Amore, Khan - The Divine Proportion

The Divine Proportion

1

Khan Amore’s Commentary on


[ = 1.6180339887498948482045868343656381177203…]

(Also known as the Golden Section, the Golden Ratio, the Golden Mean, or the Mean Proportional)

Contents:

§01: Definition of Divine Proportion (or Golden Section, or Golden Ratio, or

Mean Proportional), History of the Divine Proportion, and Symbolic

Representation.

Page

2

§02: Algebraic Derivation of Divine Proportion (or Golden Section, or Mean

Proportional). The derivation is the solution of a simple quadratic equation.

Page

4

§03: How to Geometrically Construct the Divine Proportion (or Golden Section,

or Mean Proportional) using only a Compass and Straight Edge.

Page

14

§04: Analysis of the Construction of the Divine Proportion (or Golden Section, or

Mean Proportional). The construction is based on the Pythagorean

Theorem.

Page

25

§05: The Golden Rectangle (having Height-to-Width Ratio in Divine Proportion)

and How to Construct the Golden Rectangle From a Square.

Page

28

§06: The Divine Proportion’s Connection With the Logarithmic Spiral, and with

the Chambered Nautilus. How to Draw a Spiral Using Golden Rectangles.

Page

31

§07: The Divine Proportion’s Connection with the Fibonacci Numbers and with

Spiraling Natural Structures.

Page

36

§08: The Divine Proportion’s Connection with the Pentagon, the Pentagram, the

Decagon, and with Various Polyhedra.

Page

42

§09: The Divine Proportion’s Representation as a Continued Fraction. Page

50

§10: Miscellaneous Properties, including Reciprocals and Powers, of the Divine

Proportion.

Page

51

§11: Divine Proportions to be Found in Art, Architecture, and in the Most

Beautiful, Harmoniously-Proportioned Human Bodies (Including a Pictorial

Chart.)

Page

54


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§1

Definition of Divine Proportion

(or Golden Section, or Mean Proportional, or Golden Ratio):

A line segment is said to be divided in golden section (or in divine proportion)

if the larger segment is the mean proportional between the complete segment

and the smaller segment. In other words, two line segments are in divine

proportion when the ratio of the length of the smaller segment to

the length of the larger segment is equal to the ratio of the length of

the larger segment to the sum of the lengths of the smaller and

larger segments, taken together.

The History of the Divine Proportion:

Although Pythagoras is believed to have been the discoverer of the divine

proportion, and the ancient Greeks that followed him clearly knew how to

construct it, surprisingly, the ancients seem not to have had a special name for

this fundamental proportionality of Nature. In that definitive masterpiece of

geometry, The Elements, Euclid (as well as Hypatia) merely call this a division

or section “in the extreme and mean ratio” and this mean proportional was

used by them to construct the regular pentagon, the pentagram, the

dodecahedron and its dual, the icosahedron. Indeed, this division in mean

and extreme ratio appears many times in The Elements, particularly in Book

XIII, starting with Proposition 1 of that section of this most important book.

The ancient Greeks (particularly the Pythagoreans) regarded the pentagon,

the pentagram, and the Platonic Polyhedra with reverence and awe, and so it

should come as no surprise that they regarded with similar veneration the

“golden mean” proportionality which underlies these shapes, enables their

construction, and makes them what they are.

There is some evidence that the golden ratio was also important to the ancient

Egyptians, for the Rhind Papyrus refers to a “sacred ratio,” and the ratio in the

Great Pyramid at Gizeh of the altitude of a face to half the side of the base is

almost exactly 1.618 (i.e., the golden ratio).


3

It is quite likely that the ancient Greeks used the divine proportion in their

architecture, for one has but to measure the dimensions of their most beautiful

structures (like the Parthenon) to find the golden ratio hiding there, but no

documentary proof remains that this was the result of deliberate calculation

rather than of heightened intuitive esthetic sensibilities. There is no doubt,

however, that this harmonious proportionality was consciously exploited by

Renaissance artists who knew it as the divine proportion. In 1509 Fra Luca

Pacioli published De Divina Proportione, illustrated with drawings of the

Platonic solids made by his friend, Leonardo da Vinci. Da Vinci was

probably the first to refer to the mean proportional as the “sectio aurea” (i.e.,

the golden section.)

Johannes Kepler (1571-1630), the brilliant discoverer of Kepler’s Laws and

of the ellipticity of planetary orbits, also rhapsodized over the Divine

Proportion, declaring that “Geometry has two great treasures: one is the

Theorem of Pythagoras, and the other the division of a line into extreme and

mean ratio; the first we may compare to a measure of gold, the second we

may name a precious jewel.”

Renaissance artists regularly used the golden section in composing paintings

into the most pleasing proportions, just as architects both ancient and modern

have used the golden ratio to plan and analyze the proportions of buildings.

For example, Vitruvius’ De Achitectura uses the golden ratio to analyze the

elevation of the Milan Cathedral.

The ancient Greeks realized that the most esthetically-pleasing rectangle to the

human eye is one in which the height-to-width ratio is in divine proportion;

that they realized this is evident from the many golden ratios which are to be

found in their sculptures and temples. There have been skeptics, however,

who regarded this connection between esthetics and mathematics as akin to

numerology — an occult pseudo-science. In order to see if there was any

empirical evidence to support any such connection, the German psychologist

Gustav Fechner made a serious and thorough study of the matter. He

made literally thousands of ratio measurements of common rectangular

objects, such as playing cards, windows, writing-paper pads, and book covers,

and found that the average was very close to the golden ratio (1.618…) He

also did an extensive statistical testing of personal preferences and found that

most people prefer a rectangle whose proportions lie between those of a

square and those of a double square. Despite the fact that most of those

tested had never even heard of the divine proportion, the plot of the results of


4

these tests shows a sharp spike precisely at a length-to-width ratio of 1.618

(the divine proportion). Of those tested by Fechner, fully 75.6 percent voted

for it (or for a rectangle differing from it by no more than 5%) out of ten

different rectangular shapes having width-to-length ratios ranging from 0.40 to

1.00. These tests have been repeated independently by at least three other

investigators (Witmar, Lalo, and Throrndyke), each in a different decade, and

the results were similar in each case. It seems, there can be little doubt about

it: the divine proportion is quite simply the most pleasing proportionality to the

human eye. [For further corroboration of this connection between esthetics

and mathematics, check out Khan Amore’s own startling findings on the

plurality of divine proportions to be found in his composite of the ideal

woman, to be found at the end of this article.]

A Note on the Symbol Used to Represent the Divine Proportion:

Mathematicians today symbolize the divine proportion either by the lower-

case Greek letter, tau (τ), the first letter of tome (“to cut”); or (more

commonly) they use the lower-case Greek letter phi () (pronounced “fee” in

Greek), following the example of the American mathematician, Mark Barr,

who chose this letter because it is the first letter of the name of the greatest

sculptor in history, Phidias, whose masterpieces of sculpture and architecture

(including the Parthenon) seem to have been based upon the divine

proportion.

§2

Algebraic Derivation of the Divine Proportion


The algebraic derivation of the Divine Proportion proceeds in a

straightforward manner from its definition:


5

“A line segment is divided in Divine Proportion or Golden Ratio when the

short part is to the long part as the long part is to the whole segment.” This

can be written mathematically as:

01) (Equation

b)(a

b

b

a

Where:

a = the shorter of the two segments which are in Divine Proportion

b = the longer of the two segments which are in Divine Proportion

(a + b) = the length of the whole segment before it is divided into sections

In order to solve for a or b in Equation 01, we must first move the variables

out of the denominators. This can be rather cleverly be accomplished by

multiplying each side of the equation by a fraction that has the same

expression in the numerator as in the denominator (which is permissible

because it is equivalent to multiplying by one), but these “unity-fractions” are

carefully chosen so as to give the expressions on both sides of the equal sign

the same denominator:

02) (Equation

b

b

ba

b

b

a

ba

ba


6

Now that we have made the denominators the same on either side of the

equal sign, performing the multiplication indicated in Equation 02, leaves us

with:

03) (Equation

ab) (b

b

ab) (b

ab a

2

2

2

2

Now we can get rid of the denominators by multiplying both sides of Equation

03 by the denominators, (b

2

+ ab):

04) (Equation ab b

ab) (b

b

ab) (b

ab a

ab b

2

2

2

2

2

2

Because multiplying a fraction by its denominator in effect removes the

denominator, this has the effect of canceling out both denominators while

leaving us with an equation having equivalent — although more tractable —

expressions:

05) (Equation b ab a

22

If we now subtract b

2

from both sides of Equation 05, this leaves us with an

equation which lies at the very heart of the Divine Proportion:

06) (Equation 0 b ab a

22


7

It is important to note that this equation (i.e., Equation 06) must hold true if

sections a and b are to be in divine proportion. This leaves us with one

equation of two variables (a and b), but since these variables are always in the

same ratio we can figure out what the numerical value of the ratio is by setting

one of the variables (in this case, the shorter section, a) equal to one. Then

we can solve for b under the condition when a = 1:

07) (Equation 0 b (b)(1) 1

22

08) (Equation b b 1

2

09) (Equation 0 1 b b

2

This equation (Equation 09) is quadratic in b and, if only we substitute x for b

wherever it appears, this is equivalent to the standard-form quadratic

equation:

10) (Equation 0 c bx ax

2

Which (if we let a = +1, b = -1, and c = -1) can be solved using the well-

known Quadratic Formula:

11) (Equation

2a

4ac b b

x

2


8

Bearing in mind that when we solve for x in Equation 11 we are really solving

for b in Equation 09, we substitute +1 for a, -1 for b, and -1 for c in Equation

11, which yields:

12) (Equation

2

411

1)2(

1)1)(4(11)

x

2

(

Which reduces to:

13) (Equation

2

5 1

x

but since using the minus sign in the numerator of Equation 13 would yield a

negative value for x (specifically, - 0.6180339887…) we will discard this as an

extraneous root of the quadratic equation, since x, in this case, is really the

length of a section of a line segment, and negative lengths are regarded as

meaningless. This leaves us with only a single solution:

14) (Equation

2

1 5

x

Recalling that, in order to use the Quadratic Formula, we substituted x for b to

solve Equation 09 (and that the x in Equation 14 is really b), we now

substitute b back for x in our solution:

15) (Equation 8749891.61803398

2

1 5

b

...


9

Behold, we have just derived the Divine Proportion, algebraically! This means

that, given a short section that is one unit in length, the corresponding long

section will be in Divine Proportion if the long section is 1.6180339887 …

units in length (and the three dots at the end of this number of course signify

that the number is irrational, and the digits go on forever.) And since this is a

constant proportionality, we can say that if the short section is any arbitrary

length, a, (not necessarily one unit in length), then the long section, b, in

order to be in divine proportion with the short section, must have a

corresponding length of:

16) (Equation

2

1 5

a b

But what if we had let the longer section (i.e., b) equal one? What, then,

would the length of the shorter segment have to be in order for the two

sections to be in divine proportion? To answer this, we proceed as before,

from Equation 06, except this time we let b (instead of, as before, a) equal

one:

17) (Equation 0 )(1 a(1) a

22

Or,

18) (Equation 0 1 a a

2

This equation is quadratic in a, and is equivalent to the standard-form

quadratic equation (appearing here for the second time):

10b) (Equation 0 c bx ax

2


10

Which (if we let a = +1, b = +1, and c = -1) can be solved using the well-

known Quadratic Formula (appearing here for the second time):

11b) (Equation

2a

4ac b b

x

2

Bearing in mind that when we solve for x in Equation 11b we are really

solving for a in Equation 18, we substitute +1 for a, +1 for b, and -1 for c in

Equation 11b, which yields:

19) (Equation

2

411

1)2(

1)1)(4(11

x

2

)(

Which reduces to:

20)(Equation

2

5 1

x

But since using the difference between the two terms in the numerator would

yield a negative value of x (which in this case is really the length of segment

a), we will discard the “minus” part of the “plus-or-minus sign” as an

extraneous root of the quadratic equation, since negative lengths are held to

be meaningless. This leaves us with only a single solution:

21)(Equation

2

1 5

x

Recalling that, in order to use the Quadratic Formula, we substituted x for a to

solve Equation 18 (and that the x in Equation 21 is really a), we now

substitute a back for x in our solution:


11

22)(Equation 8749894...0.61803398

2

15

a

Behold, we have algebraically derived the reciprocal of , and amazingly, the

difference between and its reciprocal is exactly one! This means that, given

a long section that is one unit in length, the corresponding short section will be

in Divine Proportion if the short section is 0.618033988749894… units in

length (and the three dots at the end of this number of course signify that the

number is irrational, and the digits go on forever.) And since this is a constant

proportionality, we can say that if the long section is b units in length, then the

short section (a), in order to be in divine proportion, must be:

23)(Equation

2

1 5

b a

But what if we had let the whole of the original segment [i.e., (a+b)] equal a

given value, c? What, then, would a and b have to be in order to be in

divine proportion?

To find out the answer to this, we let c = (a+b), then we can re-write

Equation 01 as:


12

24)(Equation

c

b

b

a

But, from Equation 23 we know that:

23b)(Equation

2

1 5

b a

So we may substitute this expression for a into Equation 24:

25)(Equation

c

b

b

2

1 5

b

Or,

26)(Equation

2

1 5

c

b

Or,

27)(Equation

2

1 5

(c) b


13

Where:

b = the length of the longer section, and

c = the length of the whole segment [i.e., c=(a+b)]

This means that if the whole segment is c units in length, then, to be in divine

proportion, the longer section must be 61.8033989…% the length of the

whole segment, just as the smaller section must at the same time be

61.8033989…% the length of the larger section. Since c=(a+b) and since…

27b)(Equation

2

1 5

(c) b

We can write:

28)(Equation

2

1 5

c a c

29)(Equation

2

1 5

1 (c) a

Which can be simplified as:

30) (Equation

2

1 5 2

(c)

2

1 5

2

2

(c) a


14

Which, upon adding the terms in the final numerator, becomes:

31) (Equation 2...0.38196601 (c) a ;

2

5 3

(c) a

Note that this last decimal factor (0.381966012) is just [1 - 0.618033989…] or

[1 - 1/]. The upshot of this result is that, if the whole segment is c units in

length, then the shorter section, to be in divine proportion, must be (c times

0.381966012…) units in length — which is simply the length of the whole

segment minus 61.8033989…% of the whole segment.

This concludes Khan Amore’s algebraic derivation of the divine proportion.

§3

Geometric Construction of the Divine Proportion


The applications of the divine proportion are chiefly geometric and esthetic.

This harmonious proportionality is built into Nature herself, and in any field

which seeks to create or capture natural beauty — such as in the fields of art,

photography, sculpture, or architecture — it is well to be able to construct the

golden mean geometrically, as the ancient Greeks did it. Here, then, is the

complete procedure for constructing the Golden Section or Divine Proportion

using a straight-edge and compass alone:


15


16


17


18


19


20


21


22


23

The full construction of the Golden Section or Divine Proportion is

summarized (with more mathematical conciseness) and shown with all

construction marks on the next page …


24

Geometric Construction of the Golden Section or Divine Proportion

(Summary of Procedure):

Given a line segment, AB, which is to be divided into the Golden Section,

1. Bisect line segment AB with bisector m (the length of AB is a, and

a/2=c).

2. Construct perpendicular line n at point A.

3. Mark off distance c along perpendicular line n at A. Label this point “O”.

4. Draw circle, with center O and radius c (recalling that c = AB / 2),

5. Draw secant BO which intersects at point D the circle centered on point O.

6. BD (which has a length of x) is the larger segment of the golden section.

7. Mark off this distance x from point B on the original line segment, AB.

The two sections of AB thus produced are in divine proportion.

Q.E.F.


25

§4

Analysis of the

Geometric Construction of the Divine Proportion


In Section 2 of this article we saw how the Divine Proportion could, from its

definition, be derived algebraically as the positive root of a simple quadratic

equation. Now we will analyze how the geometric construction of Section 3

produces the same results geometrically, using the Pythagorean Theorem.

Here is a diagram which shows the essentials of the construction:


26

In this diagram, note that angle OAB is a right angle, and triangle OAB is

consequently a right triangle; note furthermore, that c = a/2 and z = (c+x)

Because this is a right triangle, the Pythagorean Theorem assures us that

(AB)

2

+ (OA)

2

= (OB)

2

(Equation 32)

But since

AB = a (Equation 33)

And

OA = a/2 (Equation 34)

And

OB = (a/2 + x) (Equation 35)

We can re-write Equation 32 as:

36) (Equation x

2

a

2

a

a

2

2

2

37) (Equation x

2

a

4

1

1 a

2

2

38) (Equation x

2

a

4

5

a

2

2

39) (Equation x

2

a

5

2

a


27

Subtracting (a / 2) from each side of Equation 39 then leaves us with:

40) (Equation x

2

a

5

2

a

41) (Equation 1 5

2

a

x

42) (Equation

2

1 5

(a) x

Note how this equation (Equation 42) is of exactly the same form as Equation

27, except what we have here called “x” (i.e., the longer section, or Mean

Proportional) was called “b” in Equation 27; and what we have here called

“a” (i.e., the length of the whole line segment before it was sectioned) was

called “c” in Equation 27. If we substitute x for b, and a for c, the results of

the algebraic derivation of the Divine Proportion from its definition is identical

with the results we obtain by applying the Pythagorean Theorem to the

triangle which is used in the classical construction of the Golden Section, using

only a straight-edge and compass. This proves that the geometric construction

of the Golden Section shown above really does section a given line segment

into two parts which are in theoretically perfect Divine Proportion. The

Golden Ratio is an irrational number — a number which cannot be exactly

represented by any fraction or any finite string of digits — but the power of

classical construction technique and the power of the Pythagorean Theorem

together enable us, without calculation or measurement, to exactly section any

given line segment into the Divine Proportion. Rather amazing, wouldn’t you

say? There was an unassailable brilliance and a crystalline purity to the

ancient Greek mode of rational thought that is sadly fading from this world.


28

§5

The Golden Rectangle

And Its Construction From a Square:

The Golden Rectangle is just a rectangle whose width-to-height ratio is in

Divine Proportion. In other words, the golden rectangle’s height is to its width

as its width is to its width plus its height. As was already mentioned in the

history of the Divine Proportion (in Section 1), the golden rectangle is the

most pleasing aspect-ratio to most people, and has been much used in art and

architecture ever since the ancient Greeks discovered this harmonious

proportionality of Nature. Even designers of computer monitor screens seem

to be aware of the optimally-pleasing qualities of the Golden Rectangle, for

the widescreen 1280 x 800 pixel (16:10 aspect ratio WXGA) computer screens

approximate the golden rectangle to within about 1.127 percent — really

quite a good approximation considering how “round” these numbers are

(1280 = 5 x 2

8

, and 800 = 5

2

x 2

5

). And this format has another advantage,

too: it turns out that (16:9 aspect ratio) 1280 x 720 progressive scan High

Definition Television pictures can be displayed quite well on such a (1280 x

800) screen, with 80 pixels worth of room at the top and/or bottom of the

screen for toolbars, clock, status indicators, etc. Anyway, back to the Golden

Rectangle….

In this section, we will explore how the Golden Rectangle can be constructed

from a square. Although this geometrical construction is not really any simpler

that the construction of the Golden Section which is described in Section 3 of

this article (because it necessitates the preliminary geometric construction of a

square), nevertheless, the construction of the Golden Rectangle provides us

with an interesting and enlightening alternative way of geometrically

producing the Divine Proportion. Here, then, in summary form, is how to

construct a Golden Rectangle:


29

Procedure for Constructing a Golden Rectangle From a Square:

1. Given square ABCD, bisect the square with line segment EO.

2. Draw diagonal OB (= OG). Set the radius of the compass to this

distance (OB), and draw a circle with this radius centered on point “O”.

3. Extend line segment DC to reach the circle. Label this point of

intersection “G”.

4. Erect a perpendicular upward at point “G”, and extend line segment AB

to the right, to meet this perpendicular at point “F ”.

5. The rectangle AFGD thus produced is the Golden Rectangle, in which the

height (AD) and the width (DG) are in divine proportion. Moreover, the

part (BFGC) added onto the original square is a golden rectangle, too.


30

If a rectangle is drawn, whose sides are in Golden Ratio (as above), the

“Golden Rectangle” may be divided into a square, which will leave another,

similar (but smaller) residual golden rectangle. This process may then be

repeated on the residual golden rectangle, and so on, ad infinitum. Because

the sum of the areas in this infinite series of diminishing squares is equal to the

area of the original golden rectangle (i.e., 1 x square units), we know that

must be a root of the infinite series equation:

43) (Equation 0 . . .

x

1

x

1

x

1

x

1

x

6420

Note that this equation (Equation 43) holds true when x = = 1.61803398…

If we substitute for x, Equation 43 may be re-written in sigma notation as:

0k

2k

44) (Equation

1

This implies that the Golden Ratio, , is equal to the infinite sum of the

reciprocals of all its even powers (assuming that we include zero as an even

power). [This result has been proven by Khan Amore via geometrical

induction, but the proof has not been included here, as this article is getting far

too long already, and so it is left as a simple, though enlightening, exercise for

the reader.]


31

§6

The Divine Proportion’s Connection with the

Logarithmic (Equiangular) Spiral and the Chambered

Nautilus; and How to Use Golden Rectangles to Draw

an Approximation to an Equiangular Spiral:

As mentioned in Section 5, when a golden rectangle is divided into a square

and a left-over piece, the left-over piece is itself a golden rectangle (albeit a

smaller one, rotated by 90°). This process may then be repeated with similar

results on the residual golden rectangle, ad infinitum. When a golden

rectangle is divided into a series of diminishing squares like this, it is possible

to draw an equiangular spiral through successive vertices of the sequence of

squares. A good approximation to this (equiangular or logarithmic) spiral can

be produced by a sequence of quarter-circles of diminishing radius. The spiral

recedes inward, converging toward the point where the diagonals of all the

golden rectangles meet:


32

Referring to the immediately-preceding figure, the equiangular (or logarithmic)

spiral may be approximated (using a compass and straight-edge alone) from

the golden rectangle in the following manner:

1. Construct a golden rectangle, in the manner outlined in Section 5.

2. Partition the golden rectangle into the largest possible square and a

left-over rectangle (Note: this left-over rectangle also turns out to be in

divine proportion — automatically!) This square is, of course,

produced by marking off with the compass the (shorter) height of the

golden rectangle along its (longer) width, and then drawing the

partitioning segment, BC.

3. Now repeat Step 2 on every successive residual golden rectangle as

many times as you can before the squares and rectangles become too

small to work with. [As a practical matter, inaccuracies grow with

every successive partitioning, but if you draw the diagonals AF and

CE, this will help keep you from going too far astray, for three of the

corners of every golden rectangle in the figure fall upon these two

lines, as do two of the corners of every square. The diagonals are

shown as the two crossing red line segments in the figure.]

4. After you have partitioned the original golden rectangle into a total of

six (or so) squares, set the compass radius to span the distance from

point “C” to point “D”, and then, using point O

1

(at point “C”, or the

bottom right corner of square #1) as center, swing a circular 90° arc

from point “D” to point “B”.

5. Next, reset the compass radius to span the distance from point O

2

to

point “B” and, using point O

2

(the bottom left corner of square #2)

as center, swing a circular 90° arc from point “B” to the bottom right

corner of square #2.

6. Now, reset the compass radius to span the distance from point O

3

to

the upper right corner of square #3 and, using point O

3

(the upper

left corner of square #3) as center, swing a circular 90° arc from the

upper right corner of square # 3 to the bottom left corner of the same

square.

7. Once again, reset the compass radius, this time to span the distance

from point O

4

to the bottom right corner of square #4; then, using

point O

4

(the upper right corner of square #4) as center, swing a

circular 90° arc from the bottom right corner of square #4 to the

upper left corner of the same square.

8. As before, reset the compass radius, this time to span the distance

from point O

5

to the bottom left corner of square #5; then, using


33

point O

5

(the bottom right corner of square #5) as center, swing a

circular 90° arc from the bottom left corner of square #5 to the upper

right corner of the same square.

9. Finally, reset the compass radius again, this time to span the distance

from point O

6

to the upper left corner of square #6; then, using point

O

6

(the bottom left corner of square #5) as center, swing a circular

90° arc from the upper left corner of square #5 to the bottom right

corner of the same square. The resulting figure should be a fairly nice

geometrically-constructed approximation of an equiangular or

logarithmic spiral, based upon the properties of the divine

proportion.

Every part of this equiangular spiral is similar to every other part (differing

only in scale), so it should not be surprising to find that it occurs frequently in

Nature, in the arrangement of the seeds in sunflower heads, in the

arrangements of leaves on branches, and in the graceful convolutions of spiral

shells, such as the shell of the Chambered Nautilus:

This shell’s similarity to the construction of the equiangular spiral via the

golden rectangle is particularly evident in the cutaway view:


34

And just what does the creator and inhabitant of this mathematical masterpiece look

like? This cephalopod, floating at neutral-buoyancy in the aphotic depths of the

Pacific Ocean, may well take the prize for “most alien-looking life form on Earth”:


35

The Chambered Nautilus (Nautilus Pompilius):

Surviving today only in the depths of the Pacific ocean between Fiji and the

Philippines, the four species of the Chambered Nautilus (or “pearly nautilus”)

are the most ancient of the living cephalopods and are related not only to the

squids, octopi, and cuttlefish, but also to the ammonites which thrived in the

Jurassic period (195 million to 135 million years ago), and became extinct

in the Cretaceous period (135 million to 65 million years ago). An odd yet

gregarious animal, the chambered nautilus lives in droves with others of its

species, hiding near the dark bottom of the ocean depths in the day, and

floating up to shallow water at night to feed. In contrast to the octopus’ eight

strong tentacles — and the squid’s ten — the chambered nautilus has about

90 comparatively small and weak tentacles encircling its mouth, and unlike its

relatives, the octopi, squids, and cuttlefish, the nautilus has no evasive ink or

sepia to disappear behind, but relies instead upon its shell and upon darkness

for safety. The inside of the shell is a nacreous, iridescent, mother-of-peal,

and as the animal grows it enlarges the mouth of its shell in accordance with a

strict mathematical scheme. The animal lives in the outermost chamber of the

shell, and every so often it seals off the earlier, smaller chambers (called

“septa”) with a layer of nacre or pearl, so that the chambered innermost parts

of the shell may be filled with just enough air to cancel out its weight in water,

thus enabling it to float at neutral buoyancy, without the expenditure of any

effort. Perhaps to enable such ballasting, a thin tail-like siphuncle extends

from the animal in the outermost chamber, through smooth holes in the

partitions, all the way to the origin of both the spiral shell and its inhabitant.

This alien-looking creature is a true “blue-blood,” as its circulatory fluid

contains the blue copper-based respiratory pigment, hemocyanin, as opposed

to our red iron-based counterpart, hemoglobin. As if this were not strange

enough, the chambered nautilus has four kidneys, as opposed to our single

pair, and its eyes are lens-less open pits, which operate like pinhole cameras.


36

§7

The Divine Proportion’s Connection with the Fibonacci

Numbers and with Spiraling Natural Structures :

The Fibonacci Numbers (so-named after Leonardo of Pisa, also known as

filius Bonacci, meaning “son of Bonacci,” which was contracted to

“Fibonacci”) are members of a sequence of integers having the property that

each number of the sequence is the sum of the two preceding members of that

sequence. The sequence begins as follows (the zero is usually omitted):

(0), 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597 …

The Fibonacci sequence often turns up in combinatorial problems. The

sequence first arose in Leonardo of Pisa’s book, Liber Abaci (meaning

literally, “the Book of the Abacus”), as the solution of the following problem:

If a single mating pair composed of one male rabbit and one female rabbit are

placed in an enclosure, how many pairs of rabbits can be produced from that

pair in a year if it is supposed that every month each pair reproduces a new

mating pair which from the second month becomes productive? Neglecting

for the moment whether or not this is an accurate reflection of the leporine

rates of gestation and maturation, that exactly one male and one female are

born per monthly litter — and assuming that none of the rabbits die — the

number of pairs at the end of each month would be

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233

This sequence was named the Fibonacci sequence by Eduard Lucas in

1877. Another example of a combinatorial problem whose solution turns out

to be the Fibonacci sequence is this: In how many different ways can you seat

males and females in a row of n chairs such that no two females are allowed

to sit next to each other? When n = 1, 2, 3, 4, … the answers are 2, 3, 5, 8,

… and so on, in Fibonacci sequence.

At this point you may be thinking, “This is all well and good, but what does

this have to do with the Divine Proportion?” As it turns out, the ratio of any


37

Fibonacci number (after 3) to its predecessor in the sequence just happens to

approximate the divine proportion. What is more, the successive ratios are

alternately less than and greater than the golden ratio, getting closer and closer

to the true value the farther one goes down the sequence:

Given these two consecutive

Fibonacci Numbers:

The ratio of these two

consecutive numbers will be:

Percent deviation of this ratio

from the true value of :

5/3 = 1.666666666… +3.005664796%

8/5 = 1.600000000… - 1.114561815%

13/8 = 1.625000000… +0.43052316%

21/13 = 1.615384615… - 0.163740318%

34/21 = 1.619047619… +0.062645841%

55/34 = 1.617647059… - 0.023913527%

89/55 = 1.618181818… +0.009136396%

144/89 = 1.617977528… - 0.003489420%

233/144 = 1.618055556… +0.001332976%

377/233 = 1.618025751… - 0.000509075%

610/377 = 1.618037135… +0.000194495%

987/610 = 1.618032787… - 0.000074226%

True Value of =

1.61803398874989…

(for comparison)

Thus, the ratios of two successive Fibonacci numbers give us a series of

rational approximations of , to any desired degree of accuracy (since these

ratios approach a limit which is ), and the Fibonacci number sequence also

gives us a very simple algorithm for calculating to any desired number of

digits. As you can see, there is an intimate connection between the Divine

Proportion and the Fibonacci Numbers, and this is not the only such

connection; as will be shown later (in Section 10 of this article), the Fibonacci

Numbers also pop up in the powers of the Golden Ratio.

Binet’s formula allows direct calculation of the n

th

Fibonacci Number (F

n

):

) ... 3, 2,1, 0, n :(where

2

51

5

1

2

51

5

1

F

nn

n

One has but to look at Binet’s Formula to see the Divine Proportion hiding

within it. It should come as no surprise, then, to find that, as n approaches


38

infinity, the limit of the ratio of the (n+1)

th

Fibonacci number (F

n+1

) to the n

th

Fibonacci number (F

n

) will approach the exact value of the golden ratio ():

46) (Equation

F

F

lim

n

1n

n

Now that we have some idea of how intimately related the divine proportion

and the Fibonacci Numbers are, we are better able to see that the divine

proportion is hiding in plain sight, all around us. Consider the case of the

pineapple:


39

As it turns out, the pineapple naturally grows with 13 grooves spiraling

downward from the crown in a clockwise direction, and 8 grooves spiraling

downward in a counter-clockwise direction; and, of course, 8 and 13 just

happen to be successive numbers in the Fibonacci sequence, and the ratio of

13/8 just happens to approximate the divine proportion. Coincidence? If so,

it is certainly not a rare coincidence. As another example of such a

“coincidence,” consider the case of the sunflower:

Sunflowers are amazing enough in their size, robustness, rate of growth, and

nutritional value of their seeds. Perhaps even more amazing, though, is the fact that,

throughout the day these giant flowers slew their heads slowly, like miniature

heliostats, to track the sun as it moves across the sky. I discovered this arresting

phenomenon one sunny day, years ago, while making a skydive (for class D

“expert” license qualification) into a lake in Wisconsin. As the jump aircraft climbed

to altitude it just happened to circle over a large field of sunflowers, and the sight was

simply breath-taking. The whole field blazed with yellow when we looked at it with


40

the sun at our backs, but as soon as the airplane came out of alignment with the sun,

the heads of the sunflowers were no longer oriented toward us, and so the field

suddenly looked a drab green instead of bright yellow. The field was only yellow

when viewed from the vantage point of the sun, and out of perhaps a million

sunflowers, there wasn’t a straggler in the bunch — every one of them had its head

turned in exactly the same direction. But if you think that this degree of order is

amazing to find in such a lowly life-form, take a close look at the exquisitely intricate,

yet very orderly, seed pattern in the head of the sunflower in the picture above.

Most people pass by such a mundane plant every day without giving it a glance, and

in so doing they zip through their lives blind to the beauty and wonders which lie

right under their noses. Fortunately for us, though, some solitary lover of flowers

and mathematics (perhaps with too much time on his hands) has stopped, and

looked, and thought, and in so doing has detected a remarkable pattern in the

arrangement of seeds in the head of the sunflower. As it turns out, the seeds

naturally grow in two sets of equiangular spirals, superposed or entwined, one being

a right-handed, and the other a left-handed spiral, with each floret filling a dual role

by belonging to both spirals. This is amazing enough, but even more astonishingly,

the number of spirals just happen to be adjacent Fibonacci Numbers (again!): a

medium-sized specimen of the sunflower has 21 clockwise and 34 counter-clockwise

spirals, and the ratio of 34/21 approximates the golden ratio with an accuracy of

99.94 percent. Large specimens of the sunflower tend to have more spirals, but

amazingly, when they do, the number of spirals each way jumps to a higher pair of

Fibonacci Numbers! A ratio of 89/144 has been found in some large specimens, and

a ratio of 144/233 has been claimed for one giant sunflower. Clearly, the beauty of

the sunflower is largely mathematical, and the mathematical relationship at play here

is once again intimately related to the divine proportion. Moreover, the more you

look around you, the more you find the divine proportion po pping up. It has even

been asserted that naturally-occurring opposing logarithmic spirals always occur with

a pre-determined number of each kind of spiral. Just as the sunflower does, most

daisies have 21 spirals radiating outward anticlockwise and 34 spirals radiating

outward clockwise (and these are successive Fibonacci Numbers, whose ratio

approximates the Divine Proportion.) Similar arrangements of opposing spirals are

found on pine-cones and Hazelnut bush (Coryllus Avellana) catkins (both of which

have 5 spirals one way, 8 the other — again adjacent Fibonacci Numbers whose

ratio approximates the Divine Proportion), and this also holds true with many other

plants with spiral growth patterns — such as the bud formations on alders and

birches — and of course we have already mentioned the case of the Pineapple

(having a 13:8 ratio of Spirals). Logarithmic spirals don’t just occur in plants and in

the shell of the Chambered Nautilus. They also occur naturally in the curve of

elephants’ tusks, in the horns of wild sheep, and even in canaries’ claws. Before we

move on, here is the mathematical scheme which underlies the orderly arrangement

of seeds in the sunflower:


41

[Note that in this illustration “clockwise” means “clockwise, spiraling inwards”, whereas

elsewhere in the text “clockwise” means “clockwise spiraling outwards.”]


42

§8

The Divine Proportion’s Connection with the Pentagon, the

Pentagram, the Decagon, and with Various Polyhedra:

— Divine Proportions Appearing in the regular Pentagon —

In the regular pentagon, (i.e., a five-sided polygon in which all sides are of

equal length) any diagonal of the pentagon is in divine proportion with the

length of any of that pentagon’s sides (or edges):

Also, again in the regular pentagon, twice the radius of the inscribed circle is in

divine proportion with the radius of the circumscribed circle.


43

— Divine Proportions Appearing in the Pentagram —

As proven in Euclid’s Elements (Book 13, Proposition 8,) in the regular pentagram

(i.e., the five-pointed star made up of all five diagonals of the regular pentagon),

each diagonal of the pentagon (in this case, AB) intersects two others such that the

diagonal is sectioned into the divine proportion in four different ways:

According to the Encyclopædia Britannica, the pentagram contains a

staggering two hundred golden ratios! Are you beginning to see why the

Pythagoreans were in awe of the Pentagon, the Pentagram, and the polyhedron

based upon them — the dodecahedron? When they discovered this mystical

proportionality popping up over and over again, they felt they had touched divinity.

Hence, the name.


44

— Divine Proportionality of Nested Pentagons —

If a series of nested pentagons and pentagrams is drawn such that two of the

indented edges of one pentagram also define two of the protruding edges of

the next (smaller) pentagon in the series, then the sides of any two adjacent

pentagons nested in such a diminishing series will be in Divine Proportion:

This, of course, also implies that the distance between any two adjacent points

of a pentagram (i.e., neighboring “points of the star”) is in divine proportion

with the length of the long sides of the isosceles triangles which point outward

from the central pentagon produced by the intersections of the five diagonals.

It would seem, the divine proportions crop up in such profusion in the

pentagon and pentagram, that they are literally piled one upon the other!


45

— Divine Proportions Appearing in the Decagon —

A regular decagon (i.e., a 10-sided polygon having all sides equal, and all

angles equal) can be constructed by laying together ten acute isosceles golden

triangles (i.e., acute triangles two of whose sides are in divine proportion to

the third) in such a way that the vertices opposite the short side of the triangles

meet at a point, which becomes the center of the decagon:

Another way of looking at this would be that, in the regular decagon, the

circumcircle radius is in divine proportion with any side of the inscribed

decagon. One would expect the central angle of each side of the decagon

(and hence the triangles of which it is composed) to subtend 36° — since ten

equal angles must add up to 360° — but what it is perhaps surprising is that

the 72° - 36° - 72° isosceles triangle just so happens to have sides which are in

divine proportion, and, therefore, five such triangles can be arranged to

produce a regular pentagram.


46

The Divine Proportions Appearing in the Regular Icosahedron,

Regular Dodecahedron, and Rhombic Triacontahedron

In view of the fact that he was the discoverer of the Divine Proportion, it is

perhaps not surprising that Pythagoras also knew of the Golden Section’s

connection with two of the five possible regular polyhedra. The regular

icosahedron is a polyhedron having 20 equilateral triangular faces, 12

vertices, and 30 edges. Well, it just so happens that when the icosahedron’s

twelve vertices are divided into three coplanar groups of four, these groups of

four vertices lie at the corners of three symmetrically-situated, mutually-

perpendicular golden rectangles, with their one common point situated at the

center of the icosahedron. In other words, the Divine Proportion is built into

this beautiful mathematical shape! Bear in mind that this trio of golden

rectangles does not merely appear accidentally within the icosahedron, but

these golden rectangles in a sense define the icosahedron. An illustration may

help to make this relationship easier to comprehend:

In this figure, the ratio of the blue rectangle-edges to the magenta edges is exactly .


47

Because the dodecahedron is the dual* form of the icosahedron, it is

perhaps not surprising that the regular dodecahedron also has a connection

with the Divine Proportion. The Regular dodecahedron is a polyhedron

having 12 regular pentagonal faces, 20 vertices, and 30 edges. As one might

expect from this polyhedron’s duality with the icosahedron, the centers of the

12 pentagonal faces of the regular dodecahedron are divisible into three

coplanar groups of four. These tetrads lie at the corners of three

symmetrically-situated mutually-perpendicular golden rectangles, with their

common point situated at the centroid of the dodecahedron. Again, an

illustration may help to make this relationship more readily comprehensible:

Once again, a beautiful shape of Nature is based upon the Divine Proportion!

* Note: Two polyhedra are duals if the vertices of one can be put in one-to-one

correspondence with the centers of the faces of the other. Incidentally, the reader is

asked to ignore the imperfections of the figure above, and to imagine that the corners of

the golden rectangles do indeed touch the centers of the pentagons. It took Khan most

of a day to get the drawing to this state of imperfection!


48

As we have already mentioned, Johannes Kepler (1571-1630) was quite enamored of the

Divine Proportion, and indeed, upon occasion, he called it by that name. In view of his

reverence for this harmonious proportionality of Nature, it is only fitting that Kepler was also

the discoverer of the Archimedean polyhedron known as the Rhombic Triacontahedron.

The Rhombic Triacontahedron is a polyhedron having 30 identical rhombic faces, 20

vertices where 3 rhombi meet, 12 vertices where 5 rhombi meet, and 60 edges. Because it

is such an attractive polyhedron, the writer of this article could not resist the temptation to

make one, in the year 1991. When constructing the template from which to cut matte-

boards into the correctly-shaped rhombic faces (to be glued together), it was noted that the

face angles were (the arctangent of 2) and (180° minus the arctangent of 2). This

seemed to imply that the ratio between the lengths of the two diagonals of the rhombic

triacontahedron’s faces was a very simple ratio, so we did a bit of analysis to find out what

this ratio may be, and we were dumbfounded to see the Divine Proportion pop up again in

yet another beautiful mathematical shape. Although I suspect that this was known to Kepler

centuries before this humble writer re-discovered the fact, it turns out that every face of the

Rhombic Triacontahedron is a golden rhombus, i.e., a rhombus whose diagonals are in

Divine Proportion! In case the reader dares to risk catching the “polyhedron-constructing

madness bug”, here is Khan Amore’s template for the golden rhombus used to construct the

Rhombic Triacontahedron:


49

To use the template, simply cut it out along the cyan lines and then dimensionally reinforce

it with clear tape (so it won’t stretch or distort from repeated handling, and so the holes don’t

keep getting bigger). Next, place the template onto matte board (the kind used in framing

photographs) and poke the sharp metal point of a compass through the four (circled)

vertices of the golden rhombus, taking care not to move the template as you do this.

Remove the template, find the four holes you made in the matte board, and draw four fine

lines connecting these holes to delineate the rhombus. Putting many layers of newspaper

under it first (so as to avoid cutting into the table), cut out (as precisely as you can) the

rhombus using a metal straight-edge and scalpel or sharp Exacto knife (it may take many

passes of the blade in the same groove before you finally cut all the way through the rigid

cardboard). Once you have cut out a golden rhombus, repeat the operation until you have

a total of 30 identical rhomboidal pieces of matte board, then tape them all together in mid-

edge to hold them together temporarily in the shape of the polyhedron (it helps to fold each

piece of tape over onto itself on either end to facilitate removal of the tape later.) Now apply

white glue to all vertices and exposed edges of the polyhedron. After the glue has dried on

the vertices, take off the pieces of tape which served to hold the faces together temporarily,

and apply a thick, gap-filling fillet of white glue along all of the edges and allow the glue to

dry, then repeat with a second fillet of glue, if necessary. As a finishing touch, you may wish

to spray-paint the polyhedron with a faux-granite spray. The end result is a remarkable-

looking, yet inexpensive (and light) objet d’art that looks something like this:


50

§9

The Divine Proportion’s Representation as a Continued

Fraction:

As we have seen (in Section 2) the Divine Proportion can be algebraically

derived as the solution of an extremely simple quadratic equation. As yet

another simple algebraic approach, the Golden Ratio or Divine Proportion ()

can also be expressed as a continued fraction:

47) (Equation

(etc.) ...

1

1

1

1

1

1

1

1

1

This is the simplest continuing fraction in existence, but it is also the slowest of

all continuing fractions to converge to its limit. The successive convergents of

this continuing fraction are:

1/1, 2/1, 3/2, 5/3 …

As the reader will no doubt note, the numerators and denominators of the

convergents of the continuing fraction representation of follow the

Fibonacci sequence! Will these wonders ever cease?


51

§10

Miscellaneous Properties, Including Reciprocals,

Powers, and Rational Approximations of the Divine

Proportion:

Golden Section Property #1:

Given a line segment that is sectioned into two segments which are in Divine

Proportion, if the smaller segment is marked off on the larger segment, then

the larger segment is itself divided in golden section.

Golden Section Property #2:

Given a line segment that has been sectioned into two segments which are in

Divine Proportion, if the whole segment (i.e., the sum of the two sub-

segments) is extended by the length of its larger sub-segment, then the original

segment divides the extended segment in golden section.

[With the help of the two above theorems, an arbitrary number of golden

sections can easily be constructed, from an initial golden section.]

Irrationality of the Golden Ratio, and Rational Approximations:

Because the golden ratio is based upon the square root of 5, and the square

root of five is, or course, an irrational number, it should come as no surprise

that the golden ratio is also an irrational number. However, as was pointed

out earlier, there is an intimate connection between the Divine Proportion and

the Fibonacci Number Sequence — a connection which allows the golden

ratio to be rationally-approximated to any desired degree of accuracy: The

ratio of two adjacent numbers in the Fibonacci Number series approximates

the golden ratio quite well. Indeed, the farther we go down the Fibonacci

series (i.e., the bigger the two consecutive Fibonacci Numbers used,) the

better the approximation. For example, the simple rational (Fibonacci)

approximation 377/233 approaches the golden ratio to better than a

99.99949% accuracy!


52

Reciprocal and Powers of the Golden Ratio :

If the greater segment of the golden section has a length of and the lesser

part is 1, then, by definition of the golden ratio, the following mathematical

relationship must hold true:

48) (Equation

1

1

Which may be re-written either as:

49) (Equation 1

2

Or as:

50) (Equation 1

1

In other words, is squared by adding unity, and its reciprocal is found by

subtracting unity! Indeed, the golden ratio ( = 1.61803398874989…) and

its negative reciprocal (-1/ = -0.61803398874989…) are the only two

numbers whose reciprocals are exactly one less than themselves. That is to

say:

[1.61803398874989…]

-1

= 0.61803398874989…

and

[-0.61803398874989…]

-1

= -1.61803398874989…


53

As it turns out, the higher powers of the golden ratio can also be expressed

simply in terms of :

-1

= 0.6180339887… 1() – 1 = 0.6180339887…

0

= 1.0000000000… 0() + 1 = 1.0000000000…

1

= 1.6180339887… 1() + 0 = 1.6180339887…

2

= 2.6180339887… 1() + 1 = 2.6180339887…

3

= 4.236067971… 2() + 1 = 4.236067971…

4

= 6.854101953… 3() + 2 = 6.854101953…

5

= 11.09016992… 5() + 3 = 11.09016992…

6

= 17.94427186… 8() + 5 = 17.94427186…

In the left column of the table, note that each power is not only the product of

and the previous power (for example, 1.6180339887 x 2.6180339887 =

4.236067971), but each power is also the sum of the two previous powers (in

the same example, 1.6180339887 + 2.6180339887 = 4.236067971 as well).

There is one and only one series of numbers that has this unique property,

and that is the power series of , the Divine Proportion.

Note also, in the right column of the table, that the Fibonacci Number Series

makes yet another appearance, this time in the golden ratio’s power series.

The coefficients of appear in the by-now-familiar Fibonacci sequence, (1, 1,

2, 3, 5, 8…) and so do the addends, although the two Fibonacci series are

offset by one, such that the coefficient of one power becomes the addend of

the next. Note how the inclusion of zero in the Fibonacci number sequence

makes for a seamless transition between positive and negative powers of .

This is why, at the beginning of Section 7 of this article, we included zero at

the beginning of the Fibonacci number sequence (0, 1, 1, 2, 3, 5, 8 …) when

it is usually omitted from the sequence.

These results are remarkable, perhaps even fascinating, yet in this

commentary we have only scratched the surface. There is much more that is

known about the divine proportion, and no doubt, there is much, much, more

waiting to be discovered. Those wishing to learn more about this absorbing

subject would be well advised to check out H.E. Huntley’s excellent book, The

Divine Proportion, A Study in Mathematical Beauty (Dover). And now, for

the most useful and exciting Divine Proportions ever discovered …


54

§11

Divine Proportions to be Found in Art, Architecture,

and in the Most Beautiful, Harmoniously-Proportioned

Human Bodies:

As we have already mentioned, artists both ancient and modern have based

many of their most beautiful works on the divine proportion. For example,

the Parthenon, before it had its roof blown off, had an aspect ratio of , and

the ratio in the Great Pyramid at Gizeh of the altitude of a face to half the side

of the base is also almost exactly 1.618 (i.e., the golden ratio). But did you

know that the some of the greatest Greek statues of the human form were

apparently based on the divine proportion, too? After a proportional analysis

of the human body, the Swiss-born, French-trained revolutionary architect

and artist, Charles Le Corbusier concluded that the most beautiful,

harmoniously-proportioned human bodies (or sculptural representations of

them) contain many Divine Proportions, just as the pentagram does. For

example,

It has been determined that in the most Beautiful, Harmoniously -

Proportioned Human Bodies the following mathematical relations

hold true:

1. The ratio of [the breadth of the lips] to [the breadth of the nose (at its

widest point)] is equal to .

2. The ratio of [the breadth of the lips] to [the distance between the edge

of mouth and the edge of face (nearest jaw line)] is equal to .

3. The ratio of [the distance between the top of the head and the bottom

of the chin] to [the distance between the bottom of the chin and the

navel] is equal to .

4. The ratio of [the distance between the top of the head and the navel]

to [the distance between the navel and the bottoms of the feet] is

equal to .

5. The ratio of [the distance between the suprasternal notch (i.e., bottom

of the neck) and the knees] to [the distance between the knees and

the bottom of the feet (perhaps while raised up on the toes)] is equal

to .


55

6. The ratio of [the distance between the suprasternal notch (i.e., bottom

of the neck) and the navel] to [the distance between the navel and

the perineum (or vulva)] is equal to .

7. The ratio of [the distance between the perineum (or vulva) and the

knees] to [the distance between the navel and the perineum (or

vulva)] is equal to .

8. The ratio of [the breadth of the lips] to [the breadth of each eye] is

equal to .

9. The ratio of [the breadth of the central incisors (front teeth)] to [the

breadth of the lateral incisors (smaller teeth which flank the front

teeth] is equal to .

10. The ratio of [the length of the middle phalanx of any finger] to [the

length of the distal phalanx of that same finger] is equal to .

11. The ratio of [the length of the proximal phalanx of any finger] to [the

length of the middle phalanx of that same finger] is equal to .

12. The ratio of [the distance between the level of the eyes and the top of

the upper lip] to [the distance between the top of the upper lip and

the bottom of the chin] is equal to .

13. The ratio of [the width of the shoulders] to [the width of the

narrowest part of the waist] is equal to .

14. The ratio of [the length of the forearm, from wrist to elbow tip] to [the

length of the extended hand, from the wrist to the tip of the longest

(i.e., middle) finger] also approximates .

Khan Amore’s Illustration of the Divine Proportions Which Can Be

Found in the Most Beautiful, Harmoniously-Proportioned Bodies:

When Khan Amore set forth to write this commentary, he decided that no

such article would be complete without an illustration of the plurality of divine

proportions which are to be found in Nature’s greatest work of art: the body of

a beautiful woman. Braced for many expected hardships, he set aside an

entire week for the purpose of dissecting his composite image of history’s

greatest woman (the first female mathematician), then distorting it to be in

conformity with the divine proportions listed above, finally to reassemble the

figure, and thus to see what a body based entirely upon the divine proportion

would look like. He was really quite fond of his original artwork and felt that

any change in proportions would be a change for the worse, but he was

curious to see what Hypatia (who loved the Divine Proportion) would look

like if her bodily proportions were based entirely upon this harmonious


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proportionality of Nature. To begin this process, Khan had to check the pre-

existing proportions of his photograph-based composite of Hypatia so that he

could plan what promised to be a very challenging dissection, distortion, and

reassembly. Using the Corel PhotoPaint program, he quickly created a cyan

square 1000 x 1000 pixels in dimension, and then, by changing the paper size

and color (to magenta), he enlarged one dimension of the image by 618

pixels to produce a golden rectangle measuring 1618 by 1000 pixels with a

color change occurring at the 1000 pixel line. This divine proportion scale bar

he then copied and pasted (as a movable, scalable “object”) onto his digital

composite of Hypatia. He was surprised and pleased when his first scale

measurement of the proportions of his composite of Hypatia turned out (quite

accidentally) to be exactly in divine proportion. He was relieved that at least

there would be one fewer proportion he would have to tamper with. He

moved on to check the next proportion, still braced for the worst, and, guess

what? By the time he had checked all of the proportions, he found, to his

utter delight, that every single divine proportion that he had grimly resolved to

distort into his composite of Hypatia was already there! His astonishment

could not have been greater, for he had originally constructed the figure of

Hypatia without any measurement or calculation whatsoever — simply by

scanning photographic elements and then assembling them in Frankensteinian

fashion. Khan Amore’s composite of Hypatia was put together using only his

esthetic sensibilities, with an eye to making the figure as beautiful as he could

possibly make it, and yet once again, the divine proportion popped up on its

own. Perhaps Pythagoras was right, after all: Mathematics underlies all, and

serves as the very basis of beauty.

As a final send-off, here, then, (on the next page) is Khan Amore’s conception

of the perfect woman. Even if this artist’s conception did not resemble her,

perhaps Hypatia herself would have appreciated the plurality of divine

proportions which exist in such a celebration of Nature’s greatest work of art,

the female human body:


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Amore, Khan - The Divine Proportion

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Transcript of Amore, Khan - The Divine Proportion