

find that transforming by this adds some multiple (which can be odd) of(f, g) to (α,β), so the only further relation is s,α,β ≈ s,α+ f,β + g .

To summarize, we have for this group

Variables Conditions Equalities

α (mod 2) s2 = fg + 1 s,α,ββ (mod 2) αg + βf ≡ 0 (mod 2) ≈ −s,α,−βs (mod 2f) ≈ s+ f,α,β − α

≈ s,α+ f,β + g ,

4 In the second case we can choose new generators to simplify the group; namely,conjugation by [1, eλ] fixes E and replaces ∗[eλ, eµj] by ∗[1, eµ−λj] = ∗[1, J ], and thenJ can replace j, since (∗[1, J ])2 = [J, J] must be in H.

Appendix: Completeness of the Tables 53

while for ±[D ×D(s)] · 2 we have

Variables Conditions Equalities

s (mod f) s2 = fg − 1 s ≈ −s .

Equalities in the other cases are summarized as:

Group Variables Conditions Equalities

+[D ×D(s)] · 2(α,β) α (mod 2) s2 = fg + 1 s,α,ββ (mod 4) αg ≡ 0 (mod 4) ≈ −s,α,−βs (mod 4f) n odd, g even ≈ s+ 2f,α, β − 2α

≈ s,α, β + 2h

+[D ×D] · 2 s (mod 2f) s2 = fg − 1, g = 2h even s ≈ −s

±[C × C] · 2(γ) s (mod f) s2 = fg − 1 s, γγ (mod 2) (g, s− 1)γ ≡ 0 (mod 2) ≈ s,−γ∗[1, eγ(f,s+1)2 ] (f, s+ 1)γ ≡ 0 (mod 2)

g even

+[C × C] · 2(γ) s (mod 2f) s2 = fg − 1 s, γγ (mod 2d) (g, s− 1)γ ≡ 0 (mod 4) ≈ s,−γ∗[1, eγ(f,s+1)2 ] (f, s+ 1)γ ≡ 0 (mod 2)d = (2f,s+1)

(f,s+1)n odd, g = 2h even

Table 4.4 summarizes the different achiral groups among the last fourlines of Table 4.3. In the last eight lines, it is always permissible to replaceD2 by C2 and D4 by D4.

f, g even : ·2(0,0), ·2(0,1), ·2(1,0), ·2(1,1) and · 2else ·2 and · 2

f, h even : ·2(0,0), ·2(0,2), ·2(1,0), ·2(1,2) and · 2else ·2 and · 2

g even : ·2(0), ·2(1) and · 2else ·2 and · 2

h even : ·2(0), ·2(d) and · 2 d = (2f,s+1)(f,s+1)

else ·2 and · 2Table 4.4. Different achiral groups.

5

5

5

5

The Hurwitz IntegralQuaternions5.1 The Hurwitz Integral Quaternions

How shall we define what it means for the quaternion q = a+ bi+ cj + dkto be integral? One obvious possibility (following the example of theGaussian integers) is to demand that a, b, c, and d be ordinary rationalintegers. Technically, we say that a+ bi+ cj + dk is a Lipschitz integerexactly when a, b, c, d ∈ Z. This is equivalent to q = z1 + z2j, wherez1 and z2 are Gaussian integers. However, Hurwitz later found a differentdefinition that turns out to have nicer properties. We say that a+bi+cj+dkis a Hurwitz integer just if either all of a, b, c, d are in Z or all are inZ+ 1

2 .When we studied the arithmetics of the Gaussian integers in Chapter 2,

an important role was played by “division with small remainder,” namelythat one can divide any (Gaussian) integer, Z, by any non-zero one, z,so as to have an integer quotient, Q, and a remainder, R, strictly smallerthan the divisor. Recall how this was proved: If Z/z = a + bi and we letQ = A+Bi, where A, B are nearest integers to a, b, then

R/z = Z/z −Q = (a−A) + (b−B)iso

N(R/z) = (a−A)2 + (b−B)2 ≤ (1/2)2 + (1/2)2 = 1/2 < 1.However, if Z and z = 0 are Lipschitz-integral quaternions with q =Zz−1 = a + bi + cj + dk, and we let A, B, C, D be the nearest inte-

55

56 5

5

5

5 The Hurwitz Integral Quaternions

gers to a, b, c, d and define Q = A+Bi+ Cj +Dk and R = Z − qz, thenwe find

N(Rz−1) = (a−A)2 + · · ·+ (d−D)2 ≤ (1/2)2 + · · ·+ (1/2)2 = 1,showing only that N(R) ≤ N(z). Unfortunately, the fact that this inequal-ity is not strict makes all the difference in the world, because the argumentsdepend on repeated norm-reductions.Note that the only case when N(R) = N(z) is when

|a−A| = |b−B| = |c− C| = |d−D| = 1/2,that is to say, when a, b, c, d are all in Z+ 1

2 . This is what makes Hurwitz’definition preferable, for if we suppose Z, z = 0 are Hurwitz-integral anddefine q, Q, and R as above, then either we have

Z = Qz +R with N(R) < N(z)

or

N(R) = N(z),

in which case the above argument shows q is actually a Hurwitz integerand Z = qz + 0 (in which N(0) < N(z)).We have proved: The Hurwitz integers, but not the Lipschitz ones, have

the “division with small remainder” property. This makes it easier to workin H rather than in L. Theorems about L are usually proved by reductionto theorems about H.

5.2 Primes and Units

We now turn to the theory of prime factorization of Hurwitz integers. Aprime Hurwitz integer P is one whose norm is a rational prime p. Anal-ogously to the fact that p = p × 1 and p = 1 × p are the only ways p isthe product of two rational primes, its only factorizations into two Hurwitzintegers must have the form

P = P × U and P = V × P ,

where N(P ) = N(P ) = p and N(U) = N(V ) = 1. So, we must alsostudy the Hurwitz units, namely the Hurwitz integers of norm 1.

Theorem 1. There are precisely 24 Hurwitz units, namely the eightLipschitz units ±1, ±i, ±j, ±k, and the 16 others ±1

2 ± 12 i± 1

2j ± 12k.

5.2. Primes and Units 57

Proof. For a Lipschitz unit, we must have one of |a|, |b|, |c|, |d| ≥ 1,and the equation a2 + b2 + c2 + d2 = 1 implies that that coordinate is ±1and the rest are 0. In the remaining cases, we must have |a|, |b|, |c|, |d| ≥ 1

2 ,and since a2 + b2 + c2 + d2 = 1, all are ± 1

2 .The Hurwitz units completely solve the problem of factorizing a Hurwitz

prime. Namely, if P is a Hurwitz prime, then its only factorizations as aproduct of two Hurwitz integers are

P = PU−1 × U and P = V × V −1Pas U and V run over the 24 Hurwitz units. These are the only possibilities,since we already showed that one factor must be a unit, and either factordetermines the other, and indeed they are factorizations since if U , V areunits, so are U−1, V −1, so that PU−1 and V −1P are Hurwitz integers.The way a Hurwitz integer factors into primes depends heavily on

whether that integer is imprimitive (i.e., divisible by some natural numbern > 1). In the primitive case, we have

Theorem 2. To any factorization of the norm q of a primitive Hurwitzinteger Q into a product p0p1 · · · pk of rational primes, there is a factoriza-tion

Q = P0P1 · · ·Pkof Q into a product of Hurwitz primes with N(P0) = p0, . . . , N(Pk) = pk.We shall say that “the factorization P0P1 . . . Pk of Q is modelled on thefactorization p0p1 . . . pk of N(Q).” Moreover, if Q = P0P1 · · ·Pk is anyone factorization modelled on p0p1 . . . pk, then the others have the form

Q = P0U1 · U−11 P1U2 · . . . · U−1k Pk

i.e., “the factorization on a given model is unique up to unit-migration.”

Thus, a primitive Hurwitz quaternion of norm 60 has, up to unit-migration, just 12 factorizations into prime quaternions, namely those mod-elled on the 12 factorizations

2.2.3.5 2.2.5.3 2.3.2.5 2.5.2.3 2.3.5.2 2.5.3.23.2.2.5 5.2.2.3 3.2.5.2 5.2.3.2 3.5.2.2 5.3.2.2

of 60 into rational primes. The full number of factorizations will be 243 ×12 = 165888, since the 24 units can migrate across any one of three places.

58 5

5

5


Proof. The ideal p0H+QH must be principal, so that we have

p0H+QH = P0H

for some P0. Here, [P0] must divide [p0] = p20, so it is one of 1, p0, p

20.

However, if [P0] = 1, then p0H + QH would be all of H, which cannotbe, since its typical element p0a+Qb has norm

[p0a] + 2[p0a,Qb] + [Qb] = p20[a] + 2p0[a,Qb] + p0 · · · pk[b],

divisible by p0. Nor can P0 have norm p20, for then since P0 divides p0 weshould have p0 = P0U , where U is a unit (since [U ] = 1), which shows thatp0 divides Q (since p0U

−1 = P0 does). The only possibility that remainsis [P0] = p0, showing that P0 is a Hurwitz prime dividing Q.So we have Q = P0Q1, where [Q1] = p1 · · · pk, and P0 is unique up to

right multiplication by a unit. Repeating the argument, we can producesimilar factorizations

Q1 = P1Q2, Q2 = P2Q3, . . .

in which P1, P2, . . . are Hurwitz primes of norms p1, p2, . . .. This givesQ = P0P1 · · ·PkQ , where Q must be a unit (and thus can be absorbedinto Pk). The argument has shown that the factorization is unique up tounit-migration.

5.3 Quaternionic Factorization of Ordinary Primes

Before we complete the discussion, we must study the quaternionic factor-izations of rational integers and, in particular, of a rational prime p. Recallthat

(i) a quadratic residue r satisfies r ≡ a2 (mod p),for some a ≡ 0 (mod p),

while the quadratic non-residues are the other numbers ≡ 0 (mod p).We show that

(ii) each quadratic non-residue n satisfies n ≡ a2 + b2 (mod p),for some a, b ≡ 0 (mod p),

and

(iii) 0 ≡ a2 + b2 + c2 (mod p),for some a, b, c ≡ 0 (mod p).

5.3. Quaternionic Factorization of Ordinary Primes 59

Since the non-residues are the multiples of any fixed one by the residues,it suffices to prove (ii) for n, the least possible non-residue. But thenn = r + 1 for a quadratic residue r, and so for r ≡ a2 (mod p) we deducen = a2 + 12 (mod p). Finally, 0 = −1 + 12, which is congruent to the sumof ≤ 3 squares (including 12), since −1 is congruent to the sum of ≤ 2squares. This enables us to prove the following

Theorem 3. Each rational prime p admits at least one quaternionicfactorization

p = P0P0.

Proof. Since x2 ≡ (p − x)2 (mod p), we can suppose in (iii) that0 ≤ a, b, c ≤ p/2, thereby obtaining a solution of a2 + b2 + c2 = mp with0 < m < p, and a quaternion Q = a + bi + cj of norm mp. Then ifpH+QH = P0H, we see as before that [P0] = p.In fact p has as many such factorizations as there are quaternions P

of norm p. By analytic methods it can be shown that the total number is24(p+ 1) (if p > 2) or 24 (if p = 2).What happens to unique factorization when Q need not be primitive?

We shall suppose that Q of norm 2n0pn11 pn22 . . . pnkk is exactly divisible by

the rational integer 2s0ps11 ps22 . . . p

skk . Then the answer is given by

Theorem 4. The number of factorizations (counted up to unit-migra-tion) that are modelled on a given factorization of N(Q) into primes is theproduct

i≥1Cni,si(pi)

of “truncated Catalan polynomials,” evaluated at the odd primes involved.

The first few Catalan polynomials are

C0(x) = C1(x) = 1C2(x) = x+ 1C3(x) = 2x+ 1C4(x) = 2x

2 + 3x+ 1,

and their coefficients in general can be read from the “Catalan triangle”(Figure 5.1 – see also [22]). The truncated Catalan polynomial Cn,s(x)consists of the terms of Cn(x) that have degree at most s.

60 5

5

5


Proof. It will suffice to consider factorizations of the form

Q = P1P2 . . . Pn

where all the Pi have the same prime norm p.To this end, suppose Q (of norm pn) is divisible by ps but not ps+1,

and write Q = psP . Then since P is primitive, it has a factorization

P = P1P2 . . . Pn−2s

that is unique up to unit-migration.We quote the result that (up to unit right multiplications) there are just

p+1 quaternions of norm p and ask (still up to unit right multiplications)whether P1 is or is not equal to P1. The one case when it is leads to thefactorization problem for a quaternion of norm pn−1 that is still exactlydivisible by ps, which inductively has Cn−1,s(p) solutions. The p cases whenit is not lead to problems for quaternions of norm pn−1 that are exactlydivisible by ps−1, and so each has Cn−1,s−1(p) solutions. The argumentestablishes the recursion

Cn,s(p) = Cn−1,s(p) + pCn−1,s−1(p)

that defines the Catalan polynomials. For example, the reduced problemsfor factorization of a norm p7 quaternion exactly divisible by p2 are one fora norm p6 quaternion exactly divisible by p2 (leading to C6,2 = 9p

2+5p+1solutions) and p for a norm p6 quaternion exactly divisible by p (eachleading to C6,1 = 5p+ 1 solutions), in agreement with

9p2 + 5p+ 1 + p(5p+ 1) = 14p2 + 6p+ 1 = C7,2(p).

×1

× 11 1

× 2 12 3 1

× 5 4 15 9 5 1

× 14 14 6 114 28 20 7 1

× 42 48 27 8 1

Figure 5.1. The Catalan triangle is formed in the same manner as Pascal’s triangle,except that no number may appear to the left of the vertical bar.

5.4. The Metacommutation Problem 61

For a quaternion of norm pmqn. . . exactly divisible by psqt. . . , we stillget exactly these numbers of choices for the factors of norm p, no matterwhere they are placed, so the total number of factors up to unit-migrationis the product of these numbers for the distinct prime factors.

5.4 The Metacommutation Problem

Even rational integers do not have unique factorizations; for instance,

60 = 2 · 2 · 3 · 5 = 2 · 5 · 3 · 2 = (−3) · 2 · 5 · (−2) = · · ·However, the term “unique factorization” is justified since all such factor-izations are readily obtained from any one of them.Now Theorem 2 is usually called the unique factorization theorem for

primitive (Hurwitzian) quaternions, and it really deserves this name for fac-torizations on a given model because any two such are related by the well-understood operation of unit-migration (to which we should add recombi-nation– see below–in the imprimitive case).However, this does not completely justify the term “unique factoriza-

tion” for Hurwitzians. To do so would require solving a problem we callthe metacommutation problem: How does a prime factorization PQmodelled on pq determine a corresponding factorization Q P modelled onqp? This difficult problem does not seem to have been addressed in theliterature.

There are, in fact, three basic ways in which prime factorization canbe changed. We have already met “unit-migration” which replaces PQby PU · UQ. There is also recombination which replaces a pair PPof conjugate primes of norm p by any other such pair P P . Finally, anyproduct PQ of distinct Hurwitz primes of norms p and q can be refactorizedas a product Q P of Hurwitz primes of norms q and p. We call this lastoperationmetacommutation of P and Q; the resulting factorizationQ Pis unique up to unit-migration. It is not hard to see that any two primefactorizations of a quaternion are related by repeated applications of thesethree “operations.”

5.5 Factoring the Lipschitz Integers

In order to understand Lipschitzian factorizations, we must understand howthey are related to the Hurwitzians, both additively and multiplicatively.

62 5

5

5


Geometrically, the Lipschitzians L = I4 form one copy of the 4-dimen-sional cubic lattice I4. There are two more copies of I4 inside the Hur-witzians, namely ωL = I4 and ωL = I4 . The intersection of these three I4s(which is the intersection of any two of them) is

L ∩ ωL ∩ ωL = D4,

(a copy of the four-dimensional “orthoplex lattice,” or root lattice of typeD4). The lattice of all Hurwitzians is their union

H = L ∪ ωL ∪ ωL = D∗4 ,

so-called because it is, in fact, the lattice dual to D4.Additively, D4 has four cosets [0], [1], [ω], [ω] in D

∗4 , and the lattices we

have mentioned are various unions of these (Figure 5.2).Multiplicatively, the important fact is that multiplication by ω (on ei-

ther side) has the effect

[0] [1] [ω]

[ω]

-

¢¢®AAK

and so fixes D4 and D∗4 but cyclically permutes I4, I4, I4 .

The Hurwitzians of even norm are precisely those in [0]; in fact, theywill stay in [0] upon multiplication by ω or ω (on either side). A Hurwitzianof odd norm is in just one of [1], [ω], and [ω]; equivalently, just one of Q,ωQ, and ωQ is in [1].

[0] ∪ [1] ∪ [ω] ∪ [ω]= D∗4 = H

[0] ∪ [1] [0] ∪ [ω] [0] ∪ [ω]= I4 = L = I4 = ωL = I4 = ωL

[0] = D4= L ∩ ωL ∩ ωL

¡¡ @@

@@ ¡¡

Figure 5.2. Containment among sublattices of H.

5.5. Factoring the Lipschitz Integers 63

5.5.1 Counting Lipschitzian Factorizations

This enables us to prove

Lemma 1. Any factorization of a Lipschitzian into Hurwitzians isequivalent by unit-migration to one into Lipschitzians.

Since the units we use will be powers of ω, which is not Lipschitz-integral, this means that the factorization into Lipschitzians is definitelynot “unique.”Proof. We study the effect of “ω-migration”

α = βγ → α = βω · ωγon a 2-term factorization of a Lipschitzian α. Since α is in [0] or [1],Figure 5.3 includes all possibilities for the cosets of β and γ, and there is afactorization in each triple for which both factors are Lipschitzian.

[0][0] [0][0] [ω][0] [ω][0]

[0][0] [1][0]

[0][ω] [0][ω] [ω][ω] [ω][ω]

[0][1] [1][1]

-

¢¢®AAK

-

¢¢®AAK

-

¢¢®AAK

-

¢¢®AAK

Figure 5.3. “βγ → βω · ωγ.”

A more detailed investigation shows

Theorem 5. The number of Lipschitzian factorizations equivalent byunit-migration to

P1P2 . . . Pk

is 8k−13l−1, where l is the number of factors of even norm.

For between pi and pi+1 . . . pk, we can always transfer all eight Lip-schitzian units, and also the three powers of ω just when both have evennorm.

III

The Octonions andTheir Applications to 7- and8-Dimensional Geometry

The octonions are formal expressions

x∞ + x0i0 + x1i1 + x2i2 + x3i3 + x4i4 + x5i5 + x6i6

(xt real), which constitute the algebra over the reals gener-ated by units i0, . . . , i6 that satisfy

i2n = −1 and

in+1in+2 = in+4 = −in+2in+1in+2in+4 = in+1 = −in+4in+2in+4in+1 = in+2 = −in+1in+4

(where the subscripts run modulo 2).

6

The Composition AlgebrasIn Chapter 2, we interpreted the two-square identity

(x1y1 − x2y2)2 + (x1y2 + x2y1)2 = (x21 + x22)(y21 + y22)

as saying that in the algebra of complex numbers the norm of a productequals the product of the norms, which here we write1

[z1z2] = [z1][z2].

In Chapters 3 and 4, we made heavy use of the corresponding result forHamilton’s algebra of quaternions.

In this chapter, we shall prove Hurwitz’s celebrated theorem that theonly algebras with such a composition law are the well-known ones in 1, 2,4, and 8 dimensions (exactly, for the algebras with a multiplicative identity,and up to isotopy, if not). We shall suppose at first that our algebra doespossess a two-sided identity element 1, so that 1x = x = x1, and as usual,

[x, y] =[x+ y]− [x]− [y]

2

(we are working over the reals, where 2 is invertible). We repeatedly usethe principle that if [x, t] = [y, t] for all t, then x = y.

1The use of square brackets for norms and inner products in this chapter should, ofcourse, not be confused with that for certain maps in earlier chapters.

67

68 6 The Composition Algebras

6.1 The Multiplication Laws

We first deduce some consequences of

The Composition Law: [xy] = [x][y].(M1)

The Scaling Laws: [xy, xz] = [x][y, z] (and [xz, yz] = [x, y][z]) .(M2)

Proof. Replacing y by y + z in (M1) gives

[xy] + [xz] + 2[xy, xz]M1= [x]([y] + 2[y, z] + [z]),

from which we cancel some terms and divide by 2.

The Exchange Law : [xy, uz] = 2[x, u][y, z]− [xz, uy] .(M3)

Proof. Replacing x by x+ u in (M2) gives

[xy, xz] + [xy, uz] + [uy, xz] + [uy, uz]M2= ([x] + 2[x, u] + [u])[y, z]

from which we cancel some terms and rearrange.

6.2 The Conjugation Laws

Now, we prove three laws involving the conjugation x = 2[x, 1]− x.

Braid Laws: [xy, z] = [y, xz] (and [xy, z] = [x, zy]) .(C1)

Proof. Put u = 1 in (M3) to get

2[x, 1][y, z]− [xz, y] = [y, (2[x, 1]− x)z].

(Figure 6.1 explains the name.)

Biconjugation: x = x .(C2)

Proof. Put y = 1 and z = t and use (C1) twice to get

[x, t] = [x1, t] = [1, xt] = [x1, t] = [x, t] for all t.

6.3. The Doubling Laws 69

Product Conjugation: xy = y x . (C3)

Proof. Repeated use of (C2) gives

[y x, t] = [x, yt] = [x t, y] = [t, xy] = [t, xy.1] = [t xy, 1] = [xy, t].

6.3 The Doubling Laws

We now let H be an n-dimensional subalgebra containing 1, let i be a unitvector orthogonal to H , and a, b, c, . . . denote typical elements of H . Theni = −i, and [i, a] = 0, which will often cause the term 2[x, u][y, z] to vanishin applications of the exchange law. Our next three laws evaluate the innerproduct, conjugation, and product on the “Dickson double algebra” H+iHin terms of those on H .

Inner-Product Doubling: [a+ ib, c+ id] = [a, c] + [b, d] . (D1)

Proof. This is proved by the three equations

[a, id] = [ad, i] = 0, [ib, c] = [i, cb] = 0, [ib, id] = [i][b, d] = [b, d].

Conjugation Doubling: a+ ib = a− ib (so ib = −ib = −b i = bi) . (D2)

Proof. ib = 2[ib, 1]− ib = −ib.

Composition Doubling: (a+ ib)(c+ id) = (ac− db) + i(cb+ ad) . (D3)

Proof. This is proved by the three equations

[a.id, t]C1= [id, at]

M3= 0− [it, ad] C1= [t, i.ad]

[ib.c, t]C1= [ib, tc]

D2= [bi, tc]

M3= 0− [b c, ti] C1= [bc.i, t]

D2= [i.cb, t]

[ib.id, t]C1= −[ib, t.id] M3

= 0 + [i.id, tb]C1= −[id, i.tb] M2

= −[i][d, tb] C1= [−db, t].


[x y, z]

[y z, x]

[z x, y]

[x y, z]

[y, x z]

[z, y x]

[x, z y]

...

...Figure 6.1. Six inner products related by the braid law.

6.4 Completing Hurwitz’s Theorem

What this has shown is that whenever our composition algebra Z containsa proper subalgebra, it also contains its Dickson double. Therefore, if Zis finite-dimensional it must be obtained by repeated doubling from itssmallest subalgebra R, and in particular must itself be the double of analgebra Y , which might in turn be the double of an algebra X , and soon. We shall now show that the composition property can survive at mostthree doubling operations.When is Z = Y + iZY a composition algebra? Just when for all

a, b, c, d ∈ Y we have

[a+ iZb][c+ iZd] = [(ac− db) + iZ(cb+ ad)].However, since this expands to

[a][c] + [a][d] + [b][c] + [b][d] = [ac]− 2[ac, db] + [db] + [cb] + 2[cb, ad] + [ad],the answer is precisely when

[ac, db] = [cb, ad],

or equivalently, when

[ac.b, d] = [a.cb, d],

which holds just when ac.b = a.cb for all a, b, c ∈ Y .

6.4. Completing Hurwitz’s Theorem 71

So the answer is

Lemma 1. Z is a composition algebra just when Y is an associativecomposition algebra.

When is Y = X + iYX an associative composition algebra? Obviously,X must also have these properties, but since iY b.c = iY .cb for all b, c ∈ X,we shall need to have bc = cb; that is to say, X must also be commutative.But now, for a, b, c, d, e, f ∈ X, we find(a+ iY b)(c+ iY d).(e+ iY f) = [(ac− db) + iY (cb+ ad)](e+ iY f)

= (ac− db)e− f(bc+ da) + iY [e(cb + ad) + (c a− bd)f ]= [ac.e− db.e− f.bc− f.da] + iY [e.cb + e.ad+ ca.f − bd.f ],

and similarly

(a+ iY b).(c+ iY d)(e+ iY f) = (a+ iY b)[(ce− fd) + iY (ed+ cf)]= a(ce − fd)− (ed+ cf)b+ iY [(ce− fd)b+ a(ed+ cf)]= [a.ce− a.fd− ed.b− cf.b] + iY [ce.b− fd.b + a.ed+ a.cf ],

which coincide when we use the commutative and associative propertiesof X.We conclude that

Lemma 2. Y is an associative composition algebra just when X is acommutative, associative composition algebra.

When is X = W + iXW a commutative, associative composition alge-bra? Suppose a, b, c, d, e ∈ W . Since ie = ei, we’ll need e = e for all e ∈W :in other words, conjugation must be trivial in W .But now

(a+ iXb)(c+ iXd) = (ac− db) + iX (cb+ ad)and

(c+ iXd)(a+ iXb) = (ca− bd) + iX(ad+ cb),which coincide using the commutative and trivial conjugation properties ofW . Therefore,

Lemma 3. X is a commutative, associative composition algebra justwhen W is a commutative, associative composition algebra with trivial con-jugation.


We therefore have

Theorem 1. (Hurwitz) R, C, H, and O are the only compositionalgebras.

For R is a commutative, associative composition algebra with trivialconjugation. So C is a commutative, associative composition algebra withnon-trivial conjugation. So H is an associative composition algebra which isnon-commutative. So O is a composition algebra which is non-associative,and its Dickson double is no longer a composition algebra.

We have proved Hurwitz’s celebrated theorem that a composition alge-bra with identity on a real Euclidean space is one of these four algebras.We now show that this is true up to isotopy for algebras without identities.

In an arbitrary composition algebra, choose u and v of norm 1. Thenx → xv and y → uy are orthogonal maps, and so have inverses α and β,say. Now, define a new multiplication by x y = xαyβ . Then

[x y] = [xαyβ ] = [xα][yβ ] = [x][y],

showing that still gives a composition algebra; and

uv uy = (uv)α(uy)β = uy

xv uv = (xv)α(uv)β = xv,

showing that uv is a two-sided identity for the new multiplication.2

6.5 Other Properties of the Algebras

We define x−1 = x/[x] for x = 0.

Inverse Laws: x.xy = [x]y = yx.x,or equivalently x−1.xy = y = yx.x−1 .

Proof. [x.xy, t] = [xy, xt] = [x][y, t] = [[x]y, t].

2In the language of the next chapter, this shows that any norm 1 element can beconverted into an identity element by applying an isotopy, so that the monotopies aretransitive on such elements (which is the condition for a Moufang Loop).

6.6. The Maps Lx, Rx, and Bx 73

Alternative Laws: x.xy = x2y and yx.x = yx2 .

Proof. Substitute x = 2[x, 1]− x in x.xy = xx.y.

Remarks: In fact, the Bruck-Kleinfeld theorem [8] characterizes thealgebras just from the alternative property. Sometimes x.yx = xy.x iscalled the third alternative law. We shall prove it immediately after weprove the following laws.

Moufang Laws: xy.zx = x(yz).x = x.(yz)x .

Proof.

[xy.zx, t] = [xy, t.x z] = 2[x, t][y, x z]− [x.x z, ty]= 2[x, t][yz, x]− [x z, x.ty]= 2[yz, x][x, t]− [x][z y, t]= 2[x, yz][x, t]− [x][yz, t].

So xy.zx = 2[x, yz]x−[x]yz is a function of x and yz only. We can thereforereplace y and z by any two other elements with the same product, and sodeduce

xy.zx = x(yz).1x = x(yz).x and xy.zx = x1.(yz)x = x.(yz)x.

Again, since y1 = 1y = y, we can deduce the third alternative law

xy.x = xy.1x = x1.yx = x.yx.

W. D. Smith informs us that R. D. Schafer proved that all the alge-bras obtained from R by repeated Dickson doubling are power associative,satisfy (ab)a = a(ba) and (ab)a2 = a(ba2), and have the property thatevery element satisfies a quadratic equation. He also remarks that the“real parts” of octonions are associative: [ab.c, 1] = [a.bc, 1].

6.6 The Maps Lx, Rx, and Bx

The failure of the associative law suggests the study of multiplication oper-ators. Accordingly, we define the Left-multiplication, Right-multiplication,and Bi-multiplication maps:

Lx : y → xy, Rx : y → yx, Bx : y → xyx


The third alternative law shows that Bx is well-defined and is the productof Lx and Rx in either order:

yLxRx = xy.x = yBx = x.yx = yRxLx .

We shall also show that Bx has a geometrical interpretation in terms ofreflections. For comparing our formula

xy.zx = 2[x, yz]x− [x]yzwith the usual expression

ref(x) : t→ t− 2[x, t][x]

x

that represents the reflection in a vector x, we see that

xy.zx = −[x](yz)ref(x) = [x](yz)ref(1)·ref(x),showing that Bx is just a scalar multiple of ref(1) · ref(x).Let’s look at the associative law more closely. Writing it in the forms

(yz)Lx = yLxz, (yz)Rx = yzRx ,

what it tells us is that one can left or right multiply a product of twofactors by left or right multiplying one of the factors. Again, the Moufangbimultiplication law xy.zx = x(yz)x when written in the form (yz)Bx =yLxzRx shows that one can bimultiply a product by individually left andright multiplying the two factors.There are two other Moufang laws, usually written

x(y.xz) = xyx.z and (yx.z)x = y.xzx,

which makes them hard to use. We prefer to write them in the equivalentforms (“Moufang left and right multiplication laws”)

x(yz) = xyx.x−1z and (yz)x = yx−1.xzx

which makes it clear how they substitute for the associative law, becausethey show how to left or right multiply a product by performing multipli-cations on the individual factors:

(yz)Lx = yBxzLx−1 and (yz)Rx = yRx−1zBx .

We have not yet proved that our algebras obey the two new Moufanglaws. In the next chapter we shall do so by showing that all three Moufanglaws are equivalent (using the inverse laws).

6.7. Coordinates for the Quaternions and Octonions 75

6.7 Coordinates for the Quaternions and Octonions

We now turn to the problem of recovering the classical definitions of thealgebras from our description of them in terms of Dickson doubles. Wenote that two units that are orthogonal to 1 and each other–say i andj–have the property that iji = j, since Bi = ref(1) · ref(i) obviously fixesj. So,

ij = k ⇒ ki = j (and ij = k ⇒ jk = i, similarly).

Also,

k2 = ijij = jj = −1and

ji = j i = k = −k.If we take i to be the unit that extends R to C, and j that of the next

extension, we have Hamilton’s celebrated relations:

i2 = j2 = k2 = 1ij = k jk = i ki = j

ji = −k kj = −i ik = −jFor the octonions, we shall find a set of seven units i0, . . . , i6 for which

the permutations

α : in → in+1 β : in → i2n (subscripts mod 7)

are symmetries of the multiplication. We do this by calling the units of ourstarting quaternion subalgebra

i = i1 j = i2 k = i4,

and then using i0 to extend this and defining i0in = i3n, so that

i0i1 = i3 i0i2 = i6 i0i4 = i5.

Now recall that if b, c are in the quaternion subalgebra 1, i1, i2, i4 , wehave i0b.c = i0.cb. This tells us that

i6i1 = i0i2.i1 = i0.i1i2 = i0i4 = i5i5i2 = i0i4.i2 = i0.i2i4 = i0i1 = i3i3i4 = i0i1.i4 = i0.i4i1 = i0i2 = i6,

showing that ix, iy, iz behave just like Hamilton’s i, j, k for each of the seven“quaternion triplets” of subscripts

xyz = 124, 235, 346, 450, 561, 602, 013.

Since the general such system is

in+1, in+2, in+4 (subscripts mod 7),

this verifies our original assertion.


6.8 Symmetries of the Octonions:Diassociativity

The above argument also gives us information about the automorphismgroup of the octonions. Namely, any unit orthogonal to 1 could be takenas i1; any unit orthogonal to 1 and i1 could be taken as i2; and any unitorthogonal to 1, i1, i2, and i1i2 could be taken as i0.To be more precise, if we let j1 be any unit orthogonal to 1, then j2 any

unit orthogonal to 1 and j1, and finally j0 any unit orthogonal to 1, j1, j2and j1j2, then there is just one automorphism taking i1 → j1, i2 → j2,i0 → j0. The choices for j1 are points of the unit sphere in the 7-spaceorthogonal to 1; those for j2 are points of the unit sphere in the 6-spaceorthogonal to 1, j1; and those for j0 are points of the unit sphere in the 4-space orthogonal to 1, j1, j2, j1j2. These spaces are manifolds of dimensions6, 5, and 3, and so the symmetries we have found form a continuous groupof dimension 6 + 5 + 3 = 14. In the standard notation of the Lie theory,this is the Lie group G2. We shall discuss G2 further in Chapter 8.Here we use G2 to prove Artin’s “diassociativity” property of

the octonions:

Theorem 2. The algebra generated by any two octonions is associative.

For up to symmetries of the octonions, the first one can be taken as x+i1y,and the second X + i1Y + i2Z, which puts them both in the quaternionsubalgebra 1, i1, i2, i4 .

6.9 The Algebras over Other Fields

Our principal concern in this book is with composition algebras for thestandard positive definite quadratic form x21 + x

22 + · · · + x2n over the real

field. However, the proofs in this chapter actually apply almost withoutchange to the case of composition algebras for arbitrary non-degeneratequadratic forms over all fields not of characteristic 2.Recall that a quadratic form [x] is non-degenerate if the correspond-

ing bilinear form satisfies

[x, t] = 0 for all t⇒ x = 0.

This is enough to justify our method of deducing x = y from [x, t] = [y, t].The only change is that one cannot guarantee to find an element orthogonal

6.10. The 1-, 2-, 4-, and 8-Square Identities 77

toH that has norm 1. Therefore the extending element imust be permittedto have an arbitrary non-zero norm α, making the doubling rules be

(a+ ib)(c+ id) = (ac− αdb) + i(cb+ ad)and [a+ ib] = [a] + α[b].The argument “really” works also for characteristic 2, where the dou-

bling can be continued infinitely — see Albert [1](and Kaplansky [27]). Theproblem there is not with the algebras, but with the quadratic form–it isjust that one cannot guarantee to find a basis of mutually orthogonal vec-tors. However, a “Dickson double” of H can still be defined using an i notorthogonal to H–this merely complicates the formulae without affectingthe structure of the argument.There is one important way in which these more general algebras may

differ from the standard ones–there can exist non-zero elements withoutinverses. Namely, even for a non-degenerate quadratic form there can be(and in fact there necessarily is, over a finite field) a non-zero x for which[x] = 0.

6.10 The 1-, 2-, 4-, and 8-Square Identities

The existence of the four algebras R, C, H, and O is equivalent to that ofthe N -square identities

x21 y21 = (x1y1)

2

(x21 + x22)(y

21 + y

22) = (x1y1 − x2y2)2 + (x1y2 + x2y1)2

(x21 + x22 + x

23 + x

24)(y

21 + y

22 + y

23 + y

24)

= (x1y1 − x2y2 − x3y3 − x4y4)2+ (x1y2 + x2y1 + x3y4 − x4y3)2+ (x1y3 − x2y4 + x3y1 + x4y2)2+ (x1y4 + x2y3 − x3y2 + x4y1)2

.

(x21 + · · ·+ x28)(y21 + · · ·+ y28)= (x∞y∞ − x0y0 − x1y1 − x2y2 − x3y3 − x4y4 − x5y5 − x6y6)2+ (x∞y0 + x0y∞ + x1y3 + x2y6 + x4y5 − x3y1 − x6y2 − x5y4)2+ (x∞y1 + x1y∞ + x2y4 + x3y0 + x5y6 − x4y2 − x0y3 − x6y5)2+ (x∞y2 + x2y∞ + x3y5 + x4y1 + x6y0 − x5y3 − x1y4 − x0y6)2+ (x∞y3 + x3y∞ + x4y6 + x5y2 + x0y1 − x6y4 − x2y5 − x1y0)2+ (x∞y4 + x4y∞ + x5y0 + x6y3 + x1y2 − x0y5 − x3y6 − x2y1)2+ (x∞y5 + x5y∞ + x6y1 + x0y4 + x2y3 − x1y6 − x4y0 − x3y2)2+ (x∞y6 + x6y∞ + x0y2 + x1y5 + x3y4 − x2y0 − x5y1 − x4y3)2


Hurwitz’s theorem shows that such an identity, say

(x21 + · · ·+ x2N )(y21 + · · ·+ y2N) = z21 + · · ·+ z2N ,in which the zk are bilinear functions of the xi and yj , can exist only forN = 1, 2, 4, 8 (and in these cases is then necessarily “isotopic” to one ofthose above).

6.11 Higher Square Identities: Pfister Theory

In view of these, it came as a great surprise when in 1967 A. Pfister [34]proved that

Theorem 3. In any field and for every n, the product of two sums of2n squares is always another such sum.

He did this by showing that there is an N -square identity

(x21 + · · ·+ x2N )(y21 + · · ·+ y2N) = z21 + · · ·+ z2N ,which the zk are rational functions of the xi and yj whenever N is a powerof 2 (and, unless the characteristic is 2, only when N is a power of 2). Infact, the zk can always be made linear in either the xi or the yj .It does not seem to be generally known that such identities can be

produced by various modifications of the Dickson doubling procedure.3

For example, we obtain a 16-square identity (for ordered pairs a+ ib, c+ idof octonions) from the “definition”

(a+ ib)(c+ id) = ab.b−1c− db+ i(cb+ ad)= X + iY, say,

in which “ab.b−1c” is to be interpreted as ac if b = 0.This is because the norm of the right-hand side is

[X] + [Y ] = [ab.b−1c] + [db] + [cb] + [ad]− 2[ab.b−1c, db] + 2[cb, ad]which equals

[a][c] + [d][b] + [c][b] + [a][d] = ([a] + [b])([c] + [d])

since the inner product terms cancel:

3Pfister did not originally obtain his identities this way.

6.11. Higher Square Identities: Pfister Theory 79

[ab.b−1c, db] = [(ab.b−1c)b, d] = [a.cb, d] = [cb, ad],

where the middle step uses a Moufang law (Chapter 7)

(xy)z = xz−1.zyz.

This is only one of various ways to replace any term ac, db, cb, ad inthe formula

(a+ ib)(c+ id) = (ac− db) + i(cb+ ad)

that produce 16-square identities. Some of them can be repeated indefi-nitely to produce 2n-square identities for all n. One such formula is

(a+ bi)(c+ di) = (ac− bd) + (bc+ b(a.b−1d))i.

Pfister went on to create an entire theory of sums of squares and othermultiplicative quadratic forms. We confine ourselves to an easy result.

Theorem 4. If in a certain field F, the number −1 can be expressedas the sum of N squares, then the least possible value of N (which Pfistercalls the “Stufe” of F) is a power of 2.

For suppose −1 is the sum of 2n + k squares, where 0 < k < 2n, say

−1 = x21 + · · ·+ x22n + y21 + · · ·+ y2k,

but not the sum of fewer (so that x21+ · · ·+ x22n = 0). We write this in theform

−(x21 + · · ·+ x22n) = y21 + · · ·+ y2k + 12 = y21 + · · ·+ y22n , say,

and multiply by s = x21 + · · ·+ x22n to get an equation of the form

−s2 = (x21 + · · ·+ x22n)(y21 + · · ·+ y22n) = z21 + · · ·+ z22n

using a 2n-square identity, from which it follows that

−1 = (z1/s)2 + · · ·+ (z2n/s)2,

a contradiction.


Appendix: What Fixes a Quaternion Subalgebra?

The automorphisms (or “symmetries”) that preserve a quaternion subal-gebra H are nicely described in the language of Chapter 4. As usual, welet i be a fixed unit orthogonal to H , and we write the general octonion asx+ iy with x, y in H, as in the Dickson doubling process. Then we assert:

Theorem 5. The automorphisms of the octonions that take H to itselfform a copy of SO4. The general one,

4 [u, v] = [−u,−v], is parametrizedby two units in H and takes x+ iy → uxu+ i.uyv.

Proof. Plainly any symmetry of H extends to the Dickson double ofH . Since all such automorphisms are the SO3 maps x → uxu, this givesus the symmetries

[u, u] : x+ iy → uxu+ i.uyu.

Modulo these, it suffices to consider the symmetries that fix H pointwise.If such a symmetry takes i to I, then it takes x + iy to x + Iy. But thenwe can write I = iv for some unit v in H , and we see that

[1, v] : x+ iy → x+ i.yv

has this effect, since i.yv = iv.y by (D3).The two types of symmetry [u, u] and [1, v] that we have just found

generate all of the symmetries [u, v] (since [u, v] = [u, u][1, uv]).The three special cases [u, 1], [1, v], [u, u] have very different properties.

Most important are the quasiconjugations Qu = [u, 1]. We can see whythis is by changing our notation: Writing X = x, Y = iy, we find that

[u, 1] : X + Y → uXu+ uY

(since uY = u.iy = i.uy by (D3)), which shows that [u, 1] is independentof the choice of i. It does however depend on the choice of subalgebra Hcontaining u. In full, we call it quasiconjugation by u for the subal-gebra H . The quasiconjugations for a given H form a normal subgroupof the automorphisms that preserve H .The H-stabilizers Sv = [1, v] are the next most important. As their

name implies, they are the elements that fix H pointwise and they formanother normal subgroup. However, the parameter v that describes a givenH-stabilizer unfortunately depends on i as well as H .The diagonal symmetries Du = [u, u] are the least important, since

whether a symmetry is or is not called diagonal can change if we apply

4Here we revert to the use of [u, v] for the general element of SO4.

Appendix: What Fixes a Quaternion Subalgebra? 81

another symmetry. This is because the conjugate of [u, u] by [a, b] is [ua, ub],which might no longer be diagonal. Algebraically, this corresponds to thefact that the diagonal symmetries form a subgroup that is not normal.We shall use these ideas in Chapter 10.

7

7

7

7

Moufang Loops7.1 Inverse Loops

An inverse loop is a set with a distinguished element 1, binary operationxy, and inverse satisfying

x1 = x = 1xx (xy) = y = (yx)x

(x ) = x

for all x, y. (We write the inverse as x rather than the usual x−1 forreasons that may soon be apparent.)A major theme of this chapter is that the basic relation xy = z can be

written in six ways (Figure 7.1). We call this a hexad of relations.Now, since the equivalence

yz = x ⇔ z x = y

xy = z

y = x z x = zy

(A)

yz = x z x = y

z = y x

©© HH

HH ©©

Figure 7.1. Duplex form of the basic hexad.

83

84 7

7

7

7 Moufang Loops

can be written

(yz )x = 1⇔ y(z x) = 1,

we see that the relations (XY )Z = 1 and X(Y Z) = 1 are equivalent andso can be written without parentheses as XY Z = 1.We’ll call xy = z and xyz = 1 the duplex and triplex forms of

the same relation. Converting the six relations of Figure 7.1 to triplexform (after replacing z by z ) makes their symmetry easier to understand(Figure 7.2).

xyz = 1

x z y = 1 z y x = 1

yzx = 1 zxy = 1

y x z = 1

©© HH

HH ©©

Figure 7.2. Triplex form of the basic hexad.

7.2 Isotopies

An isotopy of an inverse loop is a triple of invertible maps that preservethis basic relation. Two notations are appropriate according as the relationis in duplex or triplex form. We write (α,β | γ) to mean xαyβ = (xy)γ ; wewrite (α,β, γ) to mean xyz = 1⇒ xαyβzγ = 1.If (α,β | γ) is an isotopy in duplex form, then

xyz = 1⇒ xy = z ⇒ xαyβ = z γ ⇒ xαyβz γ = 1,

showing that (α,β, γ ) is its name in triplex form.Applying (α,β | γ) to the hexad (A) of relations gives the similar hexad

(B), which undoes to (C) (see Figure 7.3). This shows that when (α,β | γ)is an isotopy, so are all six members of the hexad (D) (see Figure 7.4).Once again, the symmetry is made more apparent by converting (D) intotriplex form (after replacing γ by γ ) as given in (E).

7.2. Isotopies 85

xαyβ = zγ

x αzβ = yγ zαy β = xγ

(B)yαz β = x γ z αxβ = y γ

y αx β = z γ

undoes to

xαyβ = zγ

x α yγ = zβ xγy β = zα

(C)x γ yα = z β xβy γ = z α

x β y α = z γ

©© HH

HH ©©

©© HH

HH ©©

Figure 7.3. Undoing a hexad of isotopies.

(α,β | γ)

( α , γ | β) (γ, β | α)(D)

( γ ,α | β ) (β, γ | α )

( β , α | γ )

converts to

(α,β, γ)

( α , γ , β ) ( γ , β , α )(E)

(γ,α,β) (β, γ,α)

( β , α , γ )

©© HH

HH ©©

©© HH

HH ©©

Figure 7.4. Isotopies in duplex and triplex form.

86 7

7

7

7 Moufang Loops

7.3 Monotopies and Their Companions

We define a monotopy to be any of the three maps in an isotopy. Hexads(D) and (E) show that we get the same concept whichever one we choose.So if γ is a monotopy there exist two maps α,β so that

xy = z ⇒ xαyβ = zγ .

In particular,

zα1β = zγ , so zα = zγ1β = zγb

1αzβ = zγ , so zβ = 1α zγ = azγ .

This shows that γ is a monotopy if and only if there are loop elements b anda for which (xy)γ = xγb.ayγ . We call b and a a pair of companions forγ. This shows that isotopies are remarkably similar to automorphisms (anautomorphism is just a monotopy that has 1, 1 as a pair of companions).In general, the determinations of a and b above show that the three

maps of an isotopy are very closely related: In terms of the left and rightmultiplication operations

La : x→ ax Rb : x→ xb,

we see that α = γRb and β = γLa.But even more is true! Let us take the quotient

(α,β | γ)−1 · ( α , γ | β) = (α−1 α ,β−1γ | γ−1β)of two of the equivalent isotopies from (D). Then since γ−1β = La and soβ−1γ = (La)−1 = La , this isotopy has the special form

(α−1 α , La | La).Applying this to xa yields

a(xa) = xα−1 α .a a = xα

−1 α .

This identifies the map α−1 α taking x to a(xa), and shows that

a(xa).a y = a(xy),

and in particular that

7.3. Monotopies and Their Companions 87

a(xa).a = ax, so a(xa) = (ax)a.

This allows us to use our standard notation Ba for the map taking x toa(xa) = (ax)a, so that our isotopy is (Ba, La | La). Like every isotopy, thisis one of six, the full hexad being (F) (Figure 7.5), as we see using

La = Ra , Ra = La , and Ba = Ba .

Since this hexad is symmetric in a and a , we have proved

Theorem 1. If a is the image of 1 under some monotopy, then wehave all the isotopies of (F), and so La, La , Ra, Ra , Ba, and Ba aremonotopies. In particular, if there is any monotopy that takes 1 to a, thenLa and Ra are such monotopies.

This in turn implies a theorem that explains the significance of theMoufang law:

Theorem 2. The monotopies are transitive just if the Moufang identity

zx.yz = z(xy)z

holds.

Proof. For then (Lz, Rz | Bz) must be an isotopy for every z.Such a loop is called a Moufang loop (or, informally, a “Moup”).Martin Liebeck [31] has proved that the only finite simple Moufang loops

other than groups are those obtained by reading the octonions mod p.

(La, Ra | Ba)

(Ra , Ba | Ra) (Ba, La | La)(F )

(Ba ,La | La ) (Ra, Ba | Ra )

(La , Ra | Ba )

©© HH

HH ©©

Figure 7.5. A hexad of isotopies involving La, Ra, Ba.

88 7

7

7

7 Moufang Loops

7.4 Different Forms of the Moufang Laws

Applied to xy, the six isotopies of (F) are equivalent to the three laws

z(xy)z = zx.yz (Bi-)Moufang Lawz(xy) = zxz.z y Left Moufang Law(xy)z = xz .zyz Right Moufang Law

where z is either a or a . The traditional forms of the last two laws areslightly different:

z(x(zt)) = zxz.t((tz)y)z = t.zyz

These are obtained by setting y = zt and x = tz in the left and rightMoufang laws.Moufang loops are named after Ruth Moufang, who discovered them in

connection with the coordinatization of certain projective planes — see [33].But in a sense, this was not really their first entry into mathematics, sincethe non-zero real octonions constitute the archetypical Moufang loop. In-deed, we conjecture that the loop generated by n generic real octonions isin fact the free Moufang loop on n generators.Of course, the Moufang identities can be regarded as weakened forms

of the associative law. Indeed, one of the most famous theorems aboutthis is Artin’s diassociativity theorem: Any 2-generator Moufang loop isassociative. However, we’ve seen that a better reason for studying abstractMoufang loops is that these identities are equivalent to a symmetry prop-erty, namely that the monotopies are transitive. We shall use some of theirtheory–in particular, the properties of companions –in the next chapter.

88 88 8

Octonions and8-Dimensional GeometryWe know that the multiplications by unit complex numbers or quaternionsgenerate the respective orthogonal groups SO2 and SO4, and we’ll provein this chapter the corresponding result that the multiplications by unitoctonions generate SO8.However, there are differences between the three cases, controlled by

the validity or otherwise of the commutative and associative laws. Theassociative law in the forms

Lxy = LyLx, Rxy = RxRy

for units x and y would tell us that the product of two multiplications ofthe same kind should be a third, so that when it holds, the left and rightmultiplications form subgroups of the special orthogonal groups.In the complex case, the commutative law, in the form Lx = Rx, tells

us that these groups are the same, and both are in fact the whole groupSO2. In the quaternionic case, both the left and right groups form propersubgroups, and we saw in Chapter 4 that every element of SO4 is theproduct of one multiplication of each kind. Since the associative law failsfor octonions, it is no longer true that the product of two multiplicationsof the same kind is another. We shall see in this chapter that in fact theoctonion multiplications of either kind generate the whole group SO8.

8.1 Isotopies and SO8

We first prove that the octonions are strongly non-associative, in the sensethat

89

90 88 88 8 Octonions and 8-Dimensional Geometry

Theorem 1. If x(ry) = (xr)y for all octonions x, y, then r is real.

For we have (i1i0)i2 = −i1(i0i2), so if (i1r)i2 = i1(ri2) the coefficient ofi0 in r must be 0, as must that of in(1 ≤ n ≤ 6), using (in+1in)in+2 =−in+1(inin+2).This implies that companions (Chapter 7) are essentially unique:

Theorem 2. If a, b is any pair of companions for the monotopy γ, thenany other pair has the form ar, r−1b where r is real.

Proof. The condition that A,B should be a second pair of companionsis the identity

xγa.byγ = xγA.Byγ for all x, y.

Writing xγ = X, yγ = Y , the identity becomes

Xa.bY = XA.BY for all X,Y .

In this, setting A = ar and putting X = a−1, Y = 1 gives b = rB, soB = r−1b, so the identity is now that

Xa.bY = X(ar).(r−1b)Y for all X,Y .

In this, putting Y = (r−1b)−1 = b−1r shows (Xa)r = X(ar), while puttingX = (ar)−1 = r−1a−1 shows r−1(bY ) = (r−1b)Y , transforming the identitystill further to

Xa.bY = (Xa)r.r−1(bY ) for all X,Y .

Finally, putting Xa = x, bY = ry transforms it to

x(ry) = (xr)y for all x, y,

which completes the proof by showing that r is real.We are ready now for the application to SO8:

Theorem 3. If γ is any element of SO8, there exist α,β in SO8 forwhich (α,β | γ) is an isotopy (so γ is a monotopy). Moreover, α,β areuniquely determined up to sign, the only other pair being −α,−β.

We shall prove this using the

Lemma 1. The operations ref(1)ref(a) and ref(a)ref(1) are bimultipli-cations by unit octonions.

8.2. Orthogonal Isotopies and the Spin Group 91

Proof. These are unaffected if we rescale a to have norm 1. But thenthe first one we found to be Ba in Chapter 6, and the second is its inverse,Ba−1 = Ba.

Proof of the theorem. We can write γ as the product of an evennumber of reflections, say

γ = ref(a1)ref(b1)ref(a2)ref(b2) · · · ref(a2n)ref(b2n).

Now

ref(ai)ref(bi) = ref(ai)ref(1)ref(1)ref(bi)

is the product by two bimultiplications in unit octonions, so γ is the productof 2n such unit bimultiplications, say Bc1Bc2 · · · Bc2n . Then

(α,β | γ) = (Lc1 · · · Lc2n , Rc1 · · · Rc2n | Bc1 · · · Bc2n)

is the desired isotopy. (Since c1, . . . , c2n are unit octonions, α and β areindeed in SO8.) But we also know that α and β are unique up to scalarmultiplication, and the only non-trivial scalar that keeps them in SO8 is−1, so the only other such isotopy is

(−α,−β | γ) = (α,β | γ).

8.2 Orthogonal Isotopies and the Spin Group

In the foregoing arguments, it was convenient to use isotopies in the duplexform (α,β | γ). As we approach triality, it will be convenient to switch tothe triplex form (α,β, γ), in which the isotopies we have just found takethe form

(LaLb · · · , RaRb · · · , BaBb · · ·).

We call (α,β, γ) an orthogonal isotopy if its three monotopies α,β, γare elements of SO8. What we have shown is that the group of orthogonalisotopies is a 2-to-1 cover of SO8, which we call the spin group Spin8Readers who are familiar with the usual definition of the spin groups willrecognize this one as an instance. As we remarked in Chapter 3, we preferto avoid giving a general definition in this book.


8.3 Triality

Our definition of the spin group makes it clear that

Theorem 4. The spin group has an outer automorphism of order 3,namely that taking (α,β, γ)→ (β, γ,α).

Proof. This is clearly an automorphism. That it is an outer isomor-phism (i.e., not a conjugation of the form g → h−1gh) is clear because theimages Bi0 and Li0 of the triality-related elements

(Li0 , Ri0 , Bi0 = Bi0) and (Ri0 ,Bi0 , Li0)

in SO8 are not conjugate (Bi0 fixes a 6-space, while Li0 is fixed-point-free).The special orthogonal group SO8 does not have this triality, because

γ determines α and β only up to sign. However,

Theorem 5. The projective special orthogonal group PSO8 does havea triality automorphism.

For if ϕ ∈ SO8, let us define [ϕ] = +ϕ,−ϕ, so that [ϕ] ∈ PSO8. Thenthe set of triples ([α], [β], [γ]) arising from orthogonal isotopies (α,β, γ) isplainly isomorphic to PSO8 (since now [γ] uniquely determines [α] and[β]), and has the triality

([α], [β], [γ])→ ([β], [γ], [α]).

Acting on PSO8, triality takes

[Lu]→ [Ru]→ [Bu]→ [Lu]

for every unit octonion u.

8.4 Seven Rights Can Make a Left

An immediate consequence of triality is

Theorem 6. SO8 is generated by the left multiplications by octonionunits, or equally by the right ones.

Proof. SO8 is generated by the bimultiplications, and so applyingtriality, equally by the left ones or the right ones.

8.4. Seven Rights Can Make a Left 93

In particular, any left multiplication can be expressed as a product ofright ones, or vice-versa. Just how many right multiplications are neededto represent a given Lx? The answer is given by

Theorem 7. (The 7-Multiplication Theorem) Any element ofSO8 is the product of at most seven right (or seven left) multiplications.

This is best possible:

Theorem 8. The general left multiplication is the product of sevenright ones, but no fewer.

Proof. If α ∈ SO8, then ref(1) · α ∈ GO8 \ SO8, and so is the productof at most seven reflections, showing that

α = ref(1)ref(u1)ref(u2) . . . ref(u7),

which we can write as the product of seven bimultiplications:

α = Bu1Bu2Bu3Bu4Bu5Bu6Bu7

using

ref(u)ref(v) = ref(u)ref(1)ref(1)ref(v) = BuBv.

By triality, α is equally the product of seven left multiplications or sevenright ones. In particular, any left multiplication is the product of sevenright ones; for instance, we find

Li0 = Ri2645Ri2645Ri6Ri5Ri∞013Ri∞013

Ri3

and

Li∞365 = Ri0Ri0124Ri0124Ri0124Ri∞641Ri∞512

Ri∞324,

which we shall use in Chapter 9.On the other hand, if u is a unit other than ±1, then Lu cannot be

written as a product of six unit right multiplications, since then Ru would bethe corresponding product of six bimultiplications, and so of six reflections,and so fix a vector, which it does not.Since positive scalar factors can be absorbed into all three types of

multiplication, we need not restrict to units: Lx can be a product of sixright multiplications only if x is real.


8.5 Other Multiplication Theorems

What happens for products of multiplications that may be of mixed types?If X, Y, Z are chosen from L, R,B, we define XYZX, for instance, to be the set ofall elements of SO8 that are expressible in the form XaYbZcXd for arbitraryoctonion units a, b, c, d.

Theorem 9. Sets of the types below have the respective dimensions:

X XX XXX XXXX XXXXX XXXXXX XXXXXXX

7 13 18 22 25 27 28

XY XXY XXXY XXXXY

14 20 25 28

Proof. Since octonion conjugation interchanges L and R while fixing B,while triality rotates L, R, B, we can suppose X = B and Y = R. The answerin the first line now follows from the fact that the set of products of n Bsis the set of products of n reflections, if n is even, or that set multiplied byref(1), if n is odd.We next show that the dimensions of XY, XXY, XXXY are seven more

than those of X, XX, XXX. This follows immediately from the fact thatthe equation

BaBbBcRd = Ba Bb Bc Rd

implies d = d (and so BaBbBc = Ba Bb Bc ). This is because it makes Rd Rdbelong to BBBBBB and so fix a 2-space. But if e = 0 is in this 2-space, wehave (ed )d = e, and so ed = ed, whence indeed d = d.The similar equation

BaBbBcBdRe = Ba Bb Bc Bd Re

only implies the trivial

Re Re ∈ BBBBBBBBand so does not imply e = e. However, we can still compute the dimensionof BBBBR (which is 28 rather than 29) by counting the dimension of the setof such equivalences. We suppress the details.Many other mixed products reduce to the ones mentioned using

Lemma 2. We have the equations

LR = RB = BLRL = LB = BR

8.6. Three 7-Dimensional Groups in an 8-Dimensional One 95

For example, these imply BLL = LRL = LLB showing (in two ways) thatLRL has dimension 20.

Proof. The identities

LaRb = RbaBa = BbLabRaLb = BbRba = LabBa

follow from the Moufang laws.In fact, the only set of length at most 4 not covered by our theorem is

XXYY (LLRR, RRLL, RRBB, BBRR, BBLL, LLBB are indeed all the same set, byLemma 2), while those of length 5 or more are probably the whole group;namely, we conjecture

The 5-Multiplication Proposition. If V, W, X, Y, Z are any 5 letterschosen from L, R, B that are not all equal, then the set VWXYZ contains everyelement of SO8.

Our lemma shows that every case can be reduced to XXXXY, which wealready know to have the full dimension 28, so very little is missing.

8.6 Three 7-Dimensional Groups in an8-Dimensional One

The best way to understand the relationship between Spin8 and SO8 is viathe 2-to-1 map that passes from a triple to its last coordinate:

(α,β, γ)(−α,−β, γ) γ.Hj

©*

Technically, this is an 8-dimensional representation of Spin8.The 7-dimensional spin group Spin7 is the preimage of SO7 in this map,

so the triples (α,β, γ) for which γ fixes 1 form a copy of Spin7. Obviously,by demanding instead that α or β fix 1, we obtain two other copies of Spin7in Spin8.All these groups have 8-dimensional representations obtained by re-

stricting the one above, and these three representations are very different.One of them is reducible (because 1 is fixed), and not faithful (because(−1,−1, 1) is in the kernel). The other two are both irreducible and faith-ful. This is because on one hand, the nontrivial element (−1,−1, 1) of thekernel is in neither one of those groups, while on the other hand, they aretransitive on the unit ball. For consider the particular isotopy

(α,β, γ) = (Ru, Bu,Lu)(Bu, Lu, Ru),


in which γ : x → uxu fixes 1, while α and β take 1 → u3 and u3, whichcan be any unit octonions.

Theorem 10. The intersection of any two of these copies of Spin7 inSpin8 is the intersection of all three. This is the automorphism group ofthe octonions.

Proof. Applying α,β, γ to the equation 1 · 1 · 1 = 1, we find that1α1β1γ = 1, showing that if any two of α, β, γ fix 1, then all three do. Butnow if a and b are the companions of γ, so that α = γRa, β = γLb, thisentails that a = b = 1, showing that γ is an isomorphism.This is the Lie group usually called G2. It is the smallest of the five

“exceptional” Lie groups.Figure 8.1 illustrates the relationship between the groups we have men-

tioned. To go down any line on this graph, one demands that one of α, β,γ fixes 1–this reduces the dimension by 7 since it could have taken 1 toany point of the unit 7-sphere.

Spin7 Spin7 Spin7

Spin8

G2

dim28

dim21

dim14

7 7 7

7 7 7

Figure 8.1. The automorphism group G2 of the octonions is the intersection ofany two Spin7s in Spin8.

The elements of Spin8 come in quartets

(α,β, γ) (α,−β,−γ) (−α,β,−γ) (−α,−β, γ),that are related by multiplication by elements of its center:

(1, 1, 1) (1,−1,−1) (−1, 1,−1) (−1,−1, 1).The map (α,β, γ)→ γ onto SO8 obscures some of the symmetry by fusingthese in two pairs:

(α,β, γ)(−α,−β, γ) γ

(α,−β,−γ)(−α,β,−γ) − γ.Hj

©*Hj©*

8.7. On Companions 97

We can regain it by passing to PSO8, which identifies γ and −γ to [γ]:

(α,β, γ)(−α,−β, γ) [γ]

(α,−β,−γ)(−α,β,−γ)

Hj©*

©¼HY

So Spin8 and PSO8 have a triality symmetry that SO8 lacks. We discussthis further in the next section.

8.7 On Companions

Putting together various results from this chapter and the last, we have

Theorem 11. Given any element γ ∈ SO8 there exists a pair a, b ofunit octonions called its companions such that

(xy)γ = xγa.byγ .

The companions are uniquely determined up to negation, the only otherpair being (−a,−b).

It is not hard to find the “multiplication rule” for companions:

Theorem 12. If the companions of γ and δ are a, b and c, d, then thoseof γδ are cd.aδc, dbδ.cd.

Proof.

(xy)γδ = (xγa.byγ)δ

= (xγa)δc.d(byγ)δ

= (xγδc : daδ)c.d(bδc : dyγδ)= (xγδ : c(daδ)c)(d(bδc)d : yγδ)= (xγδ : cd.aδc)(dbδ.cd : yγδ).

The first three equalities are from the definition of companions, while theremaining two use the Moufang laws.We can use these to find the companions for any combination of left,

right, and bimultiplications, starting from

Theorem 13. The companions of La, Ra, and Ba are respectively

a, a−2 a−2, a a−1, a−1.


Proof.a(xy) = axa.a−1y = ax.a : a−2.ay(xy)a = x−1a.aya = xa.a−2 : a.yaa(xy)a = ax.ya = axa.a−1 : a−1.axa

In group theory, the map Ta : x → a−1xa, called transformationor conjugation by a, is important as yielding an automorphism. In theoctonions, this is not generally true, but (after Zorn [44]) we know all caseswhen it is:

Theorem 14. Ta is an automorphism just when a3 is real.

Proof. The following argument shows that Ta has companions a−3, a3,

the last equality using diassociativity (twice):

a−1(xy)a = a−1xa−1.ay : a= a−1xa−1.a−1 : a(ay)a= a−1xa.a−3 : a3.a−1ya

In particular Tω, where ω = −12 + 12 i1 +

12 i2 +

12 i4, is an automorphism of

the octonions, and even of the integral ones that we shall define in the nextchapter.

9

9

9

9

The Octavian Integers OIn this chapter, we first find the correct way to define “the integral octo-nions” and investigate the geometry of the very interesting lattice E8 thatthey form. We deduce that the ring O = O8 of “octavian integers” has theEuclidean property–that we can divide with a remainder smaller than thedivisor. Unfortunately, this does not lead to the expected theory of idealsand unique factorization, since, as we shall show, the ideals of O are rathertrivial. Finally, a new theory that appears for the first time in this bookfinally illuminates both the arithmetic and geometry of the divisors of anarbitrary octavian integer. It was prompted by H. P. Rehm’s discovery ofan interesting new form of the Euclidean algorithm.

The elements of O have been called “The Integral Cayley Numbers”since Coxeter’s paper [11] of that title. Later, Coxeter remarked “If I hadread F. van der Blij’s ‘History of the octaves’ [42] before writing [thatpaper], I would have entitled it ‘Integral Octaves.’ ” However, the term“octonion” seems now to have become standard. By way of compromise,we propose to use “octavian” (analogous to “Hurwitzian”) for “octonioninteger,” and in further homage to Graves, “Gravesian” for the analogueof “Lipschitzian.”

9.1 Defining Integrality

We recall from Chapter 5 the two notions of quaternion integer due toLipschitz and Hurwitz, and we ask what it should mean to be an octo-

99

100 9

9

9

9 The Octavian Integers O

nion integer? Nowadays, algebraists take “integer” to mean member ofwhat’s called a maximal order, following Dickson, who used the term“arithmetic.”The concept (of order) arose first for algebraic number fields (i.e., finite

algebraic extensions of the rationals), where it means “a subring each whoseelements satisfy an equation of the form

xn + cn−1xn−1 + · · ·+ c0 = 0 with cn−1, . . . , c0 ∈ Z.”Dickson defined an arithmetic to be a maximal order.In Q(

√−3), for example, a+ b√−3 satisfies x2 − 2ax+ (a2 + 3b2) = 0,and certainly the set Q[

√−3] (consisting of those a+b√−3 with a, b ∈ Z) isan order. However, the larger ring of Eisenstein integers (with 2a, a+b ∈ Z)is also an order, so that Q[

√−3] is not a maximal order, or arithmetic.We shall use the same terminology for the “ring” Q(i0, i1, i2, i3, i4, i5, i6)

of rational octonions even though this is not associative. Since the minimalpolynomial of α = a∞ + a0i0 + · · ·+ a6i6 is

x2 − 2a∞x+ (a2∞ + a20 + · · ·+ a26) = 0,the condition is that the “trace” 2a∞ and “norm” a2∞ + a20 + · · · + a26must be ordinary integers for any element of an order or arithmetic inQ(i0, i1, i2, i3, i4, i5, i6).

9.2 Toward the Octavian Integers

The most obvious order inside Q(i0, i1, i2, i3, i4, i5, i6) consists of those oc-tonions for which all the coordinates a∞, . . . , a6 are ordinary integers–wecall these the Gravesian octaves or integers. In this section, we provethat

Theorem 1. The orders containing the Gravesian integers are preciselythe 16 integer systems of Figure 9.1.

In what follows, we shall have many occasions to refer to an octonionα whose coordinates are in 1

2Z. In this case, the coordinates that are notalso in Z form what we shall call the halving-set of subscripts for α.

Lemma 1. In an element α of such an order, the doubles 2ar of allthe coordinates of α are integers, and the size m of every halving-set is amultiple of 4.

Proof. For the first assertion, multiply α by ir. For the second, observethat

a2∞ + · · ·+ a26 ≡m

4(mod 1).

9.2. Toward the Octavian Integers 101

In view of this lemma, we introduce the notation

iabcd foria + ib + ic + id

2,

and use bars on subscripts to indicate negation, for example

iabcd foria − ib + ic − id

2.

We can specify one of the 16 systems by saying just which the halving-sets are. The multiplicative structure of the octonions already involvesa distinguished family of quadruplets, namely those that correspond toquaternion subalgebras, together with their complements. These, togetherwith the empty and full sets, we call the sixteen ∞-sets:

∅ ∞124 ∞235 ∞346 ∞450 ∞561 ∞602 ∞013Ω =∞0123456 0356 0146 0125 1236 0234 1345 2456.

∅ Ω∞5614023

∅ Ω∞6025134

∅ Ω∞0136245

∅ Ω∞1240356

∅ Ω∞2351460

∅ Ω∞3462501

∅ Ω∞4503612

all0-sets

all1-sets

all2-sets

all3-sets

all4-sets

all5-sets

all6-sets

∅Ω

∅

TheOctavianRings

TheDouble

HurwitzianRings

TheKleinian

Octaves

The

GravesianOctaves

Figure 9.1. The sixteen orders containing the Gravesian octaves, G8.

102 9

9

9


An octonion whose halving-set is an ∞-set we call an “∞-integer” or“Kirmse integer,” after J. Kirmse, who naturally but mistakenly sup-posed that they formed a maximal order.1

However, Coxeter found that

The ∞-integers are not multiplicatively closed.

For example, the product of the∞-integers i∞026 and i∞013 is i0235, whichis not an ∞-integer, and the product of this with the ∞-integer i∞235 is

−34+1

4(i0 − i1 + i2 + i3 − i4 + i5 + i6),

whose minimal polynomial x2 + 32x+ 1 is manifestly non-integral.

Dickson found the appropriate correction, which was rediscovered andcompleted by Bruck. We define the n-sets and n-integers (n = 0, . . . , 6)by interchanging ∞ with n in the definition of ∞-sets and ∞-integers.

∞124 0356∞235 1460∞346 2501∞450 3612∞561 4023∞602 5134∞013 6245∅ Ω

0124 ∞3560235 146∞0346 25∞1∞450 36120561 4∞23∞602 5134∞013 6245∅ Ω

∞124 03561235 ∞4601346 250∞1450 36∞2∞561 40231602 5∞34∞013 6245∅ Ω

∞124 0356∞235 14602346 ∞5012450 361∞2561 40∞3∞602 51342013 6∞45∅ Ω

∞− sets 0− sets 1− sets 2− sets

3124 0∞56∞235 1460∞346 25013450 ∞6123561 402∞3602 51∞4∞013 6245∅ Ω

∞124 03564235 1∞60∞346 2501∞450 36124561 ∞0234602 513∞4013 62∞5∅ Ω

5124 03∞6∞235 14605346 2∞01∞450 3612∞561 40235602 ∞1345013 624∞∅ Ω

6124 035∞6235 14∞0∞346 25016450 3∞12∞561 4023∞602 51346013 ∞245∅ Ω

3− sets 4− sets 5− sets 6− setsFigure 9.2. All n-sets, with outer n-sets in bold.

1Other people have made this very natural assumption, so it is convenient that ithas a standard name, “Kirmse’s Mistake.” The product of two randomly chosen Kirmseintegers happens to be a Kirmse integer rather more than one-third of the time.

9.2. Toward the Octavian Integers 103

Then we have

Lemma 2. The n-integers are multiplicatively closed for each n =0, . . . , 6.

Proof. The seven systems are obviously isomorphic, so we study the0-integers, for which the halving-sets are

∅ 0124 0235 0346 ∞450 0561 ∞602 ∞013Ω =∞0123456 ∞356 ∞146 ∞125 1236 ∞234 1345 2456.

This shows that the 0-integers are spanned by i∞356, i0235, i0463, and i0156over the Gravesian integers, so the result is established by the multiplicationtable:

i∞356 i0235 i0463 i0156i∞356 i∞356 i∞234 i∞461 i∞152i0235 i∞045 −1 i∞125 i∞416i0463 i∞013 i∞125 −1 i∞243i0156 i∞026 i∞416 i∞243 −1

The resulting systems are seven of the sixteen orders of Theorem 1, theseven maximal ones (see Figure 9.1). The intersections of pairs of these,which are also the intersections of certain triples, yield the seven “doubleHurwitzian” systems. (The halving-sets ∅, Ω, ∞124, 0356 for the typicalone of these show that it is obtained by Dickson doubling from a Hurwitzianring of quaternions.) The intersection of all seven maximal systems, whichis equally the intersection of any two of the double Hurwitzian systems,we call the Kleinian octaves, since they can be obtained from Graves’sinteger octaves by adjoining 1

2 (1 + i0 + · · ·+ i6), which, being of the form12 (1 +

√−7), is a “Kleinian” integer (see Chapter 2).We now complete the proof of Theorem 1. We see that every possible

halving-set is an n-set for some n = ∞, since every set of 0 or 4 or 8coordinates appears in Figure 9.2. We call it an inner or outer n-setaccording as it is or is not also an ∞-set.First we deal with the outer n-sets.

Lemma 3. Together with the Gravesian integers, any octonion α whosehalving-set is an outer n-set generates all the n-integers.

Proof. We can suppose that the halving-set is one of the eight outer0-sets. By subtracting a Gravesian integer, we can reduce α to the form

α =1

2(±ia ± ib ± ic ± id),

104 9

9

9


where, for the same reason, the choice of signs is unimportant. Then Fig-ure 9.3 shows how multiplication by three Gravesian units connects alleight outer 0-sets. Since each inner 0-set is the sum of two outer ones, theresulting octonions additively generate all the 0-integers.

52∞1

∞365

0124

4603

2530

3∞24

1065

641∞

5

5

55

6

6

6

6

3

3

3

3

Figure 9.3. The eight outer 0-sets, related by multiplication by i3, i5, and i6,ignoring signs.

Now we deal with the inner n-sets, which are also ∞-sets.

Lemma 4. In the presence of the Gravesian integers: (i) Two octa-vian integers whose halving-sets are complementary 4-element ∞-sets gen-erate each other. (ii) Two octonions whose halving-sets are distinct non-complementary ∞-sets generate the n-integers for some n (see Figure 9.4).

Proof. (i) Left multiplication by i0 converts12 (i∞ + i1 + i2 + i4) to

12 (i0 + i3 + i6 + i5), etc.(ii) We can suppose by (i) that both sets mention∞, so one is obtained

from the other by increasing subscripts by 1, 2, or 4 (mod 7), so withoutloss of generality by 1 in view of the subscript-doubling symmetry β =(i1i2i4)(i3i6i5). So, again without loss of generality, the two halving-setsare ∞602 and ∞013, and we have already seen that the product of i∞602and i∞013 has halving-set 0235, an outer 0-set.The proof that an order containing the Gravesian integers is one of our

sixteen is now easy. If it isn’t the Gravesian or Kleinian octaves, it mustinvolve a 4-element halving-set, and therefore also the complementary one,as well as ∅ and Ω. If these are all the halving-sets, it is a double Hurwitzianring. Otherwise, it must involve either an outer n-set or two distinct non-complementary inner ones, and so contain all the n-integers for some n.

9.3. The E8 Lattice of Korkine, Zolotarev, and Gosset 105

∞3460125

∞1240356

∞0451236

∞0261345

∞0132456

∞1560234

∞2350146

0

12

3

4

5

6

Figure 9.4. If two “points” in this “plane” lie on the “line” marked n, octonionswith the corresponding halving-sets generate the n-integers.

Any still larger order would, by the same argument, contain the n-integers for two distinct values of n. But applying the subscript-doublingsymmetry β allows us to suppose that these differ by 2, so without loss ofgenerality, the order contains the 0-integer i∞235 and the 2-integer i0235,which, as we have seen, give a non-integral product.

9.3 The E8 Lattice of Korkine, Zolotarev, and Gosset

We now fix our arithmetic to be a particular one of the seven maximalorders, defining the octavian integers (or briefly, “the octavians”) to bethe 0-integers. Geometrically, the octavian integers O8 form a well-knownand very interesting lattice usually called the E8 root lattice (or, since itsgeometry was investigated by Thorold Gosset [19], the Gosset lattice).2

2It is also the unique “even unimodular” lattice in eight dimensions, and as such wasproved to exist by H. J. S. Smith [39] in 1867. Korkine and Zolotarev [28] explicitlyconstructed it in their 1877 study of sphere packings. (They were anticipated to someextent by Rodrigues.) Blichfeldt [6] proved in 1935 that it does indeed yield the densestlattice packing of spheres in eight dimensions.

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Its arithmetical properties have also been much studied: We quote fromthe analytic theory of quadratic forms the fact that the number of vectorsin E8 of norm 2n > 0 is 240 times the sum of the cubes of the divisors ofn. In the following sections, we give a simple new geometrical definition ofthis lattice and use it to prove that octavians have the “division with smallremainder” property.

9.3.1 The Simplex Lattice An

An n-dimensional regular simplex is the convex hull of n + 1 pointswhose distances from each other are all equal. The n-dimensional simplexlattice An is generated by vectors along the edges of such a simplex. If wetake the vertices to be

vi = (0, 0, . . . ,i1, . . . , 0, 0)

in (n+ 1)-space, then the generators are

vi − vj = (0, 0, . . . ,i1, . . . ,

j

−1, . . . , 0, 0),which identifies An as the set of all vectors (x0, . . . , xn) of n + 1 integercoordinates with zero sum.

9.3.2 The Orthoplex Lattice Dn

The name regular orthoplex is our term for what has also been calleda cross polytope, the analogue of the 2-dimensional square and the 3-dimensional octahedron (see Figure 9.5). Its vertices (±11, 0n−1) are theends of unit vectors along the axes of an orthonormal frame, and it has one(simplicial) cell for each of the resulting orthants of n-dimensional space(so that “orthoplex” abbreviates “orthant-complex”).The n-dimensional orthoplex lattice Dn is the lattice generated by

the vectors along the edges of a regular n-dimensional orthoplex. If we takevertices of the orthoplex to be

vi = (0, 0, . . . ,i1, . . . , 0, 0) and vi = (0, 0, . . . ,

i−1, . . . , 0, 0) (i = 1, . . . , n),

then the generators of Dn are all of the vectors (±12, 0n−2) and Dn is iden-tified with the lattice of all the vectors (x1, . . . , xn) of integer coordinatesthat have even sum.

9.3. The E8 Lattice of Korkine, Zolotarev, and Gosset 107

Figure 9.5. The 2-dimensional orthoplex is a square. Its four edges correspond tothe four quadrants of the plane. The 3-dimensional orthoplex is a regular octahe-dron. Its eight triangular faces correspond to the eight octants of space.

9.3.3 Defining E8

We define E8 to be the lattice S,O generated by the edge-vectors of a sim-plex (of dimension 8) S together with an adjacent orthoplex O. Figure 9.6shows several things:

• E8 contains copies of both A8 and D8. In fact, we shall see that itcontains A8 to index 3 and D8 to index 2.

• If O is the orthoplex on an adjacent face of S, then S,O = S,O ,so the construction has the full symmetry of S: E8 is generated bythe edges of S together with those of the orthoplex on any one of itsnine faces.

• If instead O is the orthoplex on an adjacent face of O, then S,Oequals the lattice O,O generated by the edges of O and O , whichare related to each other by reflection in the plane that separatesthem. So the construction has only half of the symmetry of O: E8 isgenerated by the edges of O and a simplex (S) on any of 128 of itsfaces or of O and the orthoplex (O ) obtained by reflecting it in anyof the other 128 faces.

The figure also shows incidentally that the 8-dimensional space contain-ing an E8 is covered by simplexes and orthoplexes equivalent by symmetries

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v1v2, . . . , v8

v1

v1

v2, . . . , v8 (redundant)

v2, . . . , v8 (redundant)

vi = (0, . . . ,i

1, . . . , 0)

vi = (0, . . . ,i−1, . . . , 0)

vi = (12, . . . ,

i

−12, . . . , 1

2)

StartingOrthoplex

O

AdjacentOrthoplex

O

SSimplex

Vertices of O are v1, . . . , v8, v1, . . . , v8 and center is (08)

Vertices of O are v1, . . . , v8, v1, . . . , v8 and center is (14

8)

Vertices of S are v1, v1, v2, . . . , v8 and center is (− 16 ,

16

7)

Figure 9.6. The three lattices S,O , S,O , O,O all coincide with the latticegenerated by differences of the ten indicated vectors v1, v1, v1, v2, . . . , v8 in view ofthe identities v1 + v1 = . . . = v8 + v8 and v1 + v1 = . . . = v8 + v8.

to those mentioned. For, half of the walls of an such orthoplex lead to equiv-alent orthoplexes and the other half to simplexes, while from a simplex anywall leads to an orthoplex. Hence

Lemma 5. For any α of 8-dimensional space, there is a point β of theE8 lattice for which [α− β] ≤ 1.

For α is in either a simplex or an orthoplex, so can be no further fromthe lattice than is the center of that polytope.

To show that the octavian integers include a scaled copy of the E8lattice, it suffices to find among them the vertices of a simplex and anadjacent orthoplex. This is done in Figure 9.7. Since the norms of thesevectors are half those of our previous E8, we obtain the restated

Lemma 6. For any real octonion α, there is an octavian integer β forwhich [α− β] ≤ 1

2 .

9.4. Division with Remainder, and Ideals 109

0

i∞013i∞026i∞045i∞013i∞026i∞045i∞

i∞ + i0i∞013i∞026i∞045i∞013i∞026i∞045i0

i∞124

Simplex

Orthoplex

Figure 9.7. The octavian integers generate E8.

9.4 Division with Remainder, and Ideals

Applying this to δ−1α (or alternatively αδ−1), we immediately deduce the

Theorem 2. If α and δ are octavian integers with δ = 0, then we canwrite α = δβ + ρ (or alternatively α = βδ + ρ) for octavian integers β andρ with [ρ] ≤ 1

2 [δ].

Since in particular we have [ρ] < [δ], this entails as usual that

Theorem 3. Any left or right ideal in O8 is principal.

The usual reason to be interested in ideals is because they appear inthe standard proofs of factorization theorems. In a non-commutative ring,the left and right ideals are distinct concepts. However, O8 is an unusualring:

Lemma 7. Every ideal in O8 is 2-sided.

Proof. In particular, a right ideal I is closed under unit right mul-tiplications. In Chapter 8 we showed that the left multiplications by theparticular units i0 and i∞365 were products of seven right multiplications

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by octavian units. In Chapter 10, we shall see that it follows easily thatevery left multiplication by an octavian unit is a product of seven rightones. Therefore I is closed under left multiplication by octavian units, andso by arbitrary octavian integers, since any such is a sum of octavian units.We can say more:

Theorem 4. Any 2-sided ideal Λ in O8 is the principal ideal nO8

generated by a rational integer n.

This turns out to destroy the possibility of using ideals to prove uniquefactorization. This result was partially proved by Mahler [32], sketchedby van der Blij and Springer [43], and completed by Lamont [30]. Ourgeometrical results will make it easy.

Proof. This is because Λ is invariant under bimultiplications by unitoctavians, which generate all even symmetries of its underlying scaled E8lattice. We now apply

Theorem 5. If a sublattice Λ of E8 is fixed under all even automor-phisms of E8, then Λ = nE8.

To see this, use the “orthoplex” coordinates for E8. Let v1 =(a b c d e f g h) be an element of minimal nonzero norm in Λ. Wefirst show that some two coordinates may be supposed to be 0. For ifnot, we can apply even permutations to suppose g, h are the coordinatesof smallest absolute value, after which there is an even symmetry takingv1 to v2 = (a b c d e f −g −h), so that the norm of v3 = v1 − v2 =(0 0 0 0 0 0 2g 2h) is at most that of v1.Next we show that there is at most one nonzero coordinate. Otherwise,

if the last two coordinates are 0, there exists an even symmetry (a 3-cycle)taking v1 = (a b c d e f 0 0) to v2 = (a b c d e 0 f 0), where fis the nonzero coordinate of least absolute value, and again the norm ofv3 = v1 − v2 = (0 0 0 0 0 f −f 0) is at most that of v1. This shows thatone minimal nonzero vector of Λ is f times a minimal vector of E8, andapplying even symmetries we see that f times any minimal vector of E8 isin Λ, and so Λ = fE8.It remains to consider the case when a minimal vector of Λ has only

one nonzero coordinate. But then (being in E8) it must have the form(2f 0 0 0 0 0 0 0), which is taken to (f f f f 0 0 0 0) by a suitable evensymmetry, and we resort to the arguments of the previous case.So ideal theory will not help us study octavian factorization. A more

fruitful approach appears in the next section.

9.5. Factorization in O8 111

9.5 Factorization in O8

In this section, we show that a primitive octavian integer ρ of norm mn hasprecisely 240 left-hand divisors of norm m and 240 right-hand divisors ofnorm n, each set geometrically similar to the 240 units of O8.3 This resultis analogous to the result of Chapter 5 for the Hurwitz integers, except thatin O8, the factorizations are not unique “up to unit-migration” in view ofthe lack of associativity in O. More generally, we shall show that the setof left-hand divisors of a given octavian integer is geometrically similar tothe set of all octavian integers of a certain norm.It is remarkable that the argument does not use the full octavian form

of the division with small remainder property. It does, however, use thecorresponding property of E8 and so fails for our coarser octavian integersystems. The failure is more acute than that for the Lipschitz quaternions.We first present an intriguing “reversing” Euclidean algorithm due to

Rehm [37]. The forward stage (Figure 9.8) starts with a pair ρ1,m1, whereρ1 is an octavian whose norm is divisible by the rational integer m1 (butfor a technical reason we conjugate the remainder). We take [ρ1] = m0m1

and determine successive numbers mi and octavians γi, ρi by:

ρ1 = γ1m1 + ρ2 [ρ1] = m0m1

ρ2 = γ2m2 + ρ3 [ρ2] = m1m2 m1 > m2

......

...ρN−1 = γN−1mN−1 + ρN [ρN−1] = mN−2mN−1 mN−2 > mN−1ρN = γNmN [ρN ] = mN−1mN mN−1 > mN > 0

Figure 9.8. The forward stage of Rehm’s algorithm.

We briefly explain. Why is [ρ2] divisible by m1 (so that m2 = [ρ2]/m1

is an integer)? Because m1 divides every term on the right-hand side of

[ρ2] = [ρ1 − γ1m1] = [ρ1] +m21[γ1]−m1(2[ρ1, γ1]).

Why is m2 < m1? Because m1m2 = [ρ2] ≤ m21

2 .If m2 > 0, we can repeat the arguments above with ρ2 and m2, lead-

ing to ρ3 and m3 < m2, and so on. Since the mi are decreasing afterm0, at some point we must reach an mN+1 = 0, and we then obtain afinite collection of elements in O8 whose relationships are summarized inFigure 9.8.

3The first results along these lines are due to Rankin and Lamont. The geometricalassertion appears to be new.

112 9

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We now enter the reverse stage, displayed in Figure 9.9, which sets up1-to-1 correspondences

µN ↔ µN−1 ↔ · · ·↔ µ1 ↔ µ0,

that are in fact similarities, between certain divisors of the ρi. For let µNbe any element of norm mN . Since O is diassociative, we may write

ρN = γNmN = γN (µNµN ) = (γNµN )µN .

Set µN−1 = γNµN , so that µN−1 is a left-hand divisor of ρN of normmN−1. Then µN−1 is a right-hand divisor of both ρN and mN−1, and thusalso of ρN−1, since

ρN−1 = γN−1mN−1 + ρN = (γN−1µN−1)µN−1 + µNµN−1= (γN−1µN−1 + µN )µN−1.

Setting µN−2 = γN−1µN−1 + µN , we obtain a left-hand divisor of ρN−1 ofnorm mN−2. We can continue this procedure until we arrive at a left-handdivisor µ0 of ρ = ρ1 of norm m = m0 and a corresponding right-handdivisor µ1 of norm n = m1. Figure 9.9 summarizes this process, which canbe thought of as tracing the left-hand column of Figure 9.8 from bottomto top, factoring along the way.

[µN ] = mN

ρN = µN−1µN µN−1 = γNµN [µN−1] = mN−1ρN−1 = µN−2µN−1 µN−2 = γN−1µN−1 + µN [µN−2] = mN−2ρN−2 = µN−3µN−2 µN−3 = γN−2µN−2 + µN−1 [µN−3] = mN−3

......

...ρ2 = µ1µ2 µ1 = γ2µ2 + µ3 [µ1] = m1

ρ1 = µ0µ1 µ0 = γ1µ1 + µ2 [µ0] = m0

Figure 9.9. The reverse stage of Rehm’s algorithm.

We remark that the multiplication of the algebra is not used in anessential way in Figure 9.8. Moreover, products involving triples of elementsin Figure 9.9 occur only within associative subalgebras.

9.5.1 The Structure of the Divisor Sets

We are now able to describe the sets of all possible left-hand and right-handdivisors of ρ = ρ1 of norms m = m0 and n = m1. We shall see that the sizeof these is the number of octavian integers µN whose norm is the integermN in the last line of Figure 9.8. For the geometrical configurations ofthese sets, we use the following:

9.5. Factorization in O8 113

Lemma 8. Let the octavian integer γ be factored in two ways as γ =αβ = α β , where [α] = [α ] = 0 and [β] = [β ] = 0. Then the angle θabetween α and α is equal to the angle θb between β and β .

Proof. Taking the inner product of γ with αβ , we obtain

[α][β,β ] = [αβ,αβ ] = [γ,αβ ] = [α β ,αβ ] = [α ,α][β ],

which yields

cos θa =[α,α ]

[α]=[β,β ]

[β ]= cos θb.

Let µN, µN−1, . . . , µ0 denote the sets of all the µk that can ap-pear in the reverse stage (Figure 9.9). We have shown that in fact µN isthe set of all octavian integers of norm mN . It is obviously similar to itsconjugate set µN, but we could also deduce this by setting γ = mN inLemma 8. We indicate this relationship by

µN mN∼ µN.In a similar way, we find the similarities

µN ρN∼ µN−1 mN−1∼ µN−1 ρN−1∼ · · · m1∼ µ1 ρ1∼ µ0by alternately setting γ = ρi and mi for appropriate i. But µ0 and µ1are precisely the set of left-hand and right-hand divisors of ρ1 of norm n.This leads to the

Theorem 6. Let ρ = ρ1 be an octavian integer of norm mn, where m, nare positive rational integers. Let d be the greatest common rational integerdivisor of ρ, m, n. Then the set of left-hand divisors (µ0) of ρ of normm and the set of right-hand divisors (µ1) of ρ of norm n are geometricallysimilar to the set of all octavian integers (µN ) of norm d(= mN ).

Proof. We have done everything except identify the norm, so we haveonly to determine mN . Let gcd(η1, . . . , ηk) denote the greatest commonrational integer divisor of η1, . . . , ηk in O

8. Then:

Lemma 9. Let di = gcd(ρi,mi−1,mi) for 1 ≤ i ≤ N . Then di = di+1for 1 ≤ i ≤ N − 1.

Proof. First, see that di divides ρi+1 = ρi − γimi since di divides ρiand mi, so di divides ρi+1. It divides mi by definition. Finally, di dividesmi+1 since di divides each term in the last line of

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mi+1 =[ρi+1]mi

= ( 1mi)([ρi] +m

2i [γi]−mi(2[γi, ρi]))

= mi−1 +mi[γi]− 2[γi, ρi].Therefore, di divides di+1. By a similar argument, di+1 divides di as well,so di = di+1.Our theorem now follows, since

dN = gcd(ρN ,mN−1,mN ) = mN ,

and therefore

mN = d1 = gcd(ρ1,m0,m1) = gcd(ρ,m, n) = d.

One could use this theorem to count factorizations into more than twofactors, of arbitrary norms and “imprimitivity status.” The following sec-tions discuss only factorizations into primes.

9.6 The Number of Prime Factorizations

In Chapter 5, we counted the prime factorizations of a quaternion modelledon the factorizations of its norm. For instance, up to unit-migration thereare just 12 factorizations of a primitive Hurwitzian of norm 60 into primes,modelled on the 12 factorizations

2.2.3.5 2.2.5.3 2.3.2.5 2.5.2.3 2.3.5.2 2.5.3.23.2.2.5 5.2.2.3 3.2.5.2 5.2.3.2 3.5.2.2 5.3.2.2

of 60 into rational primes.For octonions, the failure of the associative law causes the situation

to change in two ways. On the one hand, there are now 5 × 12 = 60factorizations of 60 into ordinary primes:

60 = ((2.2)3)5 = (2(2.3))5 = (2.2)(3.5) = 2((2.3)5) = 2(2(3.5))= ((2.2)5)3 = (2(2.5))3 = · · · .

But also, the factorizations of Q modelled on a given one of its norm areno longer related by unit-migration, since αu.u−1β need not equal αβ.However, we call a change that affects just two adjacent factorsmetami-

gration, since it “imitates” unit-migration. Note that the number of fac-torizations βu.u−1γ obtained by unit-migration from βγ is the number ofunits, and the set of left-hand factors βu is geometrically similar to the setof units u. Our 2-term factorization theorem easily implies:

9.6. The Number of Prime Factorizations 115

Theorem 7. The number of factorizations of a primitive octavian, sayQ = ((P1P2)(P3(P4...Pk))), modelled on a given factorization of its normis 240k−1. Moreover, if all but Pi and Pj are fixed, then the sets of valuesfor Pi and Pj are geometrically similar to the set of 240 units.

What if Q need not be primitive?

Theorem 8. An octavian of norm pn11 . . . pnkk that is exactly divisibleby ps11 . . . p

skk has exactly

240n−1 Cni,si(p3i )

prime factorizations on any given model, where the “truncated Catalanpolynomials” Cn,s(x) are as defined in Chapter 5, and n = n1 + · · ·+ nk.

Proof. As in the quaternion case, it suffices to consider the cases thatinvolve only one rational prime p. For factorizations of the form

P1(P2(. . . (Pn−1Pn) . . .)),

the argument is exactly as in Chapter 5 except that we cannot work upto unit-migration and so must count the factors of 240 as they arise, andp must be replaced by p3 since there are 240(p3 + 1) octavians of norm p.(The prime 2 is no longer special.)A similar argument applies for factorizations such as

P1((P2P3)P4)

that can be built up by repeatedly adjoining single prime factors. To dealwith factorizations not of this form, such as

(P1P2)(P3P4),

one can use a “metassociation lemma,” according to which there will beequal numbers of factorizations of the forms (AB)C and A (B C ) providedthat the octavians in each of the pairs (A,A ), (B,B ), (C,C ) have thesame norm and are divisible by the same rational integers. We suppressthe details.Letting the model vary, we see that the total number of prime factor-

izations will be 240n−1 times the product of:

• the numbers Cn1,s1(p31), . . . , Cnk,sk(p3k) that count factorizations ona given model “up to metamigration,”

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• the multinomial coefficient nn1, . . . , nk

that counts possible or-

derings for the primes in the model, and finally,

• the Catalan number Cn−1 that counts the possible arrangements ofparentheses.

(For the corresponding result for quaternions, replace 240 by 24, p3i by pi,and omit Cn−1 in virtue of their associativity.)We can hardly call octavian prime factorization “unique” before we have

solved the metamigration, metacommutation, and metassociation problemsbelow. However, Rankin and Lamont remark that prime factorization ofan odd norm primitive octavian can be made unique by demanding thatthe factors be congruent (mod 2) to specified units.

9.7 “Meta-Problems” for Octavian Factorization

Recall that we reduced unique factorization for the Hurwitzians to themetacommutation problem: What is the relation between factoriza-tions PQ and Q P modelled on pq and qp? For octavian factorization,we also have the metassociation problem: What is the relation between(PQ)R and P (Q R ) modelled on (pq)r and p(qr)? Moreover, there is evena metamigration problem that asks for the relation between factoriza-tions PQ and P Q modelled on the same factorization pq, since they areno longer related by unit migration.There is also the problem of understanding factorizations in the

Gravesian case. The assertion that corresponds to Theorem 6 is due toC. Feaux [16]:

Theorem 9. If p, q are distinct primes, then a Gravesian integer ofnorm pq has exactly 16 factorizations

P1Q1 = (−P1)(−Q1) = . . . = P8Q8 = (−P8)(−Q8)

modelled on pq. Moreover, the left-hand factors are geometrically similarto the Gravesian units (i.e., P1, . . . , P8 are mutually orthogonal) as are theright-hand ones.

The metamigration problem for Gravesian integers asks how these 16factors can be recovered from one of them. We can partially solve it whenthe integer belongs to a Lipschitzian subring:

9.7. “Meta-Problems” for Octavian Factorization 117

Theorem 10. If L = PQ and L = Q P are factorizations of a Lip-schitzian L = a+ bi1+ ci2+di4 modelled on pq and qp, then the Gravesianfactorizations of L modelled on pq are

PU.UQ and P V.V Q ,

where U runs over ±1,±i1,±i2,±i4 and V over ±i0,±i3,±i5,±i6.

Proof. This follows immediately from the Composition Doublingformula

(a+ i0b)(c+ i0d) = (ac− db) + i0(cb + ad).

So in this case, the Gravesian metamigration problem reduces to theLipschitzian metacommutation problem.

10

Automorphisms andSubrings of OIn this chapter, we shall study the octavian units in some detail, and thenuse them to find the automorphism group of O = O8 and to study some ofits subrings.

10.1 The 240 Octavian Units

The following little table gives the quaternion triplets abc (for which ia, ib,ic behave like i, j, k) in its first column, which can be supplemented by ∞or 0 to give seven “quads” that are halving-sets for the octavians, as aretheir complements.

triplet

124 0 ∞365235 0 ∞461346 0 ∞512450 ∞ 6123561 0 ∞234602 ∞ 1345013 ∞quad

2456

co-quad

The octavian units are those octavian integers whose inverses are alsooctavian integers. How many of them are there?

119

120 10 Automorphisms and Subrings of O

Theorem 1. There are just 240 octavian units.

Proof. The units can only be of the two forms

±1, ±i0, . . . , ±i6 and1

2(±ia ± ib ± ic ± id),

where abcd is one of the 14 4-element 0-sets. Since there are 16 choices ofsign on the right, the total number is 16 + 14 · 16 = 240.It is almost as easy to classify these by their multiplicative orders. We

have

1 of each order 1 or 2, namely +1 and −1;56 of each order 3 or 6, namely ±1

2 (−1± ia ± ib ± ic),where abc is one of 356 146 125 450 234 602 013;

and 126 of order 4, namely ±in and 12 (±id ± ie ± if ± ig),

where defg is one of 0124 0235 0346 1236 0561 1345 2456.

We shall call the elements of multiplicative orders 3 and 4 “ω”s and“i”s, respectively, and use the notations

ωabc =1

2(−1 + ia + ib + ic) and idefg =

1

2(id + ie + if + ig)

for them, putting a bar over a subscript to indicate negation of a compo-nent, for example

ωabc =1

2(−1 + ia + ib − ic) and idefg =

1

2(id − ie − if + ig).

When we ask what collections of these generate, it’s natural to pairinverses with each other. Taking this point of view, what we are studyingis a beast that has 63 pairs of eyes (“i”s) and 28 pairs of arms (“oms” =“ω”s)!

10.2 Two Kinds of Orthogonality

In the next few sections, we shall study the automorphism group Aut(O) ofthe ring O of octavian integers. In a way, this is a discrete analogue of whatwe did in Chapter 6, where we found the automorphism group of the realoctaves to be the 14-dimensional Lie group G2(R), which we abbreviateto G2. We shall find that of the octavian ring to be the analogous groupG2(F2) over the field F2 of two elements, usually abbreviated by grouptheorists to G2(2).

10.3. The Automorphism Group of O 121

Our discrete theory usually follows the continuous one, but with minordifferences. For instance, in the continuous case we found that any unitorthogonal to 1 was like any other, and any orthogonal pair of such unitswas like any other such pair. The first of these remains true in the discretecase, but the second does not. Thus the 126 units orthogonal to 1–the“i”s–are all equivalent under the group, but the pairs i, j of orthogonal“i”s are of two kinds. We call i, j evenly orthogonal if 1+i+j+ij2 is also anoctavian integer, and oddly orthogonal if not. In the same circumstances,we call i, j, ij an even or odd quaternion triplet.

We shall use iabcd for an octavian integer of type12 (±ia ± ib ± ic ± id)

with an odd number of minus signs, iabcd for one with an even number,and i∗abcd for either kind.

Lemma 1. i0 is orthogonal to just 60 other “i”s, of which it is evenlyorthogonal to 12, namely ±i1,±i2, . . . ,±i6, and oddly orthogonal to 48,namely i∗1236, i∗2465, i∗4153.

Proof. Since each i other than ±in has the form i∗abcd, and since 1236,2465, 4153 are the only 0-sets that contain neither∞ nor 0, there are indeedjust 60 possibilities. As regards the type of orthogonality, the subscript-doubling symmetry (i1i2i4)(i3i6i5) makes it sufficient to prove that j = i1or i3 is evenly orthogonal to i = i0 (which they are, since

1 + i+ j + ij

2=1 + i0 ± i1 ± i3

2)

is in O8) and j = i∗1236 oddly so (which they are, since

1 + i+ (j + ij)

2=1 + i0 + (±i1 or 3 ± i2 or 6)

2

is not in O8).

10.3 The Automorphism Group of O

By joining each “i” to the six other “i”s that are evenly orthogonal to it,we obtain a graph we call the hyperhexagon.1 Figure 10.1 shows the partof this graph near i0, and Figure 10.2 shows the remainder.

1Usually called a generalized hexagon–see the end of this section.


Figure 10.1. The 15 triangles of the hyperhexagon graph near i0 (the “near side”).Each outer vertex here appears in 2 further triangles that are drawn as lines inFigure 10.2.

10.3. The Automorphism Group of O 123

_24

65

_

2465

_

24

65

_

24

65

_

24

65

_

2465

_

24

65_

2465

__

24

65__

2465

_

_24

65_

_

2465

_ _

24

65_

_

2465

24

65

2465

__

10

65

1065

_

_10

65_

_

1065

_

10

65 _

1065

_

10

65

_

1065

_

_10

24_

_

1024

__

10

24

1024

_10

24

_

1024

_

10

24_

1024

_

0436

_

0436

_

0436

_

0436

__

0436

_

_

0436

_ _

0436

0436

_

02

53

_ 02

53_

02

53

_ 02

53

0253

_

_ 0253

__

02

53_

_ 02

53

__

41

53

_

_ 4153

__

41

53

_

_ 4153

_

4153

_

4153

_

4153 _

41

53

_ _

4153 _

_ 41

53

4153

4153

_ 4153

_

41

53

_ 4153

_

41

53

_

_

1236

__

1236

__

1236

_

_

1236

1236

1236

_ _

1236

_ _

1236

_

1236

_

1236

_

1236

_

1236

_

1236

_

1236

_

1236

_

1236

1 1

33

2

2

6

6

4

4

5

5

Figure 10.2. The 48 triangles of the “far side” of the hyperhexagon graph.


We use this graph to deduce

Theorem 2. Aut(O) is transitive on “i”s.

Proof. For any ω of order 3, the map x→ ωxω is an automorphism ofoctonion multiplication by Theorem 14 of Chapter 8, and it takes O to Oif ω is in O. So for example, conjugation by ω = 1

2 (−1 + i0 + i1 + i3) is anautomorphism of O that takes i0 to i1 to i3.In an exactly similar way, we can obtain an automorphism permuting

the vertices of any triangle in the hyperhexagon, and since the graph isconnected, these establish the transitivity.What can we say about the images i†0, i

†1, i

†2 of i0, i1, i2 under the

general automorphism of O? Since i0 is evenly orthogonal to i1 and i2,which are oddly orthogonal to each other, i†0, i

†1, i

†2 must have the same

properties. On the other hand,

Theorem 3. If i†0, i†1, i

†2 are three mutually orthogonal “i”s, in which

just the orthogonalities involving i†0 are even, then there is a unique auto-morphism of O taking i0 → i†0, i1 → i†1, i2 → i†2.

Proof. The uniqueness is immediate, since when we have the imagesof i0, i1, i2 we know those of i4 = i1i2, i3 = i0i1, i6 = i0i2, i5 = i0i4. Forinstance, we shall determine the automorphism for which i†0 = i0, i

†1 = i2,

i†2 = i1. We find

i†4 = i†1i†2 = i2i1 = −i4

i†3 = i†0i†1 = i0i2 = i6

i†6 = i†0i†2 = i0i1 = i3

i†5 = i†0i†4 = i0(−i4) = −i5,

giving the map (i0)(i1i2)(i3i6)(i4−i4)(i5−i5), which is easily checked tobe an automorphism. In a similar way we find the particular symmetries

(i0)(i1 i2 i4)(i3 i6 i5)(i0)(i2)(i6)(i1 i3 −i1 −i3)(i5 i4 −i5 −i4)(i0)(i1)(i3)(i2 i4 −i2 −i4)(i6 i5 −i6 −i5)(i0)(i1)(i3)(i2 i5 −i2 −i5)(i4 i6 −i4 −i6).

We show in general that the automorphisms of the theorem exist asfollows. First, the transitivity allows us to reduce to the case i†0 = i0. Theni†1 and i

†2 must be among the ±in, since these are the only “i”s that are

evenly orthogonal to i0. The symmetries above now enable us to reduce i†1

10.4. The Octavian Unit Rings 125

to i1 (while fixing i0) and i†2 to i2 (fixing i0 and i1), and so establish the

theorem.

There is an immediate corollary:

Corollary. Aut(O) has order exactly 12096.

Proof. i†0 can be any one of the 126 “i”s. Then if i†0 = i0, i

†1 can be

any of ±i1, . . . ,±i6, the 12 “i”s evenly orthogonal to i0, and if also i†1 = i1,i†2 can be any of ±i2,±i4,±i6,±i5, the 8 “i”s orthogonal evenly to i0 butoddly to i1. So the order is

126× 12× 8 = 12096.

Theorem 4. Aut(O) is also transitive on (1) even quaternion triplets,(2) odd quaternion triplets, and (3) “ω”s.

Proof. (1) If i→ i0 and j → i1, then k→ i3. (2) If i→ i1 and j → i2,then k → i4.

(3) It will suffice to find for any ω = 12 (−1 ± ia ± ib ± ic) an even

quaternion triplet i, j, k for which ω = 12 (−1 + i + j + k); and i = ±ia,

j = iω, k = jω will do.2

Usually, the graph we illustrated in Figures 10.1 and 10.2 is called ageneralized hexagon, but we prefer the name “hyperhexagon.” An ordi-nary polygon is a connected set of vertices and edges in which each objectof either kind is incident with just two of the other. It is called an n-gonif the circuit formed by alternating vertices and incident edges containsn of each. We obtain the notion of a generalized n-gon by replacing thenumber 2 in the above by a larger number, in our case 3. The vertices ofour hyperhexagon are “i”s counted up to sign. Its “hyperedges” are evenquaternion triples i, j, k of such “i”s (again up to sign).

10.4 The Octavian Unit Rings

An octavian unit ring is a subring of O8 generated by units. We shallshow in the appendix that

2We cannot always take i, j, k to be±ia,±ib,±ic, since this might not be a quaterniontriplet. For ω = ω365 the argument of the text leads to i = i3, j = i2465, k = i2465.


G1(1)

G2(63)

G4−(252)

G8(63)

G4+

(63)

E2

(28)E4

(336)E8

(112)

H4

(63)H8

(63)

O8

(1)

Figure 10.3. The unit rings. Here, G1 is the ordinary integer Z; E2 is an Eisensteinring Z[ω] generated by an element ω of order 3 in O8; H4 is a Hurwitzian ringZ[i, j, k, 1

2(−1 + i + j + k)]; and finally, O8 is the full octavian ring. An arrow

Rn → R2n indicates that R2n is a Dickson double Rn, i for some i orthogonal toR. The number of each type of ring is given in parentheses.

Theorem 5. Up to isomorphism, there are precisely four types (G1,E2, H4, O8) of integer rings generated by odd-order elements, from whichall of the octavian unit rings can be obtained by Dickson doubling (seeFigure 10.3).

It turns out that, with one important exception, if two unit rings areabstractly isomorphic, they are related by a symmetry of O8. The exceptionis that there are two orbits of abstract type G4 = i, j, k : G4+ in whichi, j, k are evenly orthogonal, and G4− in which i, j, k are oddly so.We briefly discuss the transitivities in the nontrivial cases, paying par-

ticular attention to those that correspond to maximal subgroups. Thetransitivity on rings of type G2 and G8 follows since these are in 1-1 cor-respondence with each other3 and with the vertices of the hyperhexagon.That on those of types G4+, H4, H8 follows since these are also in 1-1correspondence with each other4 and with the “hyperedges” of the hyper-hexagon.

3The G8 corresponding to the G2 generated by i is generated by the “i”s evenlyorthogonal to that i.

4The H4 corresponding to G4+ = i, j, k is i, j, k, 12(−1 + i+ j + k) , and the H8 is

generated by this and all “i”s orthogonal to it.

10.4. The Octavian Unit Rings 127

O8±120 1

G8±8 63

G4+±4 63

G1±1 1

E8±12 112

G4−±4 252

G2±2 63

H8±24 63

H4±12 63

E2±3 28

E4±6 336 E4±6 336

K8±8 63

C1+1 1

C3+3 28

1

2863

1

1

1

4

9

9

4

3

12

1

112 63

1

3

1

3

16

16

3

1

12

33

11

14

33

3

3

Figure 10.4. Inclusions between certain subthings of O8. ±n precedes a ring thathas n pairs of opposite units. +n precedes a group consisting of n units. n followsone of n such things. Numbers against lines indicate numbers of containments –thus each E4 lies in 16 H8s, while each H8 contains 3 E4s. For simplicity, E4 appearson both sides.

The transitivity on rings G4− follows from transitivity on odd quater-nion triples. However, all automorphisms that fix i1, i2, i4 fix i0 also,

5

which with them generates a G8, so this is not a maximal subgroup. A ringG8 contains four rings G4− in this way. For example, this one contains

i1, i2, i4 , i1, i5, i6 , i2, i3, i5 , i4, i6, i3 .

5±i0 are the only “i”s evenly orthogonal to i1, i2, i4.


The transitivity on rings E2 = Z[ω] follows from transitivity on “ω”s.Since a ring of type E4 or E8 contains a unique one of type E2 (generatedby its odd-order elements), its stabilizer is not a maximal subgroup–weleave the transitivity to the interested reader.6

Figure 10.4 indicates all inclusions between these unit rings (and a fewother things). More detailed discussions of them will be found in the nextsection. With just two exceptions, every loop generated by units is theloop of units of one of these unit rings. The exceptions are the trivialgroup C1 and the group C3 of order 3, generated by an ω. Their inclusionmakes Figure 10.4 complete also for unit-loop inclusions. In Chapter 11 weshall obtain four non-unit subrings K1, K2, K4, K8 by considering octaviansmodulo 2, and we have included K8 in our figure.

10.5 Stabilizing the Unit Subrings

The following sections provide more detailed information. The stabilizersof the general quaternionic subsystem Q4 spanned by the quaternion tripleti, j, k) can be described uniformly by regarding Q8 as the Dickson doubleQ4+hQ4 (in which h is the “doubler”) and using the ideas of section 6.6 andthe notation of Chapter 4. We don’t need to discuss G4+ since it obviouslyhas the same stabilizer as the unique H4 that contains it. In the remainingcases, Q8 is generated by Q4 and units orthogonal to Q4, so Stab(Q4) iscontained in Stab(Q8).

10.5.1 Subring G4−

An example is generated by i = i1, j = i2, k = i4, for which a suitabledoubler is h = i0. The following table (in which only the subscripts areshown) shows the actions of a generating set of automorphisms on these.

i j k h hi hj hk[u, v] 1 2 4 0 3 6 5[ω,ω] 2 4 1 0 6 5 3[ζ, ζ] 2 1 4 0 6 3 5[i, i] 1 2 4 0 3 6 5[1,−1] 1 2 4 0 3 6 5

ω = −1+i+j+k2

ζ = i+j√2

6The 36 i s orthogonal to ω fall into twelve triples. Adjoining any one such tripleyields an E4, while an E8 is obtained by adjoining a suitable three of them.

10.5. Stabilizing the Unit Subrings 129

These must generate, since the first three yield all automorphisms of i1, i2, i4 ,any further generator can be supposed to fix i1, i2, i4, and must thereforetake i0 to ±i0, since these are the only “i”s evenly orthogonal to each ofi1, i2, i4.In the notation of Chapter 4, the structure of Stab(G4−) is

± 1

24[O ×O],

since −1 = [1,−1] is present, while [ω,ω], [ζ, ζ], [i, i] define the identity iso-morphism between L and R, which are both copies of the binary octahedralgroup 2O = ω, ζ, i .Any automorphism that fixes a G8, e.g., in , must fix the G

2 (here i0 )generated by its middle element, and the converse is true since the “i”sevenly orthogonal to ±i0 are the other ±in. Any G4− = i, j, k extends toa unique G8 = h, i, j, k by adjoining the middle element h of i, j, k. Thecase G4− = i1, i2, i4 yields G8 = in , which, conversely, contains just fourG4−’s:

i1, i2, i4 i1, i5, i6 i2, i3, i5 i4, i6, i3 .

So Stab(G4−) has index 4 in Stab(G8) = Stab(G2 = i0 ).

10.5.2 Subring H4

In this case, exemplified by i = i0, j = i1, k = i3, we take h = i6 and findthe following automorphisms:

i j k h hi hj hk[u, v] 0 1 3 6 2 5 4[ω, 1] 1 3 0 ← 6254∗→[ζ, ζ] 0 3 1 6 2 4 5[k, k] 0 1 3 6 2 5 4[j, j] 0 1 3 6 2 5 4[1,−1] 0 1 3 6 2 5 4[1, i] 0 1 3 2 6 4 5[1, j] 0 1 3 5 4 6 2[i, k] 0 1 3 4 5 2 6

ω = −1+i+j+k2

ζ = j+k√2

They generate a group of structure

±12[O ×D8]


since −1 = [1,−1] is present, while i, j, k, ζ generate a 2D8 that extendsby ω to 2O, and the “right kernel” is non-cyclic. It is the entire groupsince it achieves all automorphisms of H4 and is easily checked to containall H4-stabilizers.Adjoining orthogonal units, we obtain an H8 which has the same stabi-

lizer since H4 can be recovered from it as the subring gererated by elementsof order 3.

10.5.3 Subring E4

Up to sign, the units of an E4 are

1, ω, ω, i, i = ωi, i = ωi,

while the orthogonal units are

j, j = ωj, j = ωj, k, k = ωk, k = ωk,

i, j, k being an odd quaternion triplet and ω = 12 (−1+h

√−3). We displayall of these in a rectangular array

1 ω ωi i ij j jk k k

We find four automorphisms that act (up to sign) as follows:

[ω, 1]· · ·

[1,ω]· · ·· · ·

[i, i]· · ····

[h, hi]· · ·· · ·-

--

-¾

---

¾¾¾ ? ? ?6 6 6

and which generate a group7 of structure

+1

4[D12 ×D12]

in the notation of Chapter 4 (because [1,−1] is absent, and the us and vsboth generate D12’s that map to D4 modulo ω whose “cyclic coset” his interchanged with a noncyclic one hi ). That this is the whole stabilizer

7It is interesting to note that this group does not appear in [41], since its ellipticimage was omitted from Goursat’s original enumeration.

Appendix: Proof of Theorem 5 131

follows since the first map is easily checked to generate the stabilizer of E4

while the other three generate all achievable automorphisms of E4.Adjoining units orthogonal to E4, we obtain an E8 whose stabilizer is

three times as large, as we see from the example

∞ ∞365 ∞3651 0165 01652 0253 02534 0435 0436

, where this automorphism is

(124)(365)

· · ·.

? ? ?

Appendix: Proof of Theorem 5

We prove the

Theorem 5. Up to isomorphism, there are precisely four types (G1,E2, H4, O8) of integer rings generated by odd-order elements, from whichall of the octavian unit rings can be obtained by Dickson-Doubling (seeFigure 10.3).

Lemma 2. Any unit ring can be obtained by Dickson doubling fromone that is generated by ωs.

Proof. Let X be the subring generated by ωs. Then obviously any unitnot in X must be an i. But also this i must be orthogonal to every unit u inX (and so to X), since otherwise ±ui would be an ω not in X. It thereforeextends X to its Dickson double, say Y. Replacing X by Y, we can repeatthe argument, if necessary, to prove the lemma.This reduces the theorem to

Theorem 6. There are just four types of rings generated by ωs, namelyG1, E2, H4, O8.

Proof. If the ring has at most one generator, it is trivially G1 or E2.Since Aut(O8) is doubly-transitive on ω s, the ring generated by any two“ω”s (that aren’t equal or reciprocal) is again of a unique type, H4. Theproof is completed by the easy verification that adjoining any ω outside itto the particular H4 = i0, i1, i3,ω013 yields all of O8.

11

11

11

11

Reading O Mod 211.1 Why Read Mod 2?

The group Aut(O) of automorphisms of O = O8 is nearly simple–infact, it has a simple subgroup of index 2. The finite simple groups havebeen classified–which one is this? The answer is best found by readingmodulo 2.One can obtain an interesting ring by reading the octavian integers

modulo any prime p (just as when reading the ordinary integers modulo pone obtains an interesting ring that is, in fact, a field). To be precise, onesays that two octavian integers are congruent mod p provided that theirdifference is p times an octavian integer. (So, an “octavian integer mod p”is really a member of the quotient ring O/pO.)The finite group G2(p) (more explicitly written G2(Fp)) is defined to

be the automorphism group of the octonions mod p. Since this topic isperipheral to our book, we shall merely quote the fact that the order ofG2(p) is p

6(p2 − 1)(p6 − 1). Plainly, any automorphism of O can be readmodulo p to yield an element of G2(p). This gives a map

Aut(O)→ G2(p)

that is, in fact, an embedding, since only the identity automorphism of Omaps to the identity automorphism (mod p). Indeed, for p = 2 it is anisomorphism, since by the above formula, G2(2) has the same order

26(22 − 1)(26 − 1) = 12096as Aut(O).

133

134 11

11

11

11 Reading O Mod 2

So our group is G2(2). Although G2(p) is usually simple, the particularcase G2(2) is not—it has a simple subgroup of index 2 that happens toappear elsewhere in the classification–it is the group U3(3) of “unitary”3 × 3 matrices with entries in the field of order 9. We shall use G for thissimple subgroup, so that G2(2) is G · 2.Group theorists know that the best way to study a finite simple group

is to determine its maximal subgroups. The group G has (up to conjugacy)four maximal subgroups:

GG, stabilizing one of the 63 Gaussian subrings G2GH, stabilizing one of the 63 Hurwitzian subrings H4GE, stabilizing one of the 28 Eisenstein subrings E2GK, stabilizing one of the 36 Kleinian subrings K1

The group G · 2 has, in addition to the four groups GG · 2, GH · 2, GE · 2,GK · 2 just one further maximal subgroup, namely G itself. Only threeof these have appeared as stabilizers of subrings of the octavianintegers, but all four stabilize low-dimensional subrings of the octaviansmodulo 2.

To get the first three, we can read i0 , i0, i1, i3,ω013 , ω modulo 2.The fourth fixes the ring generated (modulo 2) by a Kleinian integer λ ofthe form 1

2 (−1 +√−7). It turns out that it also fixes two bases: the “j-

frame” 1, j0, . . . , j6 and the “k-frame” 1, k0, . . . , k6. Table 11.1 shows theaction of all four groups on the subalgebras of all four types.

Action ofGG GH GE GK

on the 63G2s

1 + 6 + 24 +32G2 G4+ G4− E4

3 +12+48G4+ G8 H8

27+36H4 E4

14+21+28K2 K4 E4

on the 63H4s

3 +12+48G4+ G8 H8

1 + 6 +24+32G4+ G8 H8 O8

9 +54H4 H8

21+42K4 H8

on the 28E2s

12+16H4 E4

4 +24H4 H8

1 +27E2 H4

28E4

on the 36K1s (mod 2)

8 +12+16K2 K4 E4

12+24K4 H8

36E4

1 +14+21K1 K2 K4

Table 11.1. Under the group that fixes one type of algebra, the algebras of anygiven type form orbits as shown. The two given algebras generate a third whosetype is indicated below the appropriate number.

11.2. The E8 Lattice, Mod 2 135

11.2 The E8 Lattice, Mod 2

When we study the lattice of octavian integers mod 2, it will be importantto count those of various norms. As described in Chapter 9, the octavianintegers form a scaled copy of the E8 root lattice, and it is known that thenumber of vectors of any norm n > 0 is 240 times the sum of the cubes ofthe divisors of n.

So there are exactly 240 2160 6720 . . .octavian integers of norm 1 2 3 . . . .

However, all we actually need are the numbers of short vectors, bywhich we mean those of norm at most 2. To make our book self-contained,we count these explicitly from their coordinate shapes:

norm 1 shapes: (±11, 07) and (± 12

4, 04)

number: 21.8 + 24.14 = 240

norm 2 shapes: (±12, 07) and (± 12

4,±11, 03) and (± 1

2

8)

number: 22.28 + 25.14.4 + 28 = 2160.

Now we consider the congruences modulo 2 between these short vectors.

Lemma 1. Apart from v ≡ −v, the only other congruences (mod 2)between short vectors are between orthogonal ones of norm 2.

Proof. Suppose v ≡ w, with v ≡ ±w. Then replacing w by −w ifnecessary, we can suppose the inner product [v, w] is non-negative, so forthe norms we have

[v − w] ≤ [v] + [w] ≤ 4

since v, w are short (see Figure 11.1). But v−w ∈ 2O8 implies [v−w] ≥ 4,so therefore [v − w] = 4, which implies that [v] = [w] = 2 with [v, w] = 0.

−w w

v

≤90[w] ≤ 2

[v] ≤ 2 [v − w] ≤ 4

Figure 11.1. If [v,w] ≥ 0, then [v −w] ≤ 4.

136 11

11

11

11 Reading O Mod 2

Since there can be at most eight mutually orthogonal vectors in an 8-dimensional space, it is a corollary that there can be at most sixteen norm2-vectors in any equivalence class. So, the number of equivalence classesthat contain short vectors is at least

1 +240

2+2160

16= 1 + 120 + 135 = 256 = 28.

But since O8/2O8 is an 8-dimensional space over the field of order 2, thisis the total number of classes, and we deduce

Theorem 1. Every mod 2 congruence class contains a short vector.These classes are: one class represented by the 0 vector; 120 classes eachrepresented by two vectors ±v of norm 1; and 135 classes each representedby a frame of sixteen vectors ±v1, . . . ,±v8 of norm 2. In the third case,v1, . . . , v8 are mutually orthogonal, and every norm 2 vector lies in such aframe.

From now on, we shall call these classes “vectors (mod 2).” Groupedaccording to their multiplicative structure they are

Norm Type Total0 1 element 0 11 1 element ± 1, 63 elements ± i, 56 elements ± ω 1202 63 elements 1 + i, 72 elements λ 135

256 = 28

The 16 minimal representatives of a vector of type 1+ i are four of theform ±1± i and 12 of the form ±i ± i . As an example, for 1+ i0 they are

±1± i0, ±i1 ± i3, ±i2 ± i6, ±i4 ± i5.Type λ is one we have not previously seen. Its 16 minimal representa-

tives are all of the form 12 (±1±

√−7) for various octavian square roots of−7. G2(2) is transitive on the 72 “λ”s. However, there are only 36 algebrasλ (mod 2), since the one generated by a given λ contains a second one,namely µ ≡ 1 + λ. The standard case is

λ = i∞0123456 =−1 + i0 + · · ·+ i6

2and

µ = i∞0123456 =1 + i0 + · · ·+ i6

2.

The 16 minimal representatives of µ are

±i∞0123456, ±i∞0123456, . . . (eight such).

11.2. The E8 Lattice, Mod 2 137

Every vector obtained by adding an ω to an orthogonal i is of type λ, andconversely, every number of type λ can be so expressed (in many ways).

The pair λ, µ determines and is determined by a pair of coordinateframes 1, j0, . . . , j6 and 1, k0, . . . , k6. The js and ks are found from thecongruences

jnµ ≡ µ ≡ µkm and λjn ≡ λ ≡ kmλ,

and conversely, µ is determined from them by the fact that the 28 nonzerovalues (mod 2) from the 49 products (1 + jm)(1 + kn) are congruent to µ.

The two associated frames are interchanged by an element δ that in-terchanges jm with k−m and increases the order of the group from 168 to336. The ks may be regarded as points, and the js lines, of a projectiveplane as in Figure 11.2. So the new group is the extension by duality ofthe automorphism group L3(2) of this plane, which can also be thought ofas PGL2(7). It is the fourth maximal subgroup of G · 2, and we thereforedevote the next section to it.

k5

j3

k0

k4

j2

k6 k3

j4 j1

j0k1 k2

j5 j6

Figure 11.2. Calling the ks points, the js become lines, of an order-2 projectiveplane, and vice versa.

138 11

11

11

11 Reading O Mod 2

11.3 What Fixes λ ?

By a (coordinate) frame, we mean 16 vectors of the form ±1,±v0,±v1, . . . ,±v6, where 1, v0, v1, . . . , v6 are mutually orthogonal units in O8. Just asthere are even and odd quaternion triplets, there are even and odd frames.An even frame is one that contains an even quaternion triplet; in an oddframe, all the quaternion triplets are odd.

Lemma 2. Any two oddly orthogonal “i”s belong to just three frames,of which one is even and the other two odd.

Proof. We can take the two “i”s to be i3 and i5. The only “i”sorthogonal to these are (in the notation of Chapter 10)

±i6, ±i0, ±i1, ±i2, ±i4, i∗0124,

and the only frames that can be selected from the vectors we have men-tioned are ±1,±i3,±i5,±i6 together with

±i0,±i1,±i2,±i4 or i0124 or i0124an i-frame a j-frame a k-frame.

Since i0, i1, i3 is an even triple, we see that the i-frame is even. Thecalculations that follow show that the j-frame is odd, as must be the k-frame, since j-frames and k-frames are interchanged by the symmetry thatnegates i0, i3, i5, i6.

11.3.1 The Calculations

The symmetries of the j-frame are easiest seen by expressing all of theoctonion units in it. We define

j3 = −i6, j6 = −j5, i5 = −i3, j0 = i0124, j1 = i0124, j2 = i0124, j4 = i0124.

This transformation is easily inverted: we have

i3 = −j5, i6 = −j3, i5 = −j6, i0 = j0124, i1 = j0124, i2 = j0124, i4 = j0124.

(It can be helpful to note that j0 + j1 = i2+ i4, and j0 − j1 = i1 − i0, etc.)

11.3. What Fixes λ ? 139

Then we easily find

j0j1 = j2346, j1j2 = j3450, . . . , j6j0 = j1235,

relations which are, remarkably, invariant under the maps

α : (01 . . . 6) and β : (124)(365).

Now, we can readily complete the j multiplication table, using the in-variance under β and the facts that j2n = −1 and jnjm = jmjn:

1 j0 j1 j2 j3 j4 j5 j6j0 −1 j2463 j4156 j4156 j1235 j2463 j1235j1 j2346 −1 j3504 j5260 j5260 j2346 j3504j2 j4615 j3450 −1 j4615 j6301 j6301 j3450j3 j4561 j5026 j4561 −1 j5026 j0412 j0412j4 j1523 j5602 j6130 j5602 −1 j6130 j1523j5 j2634 j2634 j6013 j0241 j6013 −1 j0241j6 j1352 j3045 j3045 j0124 j1352 j0124 −1

It turns out that any symmetry that preserves the j-frame also preservesan associated k-frame. Namely, we define

k0 = j0124 = −i0k1 = j1235 = i1263k2 = j2346 = i2456k3 = j3450 = i1263k4 = j4561 = i4135k5 = j5602 = i4135k6 = j6013 = i2456,

noticing that this is invariant under both α and β. Then we compute

k0k1 = −i0i1263 = i3621 = j0453 and j6k0 = i5i0 = i4 = j0124,

which together with the α and β symmetries, give the Tables 11.2 and 11.3.It turns out that the odd frames can be divided into two classes, the j-

frames and k-frames. An automorphism either fixes each class as a whole(when we call it “even”) or interchanges them (when we call it “odd”).The symmetries that fix a given j-frame form a copy of the simple group

of order 168 known as L3(2) or L2(7), and they also fix a particular k-frame,called the “associated k-frame.” For the standard pair of associated framesthis is generated by α, β, γ acting as follows:

α : (01 . . . 6)

β : (124)(365)on both j- and k-subscripts

γ :(56)(14) 0124

(12)(36) 0365

on j-subscriptson k-subscripts.

140 11

11

11

11 Reading O Mod 2

1 j0 j1 j2 j3 j4 j5 j6j0 −1 i52 i34 −i24 i61 −i12 −i41j1 −i52 −1 i63 i45 −i35 i02 −i23j2 −i34 −i63 −1 i04 i56 −i46 i13j3 i24 −i45 −i04 −1 i15 i60 −i50j4 −i61 i35 −i56 −i15 −1 i26 i01j5 i12 −i02 i46 −i60 −i26 −1 i30j6 i41 i23 −i13 i50 −i01 −i30 −1

1 k0 k1 k2 k3 k4 k5 k6k0 −1 i63 i56 −i61 i35 −i34 −i52k1 −i63 −1 i04 i60 −i02 i46 −i45k2 −i56 −i04 −1 i15 i01 −i13 i50k3 i61 −i60 −i15 −1 i26 i12 −i24k4 −i35 i02 −i01 −i26 −1 i30 i23k5 i34 −i46 i13 −i12 −i30 −1 i41k6 i52 i45 −i50 i24 −i23 −i41 −11 k0 k1 k2 k3 k4 k5 k6j0 ω00 i01 i02 −ω03 i04 −ω05 −ω06j1 −ω10 ω11 i12 i13 −ω14 i15 −ω16j2 −ω20 −ω21 ω22 i23 i24 −ω25 i26j3 i30 −ω31 −ω32 ω33 i34 i35 −ω36j4 −ω40 i41 −ω42 −ω43 ω44 i45 i46j5 i50 −ω51 i52 −ω53 −ω54 ω55 i56j6 i60 i61 −ω62 i63 −ω64 −ω65 ω66

Table 11.2. Multiplication tables for the js and ks. For definitions, see Table 11.3.

The action of the group GK ∼= L3(2) ·2 on various things is easy to see inTable 11.3, which displays the 240 units with respect to the j-frame (usingjabcd =

12 (ja + jb + jc + jd), etc.).

11.4 The Remaining Subrings Modulo 2

This section is devoted to a proof that the only subrings modulo 2 of O8

are the unit-rings, taken modulo 2, and four others:

K1 = λ K2 = λ, i0 K4 = λ, i0, i1, i3 K8 = λ, i0, . . . , i6 ,

where λ = −1+i0+...+i62 .

11.4. The Remaining Subrings Modulo 2 141

abcd jabcd jabcd jabcd jabcd jabcd jabcd jabcd jabcd0124 −k0 i50 i30 i60 h00 −h10 −h21 −h401235 −k1 i61 i41 i01 h11 −h21 −h32 −h512346 −k2 i02 i52 i12 h22 −h32 −h43 −h623450 −k3 i13 i63 i23 h33 −h43 −h54 −h034561 −k4 i24 i04 i34 h44 −h54 −h65 −h145602 −k5 i35 i15 i45 h55 −h65 −h06 −h256013 −k6 i46 i26 i56 h66 −h06 −h10 −h36∞653 ω00 −ω20 −ω40 −ω10 −ω00 ω20 ω40 ω10∞064 ω11 −ω31 −ω51 −ω21 −ω11 ω31 ω51 ω21∞105 ω22 −ω42 −ω62 −ω32 −ω22 ω42 ω62 ω32∞216 ω33 −ω53 −ω03 −ω43 −ω33 ω53 ω03 ω43∞320 ω44 −ω64 −ω14 −ω54 −ω44 ω64 ω14 ω54∞431 ω55 −ω05 −ω25 −ω65 −ω55 ω05 ω25 ω65∞542 ω66 −ω16 −ω36 −ω06 −ω66 ω16 ω36 ω06

hmm=jm + km m = mhmn=jm − kn m > nimn=jmkn m < nωmm=jmkm m = m

ωmn=−jmkn m > nωmn=±jmkn m ≥ nhmn=jm ± kn m ≥ nimn=jmkn m < n

Table 11.3. As in Table 11.2, we write jmkn = imn when it is an “i”; otherwise,jmkm = ωmm, jm + km = hmm, jmkn = −ωmn, jm − kn = hmn. The notation ischosen to be invariant (modulo 2) under L3(2).

Lemma 3. The ring λ1,λ2 generated by two λs (with λ1 = λ2 ) isalso generated by λ1 and some i.

Proof. We can take λ1 to be (+12

8) and λ2 to have coordinates

(±12

8

) or (±12

4

,±1, 03).

In the first case, the value is unaffected modulo 2 by changing signs on anyquad, which can be used to reduce the number of − signs to at most two,and if exactly two, to put one of them on 1, so that λ1− λ2 has the form ior i+1. For similar reasons we can suppose, in the second case, that thereis at most one “−” sign, on a 1

2 , so that λ1 − λ2 has coordinates

(04,±12

4

) or (03, 1,±12

4

)

142 11

11

11

11 Reading O Mod 2

(the coefficient of 1 being 0 or 1). The former is an i, and the latter eithera 1 + i or an i+ i (which is congruent to a 1 + i modulo 2).

Lemma 4. The ring generated by a λ and an ω is a unit-ring.

Proof. We can take

λ = (±12

8

) and ω = (±12

4

04)

and by choosing representatives of λ can control the signs so that λ − ωtakes one of the forms

(04,±12

4

) or (03,±1,±12

4

),

which is either an i or a 1 + i. Then λ,ω = ω, i for this i.Combining the two lemmas, we obtain

Theorem 2. Modulo 2, any ring not generated by units is one of thefour types K1, K2, K4, K8.

Proof. Modulo 2, any non-unit is congruent to a λ or a 1 + i, and sowe may suppose all non-unit generators are λs, and then by Lemma 1 thatthere is only one of them, and by Lemma 2, that there is no “ω.” The partof the ring generated by units therefore consists of 1 and some mutuallyorthogonal “i”s.Without loss of generality, we can now suppose that one of our gener-

ators is the above λ. From the top right-hand entry of Table 11.1, we seethat the stabilizer of this λ has three orbits on the “i”s, two of which leadto types K2 and K4, while the third can be neglected since it leads to E4,which contains an ω. So, any remaining ring can be chosen to contain thisλ and 1, i0, i1, i3 together with four “i”s orthogonal to these, which must beeither i2, i4, i5, i6, which leads to K

8, or i2645, or i2645, which are equivalentunder symmetries.

1212 122 1

The Octonion ProjectivePlane OP 2This book has been devoted to an internal study of the quaternions andoctonions. Our main aim has been to give simple and self-contained pre-sentations of their theory, their most natural arithmetics, and their appli-cations to the geometry of the spaces that contain them. For their manyapplications to other parts of mathematics and physics, we refer to theworks in the bibliography, hoping our book will be of some use in readingthem.In this final chapter, we briefly describe some “external” applications

of the octonions.

12.1 The Exceptional Lie Groups andFreudenthal’s “Magic Square”

The compact semisimple Lie (that is, continuous) groups over the realand complex numbers have been famously classified: As well as the infinitefamilies (An, Bn, Cn, Dn in the Cartan notation) that comprise the familiesof unitary, orthogonal, and symplectic groups, there are five “exceptional”groups G2, F4, E6, E7, E8. It has been realized that these are intimatelyconnected with the octonions–for instance, we have seen that G2 is theirautomorphism group.When Jordan, von Neumann, and Wigner classified “Jordan algebras”1

1These are defined by the conditions a b = b a and a (b a2) = (a b) a2, whichare satisfied by matrices over an associative ring if A B means 1

2(AB + BA). The

exceptional one is the only case that cannot be obtained from an associative ring.

143

144 1212 122 1 The Octonion Projective Plane OP 2

in 1934, they found an “exceptional” one consisting of 3× 3 “Hermitian”matrices over the octonions, whose automorphism group is F4.

In the 1950s, Freudenthal found that four particular “geometries” couldall be defined over R, C, H, O: the geometries of the elliptic and projec-tive planes, and 5-dimensional symplectic geometry, which were alreadyknown, and his new “metasymplectic” geometry. The automorphismsof the four geometries over the four rings form his celebrated “MagicSquare” (see [17] and [18]):

R C H Oelliptic plane B1 A2 C3 F4

projective plane A2 A2 +A2 A5 E65-dimensional symplectic geometry C3 A5 D6 E7

metasymplectic geometry F4 E6 E7 E8

J. Tits collaborated with Freudenthal in investigating the properties ofthese geometries.

12.2 The Octonion Projective Plane

The usual definition of a projective n-space over a field is that its pointsare the 1-spaces of an (n + 1)-dimensional vector space over the field,so can be coordinatized by non-zero (n + 1)-tuples (x0, x1, . . . , xn) withthe understanding that (λx0,λx1, . . . ,λxn) represents the same point foreach λ 6= 0. Its k-spaces are the (k + 1)-dimensional vector subspacesunder the same identification, so that in particular, its typical hyperplaneconsists of all points (x0, x1, . . . , xn) that satisfy some linear equation inthe xi.

In particular, the points of the 2-dimensional complex projective spaceare coordinatized by non-zero triples (x0, x1, x2) of complex numbers withthe understanding that (λx0,λx1,λx2) = (x0, x1, x2) for λ 6= 0 and thelines specified by linear equations in x0, x1, x2. If the 3-dimensional com-plex vector space is given an inner product defined by the usual Hermitianform x0x0+ x1x1+ x2x2, we can describe the typical line as x0y0+ x1y1+x2y2 = 0, the set of all (x0, x1, x2) orthogonal to the particular point(y0, y1, y2).

All of this works up to the quaternions, but not for the octonions.However, there is an alternative, originally suggested by some physicalideas, that does succeed in defining a 2-dimensional projective space, orprojective plane, over O.

12.3. Coordinates for OP 2 145

Let us represent a point p of the complex projective plane by the pro-jection operator onto the corresponding 1-space of C3. For p = (1, 0, 0) thishas matrix

e =

1 0 00 0 00 0 0

.In general, its matrix will be a Hermitian (eij = eji) idempotent (e

2 =e) of unit trace (e00 + e11 + e22 = 1). Also, two 1-spaces of C3 areorthogonal if and only if their projection operators e and f satisfy ef = 0,where e f is the Jordan product 1

2 (ef + fe). We can define the linef of the complex projective plane to consist of all the points e for whiche f = 0.In fact, this definition generalizes to give projective spaces of all di-

mensions over R, C, H, but over the octonions O it produces a projectiveplane but no higher projective space, because it requires the Jordan alge-bra property. We therefore define the octonion projective plane to havepoints and lines specified by 3 × 3 Hermitian idempotents of trace 1 overthe octonions, with point e lying on the line f just when e f = 0.This definition is mathematically the most satisfying one, since it makes

it immediately clear that the plane has at least the symmetries of theexceptional Jordan algebra (which constitute the Lie group F4). In fact,it is also invariant under further automorphisms that extend F4 to thelarger Lie group E6. However, it is cumbrous to work with, so the nextsection will establish its equivalence to a “coordinate definition” (due toPorteous [35] and Aslaksen [4]) that is less symmetrical but more familiarand more convenient. Our discussion of its equivalence to Freudenthal’sdefinition follows the elegant paper of Allcock [2].

12.3 Coordinates for OP 2

Our first lemma enables us to replace matrices by vectors of octonions.

Lemma 1. Any 3×3 Hermitian idempotent e of trace 1 can be writtenas ex0,x1,x2 = (xixj) for a norm 1 vector (x0, x1, x2) with x0 real.

Proof. For

e =

a γ βγ b αβ α c

and a+ b+ c = 1,

146 1212 122 1 The Octonion Projective Plane OP 2

with a, b, c ∈ R and α,β, γ ∈ O, the condition e2 = e is equivalent to thesix equations

a = a2 + [β] + [γ] γ β = (1− b− c)α = aαb = b2 + [γ] + [α] α γ = (1− c− a)β = bβc = c2 + [α] + [β] β α = (1− a− b)γ = cγ.

The three equations on the left imply that a, b, c ≥ 0, while the threeequations on the right imply that

[β][γ] = a2[α] [γ][α] = b2[β] [α][β] = c2[γ],

and so [γ] = ab and [β] = ac if αβγ = 0. It is then easy to verify the lemmaupon setting

(x0, x1, x2) = (√a, γ√

a, β√

a) if a = 0

(x0, x1, x2) = (0,√b, α√

b) if a = 0, b = 0

(x0, x1, x2) = (0, 0, 1) otherwise.

Of course, we could equally demand that either x1 or x2 be real. Theconversion is easy–for example:

Lemma 2. If (x0, x1, x2) = (r0, r1u1, r2u2) with ri ∈ R and ui units,then ex0,x1,x2 = eu1x0,r1,u1x2.

Proof. The “inner product” matrices

r0 r1u1 r2u2r0 r20 r0r1u1 r0r2u2r1u1 r0r1u1 r21 r1r2u1u2r2u2 r0r2u2 r1r2u2u1 r22

and

r0u1 r1 r2u1u2r0u1 r20 r0r1u1 r0r2u2r1 r0r1u1 r21 r1r2u1u2

r2u2u1 r0r2u2 r1r2u2u1 r22

are identical.Now we show that the coordinate condition for orthogonality works for

a point and line that have real coordinates in different places.

12.3. Coordinates for OP 2 147

Lemma 3. If x0 and y1 are real, then for e = ex0,x1,x2 and f = ey0,y1,y2we have

x0y0 + x1y1 + x2y2 = 0 ⇔ e f = 0.

Proof. Given that x0y0 + x1y1 + x2y2 = 0, we see for instance that

(ef)02 = x0((x0y0 + x1y1 + x2y2)y2) = 0

since it is the sum of

x0x0.y0y2 x0x1.y1y2 x0x2.y2y2,

which can be replaced by

x0(x0y0.y2) x0(x1y1.y2) x0(x2y2.y2),

since

x0 is real y1 is realy2y2 is real anddiassociativity.

There is just enough associativity to show in a similar way that all other(ef)ij and (fe)ij vanish, except that for the 01 and 10 entries we must firstrenormalize (x0, x1, x2) to (x

00, x

01, x

02) with x

02 real.

For the converse, we show in this way that the sum of the diagonalentries of e f (given to be 0) is the sum of three real numbers:P

i[xi((x0y0 + x1y1 + x2y2)yi), 1]

=P

i[x0y0 + x1y1 + x2y2, xiyi] = [x0y0 + x1y1 + x2y2],

and so that indeed x0y0 + x1y1 + x2y2 vanishes.Our final lemma justifies the term “projective plane,” since it shows

that any two distinct lines pass through just one point, and that any twodistinct points lie on just one line.

Lemma 4. If (y0, y1, y2) and (z0, z1, z2) are non-proportional unit vec-tors with y1 and z1 real, then the equations

x0y0 + x1y1 + x2y2 = 0 and x0z0 + x1z1 + x2z2 = 0

and the condition that x0 be real determine (x0, x1, x2) uniquely up to pro-jectivity.

Proof. The ratio of x0 to x2 is found by subtracting

(x0y0 + x1y1 + x2y2)z1 from (x0z0 + x1z1 + x2z2)y1,

and then x1 can be determined from either equation.

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Index2m-ions, 8

GO2, 12GO3, 23

GOn, general orthogonal group, 6

G2, 76G2(2), 120

G2(p), 133

H-stabilizer, 80PGOn, projective general orthogo-

nal group, 6

PSO8, 97

PSOn, projective special orthogonalgroup, 6

SO2, 12, 89SO3, 23

SO4, 89

SO7, 95SO8, 89

and multiplication maps, 92

SOn, special orthogonal group, 6, 24C, set of complex numbers, 5, 11H, set of quaternions, 5O, set of octonions, 5Q, set of rational numbers, 3R, set of real numbers, 3Z, set of rational integers, 3O = O8, set of octavian integers, 10Spin3, 24

Spin4, 42Spin7, 95Spin8, 91, 952-to-1 map, 24, 33, 42, 95

achiral, 14, 33, see also groupAlbert, A., 77algebra

Clifford, 9composition, 5, 10, 41, 67over other fields, 76

Dickson Double, 69, 70, 72, 73,75, 80

Jordan, 143normed division, 7

Allcock, D., 145Altmann, S., 10amphichiral, 33arithmetic, 5, 100Artin, E., 76, 88Aslaksen, H., 145associative, 8

2-generator Moups, 88di-, 76power, 73

Aut(O)and G2(p), 133is G2(2), 120

153

154 Index

order of, 125

transitivity of, 125

automorphism, 5

of octavians, 119

of octonions, 98

Bulow, R., 48

Baez, J., 5, 7

Baker, A., 18

Blichfeldt, H. F., 105

Brown, H., 48

Bruck, R. H., 102

Cartan, E., 5

Catalan, 59

number, 116

polynomial, 59

truncated, 59, 115

Catalan Triangle, 59

Cayley number, 9, 99

Cayley, A., 8

chiral, 14, 33, see also group

chirality

potato, 49

shoe, 49

types of, 49

chiro-, 30, see also group

Clifford, W. K., 9

companions, 86, 88, 90, 96, 97

for multiplication maps, 97

multiplication rule for, 97

complex numbers, 5

congruence, Euclidean, 11, 23

conjugation, 68

by a, Ta, 98

complex, 12

quasi-, 80

quaternionic, 42

coordinates for quaternions and oc-tonions, 75

Coxeter, H. S. M., 10, 45, 46, 99,102

cross polytope, 106

Degen, C. F., 9determinant, 6diagonal symmetry, 80diassociativity, 76

of 2-generator Moups, 88of octonions, 76

Dickson Doubling rule, 10, 69, 73,80, 103, 126

modified a la Pfister, 78non-orthogonal, 77

Dickson, L. E., 10, 69, 100, 102diploid, 33, see also groupdivision with small remainder, 55,

99, 106, 111double

cover, 24Hurwitzian, 103

doubling, see Dickson Doubling ruleDu Val, P., 48, 50duplex form

of hexad, 83of isotopy, 84, 91of relation, 84

Eisenstein integer, 16Eisenstein, F. G., 11, 16Euclid, 3Euclidean algorithm, 111Euclidean line, 3Euler, L., 9, 24Euler-Rodrigues parameters, 9

factorizationHurwitzian, 59Lipschitzian, 63modelled on another, 57octavian, 111, 114quaternionic, of an ordinary prime,

58factorizations, counting, 59, 63, 114Feaux, C., 116frame

i-, 138j- and k-, 134, 138, 139even and odd, 138

Index 155

Freudenthal, 143his Magic Square, 143metasymplectic geometry, 144

Gauss, C. F., 11, 15Gaussian

prime, 15unit, 15

generalizedn-gon, 125hexagon, 121

Gibbs, W., 8Gosset, T., 105Goursat, E., 44, 48, 50Graves, J. T., 8, 99group

G2, 120G2(2), 120G2(p), 133L3(2) ∼= L2(7), 1392-dimensional space, 183-dimensional point, 18, 233-dimensional space, 494-dimensional space, 48achiral, 14, 33, 43, 47, 53achiral diploid, 34achiral haploid, 34automorphismof octavians, 119of octonions, 76, 96

axial rotation, 30chiral, 14, 33—35, 43chiral diploid, 34chiral haploid, 34chiro-, 36chiro-icosahedral, 36chiro-octahedral, 36chiro-prismatic, 36chiro-pyramidal, 36chiro-tetrahedral, 36crystallographic, 48cyclic, 30, 32, 35cyclo-cyclic, 35cyclo-dihedral, 35dihedral, 30, 32, 35dihedro-dihedral, 35

diplo-cyclic, 35diplo-dihedral, 35diplo-icosahedral, 35diplo-octahedral, 35diplo-tetrahedral, 35diploid, 33—35, 43doubling a, 48elliptic, 34, 48exceptional Lie, 143

general orthogonal, 6, 34haploid, 33, 43holo-, 36holo-antiprismatic, 36holo-icosahedral, 36holo-octahedral, 36holo-prismatic, 36holo-pyramidal, 36holo-tetrahedral, 36

hybrid, 34, 35icosahedral, I, 30, 32, 35Lie, 76, 96, 120, 143exceptional, 143

metachiral, 44, 49octahedral, O, 30, 32, 35of quaternions, 33opposite, 48orthochiral, 46, 49orthogonal, 6, 12, 143

parachiral, 46, 49permutation, 30polyhedral, 4-dimensional, 45polyhedral (rotation), 30pro-antiprismatic, 36pro-prismatic, 36projective, 34, 43projective general orthogonal, 6,

34projective special orthogonal, 6,

34pyritohedral, 36reflection, 14, 27, 29rotation, 14, 27, 29, 30special orthogonal, 6, 24, 34

spin, 24, 42, 91, 95symplectic, 143

156 Index

tetra-octahedral, 35tetrahedral, T , 30, 32, 35unitary, 143

halving set, 100∞-set, 101, 104, 103n-set, 102inner and outer n-sets, 103

Hamilton, Sir W. R., 7, 75haploid, 33, see also groupHeaviside, O., 8Heegner, K., 18hexad

of isotopies, 84, 87of relations, 83

holo-, 30, see also groupHurley, A. C., 48Hurwitz, A., 5, 10, 55, 67, 99Hurwitzian, 5, 55

double, 103prime, 56

hyperhexagon, 121

ideal, 4, 58, 99in O, triviality of, 109principal, 4, 58, 109

identityn-square, 771-square, 7716-square, 782-square, 67, 774-square, 9, 778-square, 8, 77

imprimitive octavianfactorizations of, 115

integer∞-, 102n-, 102, 103double Hurwitzian, 103, 104Eisenstein, 5, 11, 16, 100Gaussian, 5, 11, 15, 55Gravesian, 100, 104factorization of, 116

Hurwitzian, 5, 55, 111imprimitive, 57Kirmse, 102

Kleinian, 103, 104, 134Lipschitzian, 5, 55factorization of, 61

octavian, 5, 10, 99, 105, 108,109

divisor sets of, 113primitive, 57quaternion, 5rational, 3

integrality, 10, 55, 99isochirality, 49isometry, 3, 5, 11isotopy, 49, 67, 72, 78, 84

and SO8, 89orthogonal, 91similar to automorphism, 86

Jordan algebra, 143exceptional, 145

Jordan product, 145Jordan, P., 143

Kaplansky, I., 77Kelvin, Lord, 8, 33kernels, left and right, 44Kirmse, J., 102

his mistake, 102Korkine, A., 105Kuipers, J. B., xii

Lamont, P. J. C., 110, 111, 116lattice

D4, 62D∗4 , 62E8, 99, 105, 107E8, mod 2, 135cubic, 62orthoplex, 106simplex, 106square, 15triangular, 16

lawalternative, 73, 74associative, 74, 89biconjugation, 68braid, 68, 70

Index 157

commutative, 89composition, 68composition doubling, 69conjugation doubling, 69exchange, 68inner-product doubling, 69inverse, 72, 74Moufang, 5, 73, 79, 95, 97and transitivity of monotopies,87

bi-multiplication, 74, 87, 88left-multiplication, 74, 88right-multiplication, 74, 88

product conjugation, 69scaling, 68

Liebeck, M., 87Lipschitz, R. O. S., 5, 99Lipschitzian, 55loop

inverse, 83Moufang, 72, 83, 87

Magic SquareFreudenthal’s, 143

Mahler, K., 110maximal subgroups of G2(2), 134metacommutation, 61metacommutation problem, 116

Lipschitzian, 117metamigration, 114, 115metamigration problem, 116

Gravesian, 117metassociation problem, 116model, factorization on a, 57, 114modulo 2, 5monotopy, 86, 87, 90

every SO8 element is a, 90transitivity and Moup law, 87

Moufang loop, 83Moufang, R., 72, 88Moup, see Moufang loopmultiplication map

bi-, Bx, 73, 87, 90companions for, 97left, Lx, 73, 86right, Rx, 73, 86

names for groupsalgebraic, 36geometric, 36

Neubuser, J., 48non-associative, 72, 89norm, Euclidean, 11, 100notations for groups

algebraic, 35, 36orbifold, 14, 20, 28, 36

numberCatalan, 116Cayley, 9complex, 11of prime factorizations, 115real, 3

octahedron, 106octaves, 8, 99

Gravesian, 100Kleinian, 103

octavian (integer), 99, 105unit, 119unit ring, 125

octonions, 5, 99and 8-Dimensional geometry, 89

orbifold, 20order, 100, 103

containing Gravesians, 101maximal, 100, 102, 103

orthogonality, even and odd, 120orthoplex lattice, Dn, 106

Polya, G., 19Pascal’s Triangle, 60Pascal, B., 60Peirce, B., 7Pfister, A., 78polyhedra, 36

regular, 30polytope, regular, 45Porteous, I., 145prime

Eisenstein, 16factorization, 56Gaussian, 15

158 Index

Hurwitzian, 56octavian, 115

primitive octavianfactorizations of, 115

problemmeta-, 116metacommutation, 61metamigration, 116metassociation, 116

projective plane, 88octonion, 5, 143, 144order 2, 137

quad, and co-quad, 119quadratic form

multiplicative, 79non-degenerate, 76

quadratic non-residue, 58quadratic residue, 58quasiconjugation, 80quaternion, 5, 23

finite groups of, 33subalgebra, what fixes, 80

Rankin, R. A., 10, 111, 116rational numbers, 3real numbers, 3recombination, 61reflection, 3, 11, 90

in q, 41regular

orthoplex, 106polyhedron, 26polytope, 43simplex, 106

Rehm’s algorithm, 111Rehm, H. P., 99, 111ring, 15

Kleinian, 18Rodrigues, O., 9, 105rotation, 11

simple, 6, 23, 40product of, 26

Schlafli, L., 45Schafer, R. D., 73

Seifert, H., 48similarity, 11, 23, 113simplex lattice, An, 106Smith, H. J. S., 105Smith, W. D., 73spherical geometry, 26Springer, T., 110Stark, H. M., 18Stufe of field, 79subgroups, left and right, 44symmetry, central, 34, 36symmetry, of algebra, see automor-

phism

Tait, P., 7theorem

5-multiplication?, 957-multiplication, 5, 93Euler’s on rotations, 24, 26Hurwitz classification, 67, 72,

78orbifold Magic, 20spherical excess, 28see also unique factorization

Threlfall, W., 48Tits, J., 144trace, 100transformation by a, Ta, 98translation, 3triality, 5, 6, 91

of Spin8 and PSO8, 92triplet, quaternion, 75

even and odd, 121of octavians, 119

triplex formof hexad, 84of isotopy, 84, 91of relation, 84

unique factorization, 3, 15, 16, 61,110

unit, 11Gaussian, 15Hurwitz, 56Lipschitz, 56octavian, 119

Index 159

unit-migration, 15, 57, 61, 111failure for octavians, 114

van der Blij, F., 99, 110von Neumann, J., 143

Wigner, E. P., 143Wondratschek, H., 48

Zassenhaus, H., 48Zolotarev, G., 105

On Quaternions and Octonions - The Eye · 2020. 1. 17. · rotation can be speciﬁed by a single...

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