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Hindawi Publishing CorporationISRN GeometryVolume 2013 Article ID 379074 9 pageshttpdxdoiorg1011552013379074

Research ArticleA Rabbit Hole between Topology and Geometry

David G Glynn

CSEM Flinders University PO Box 2100 Adelaide SA 5001 Australia

Correspondence should be addressed to David G Glynn davidglynnflinderseduau

Received 10 July 2013 Accepted 13 August 2013

Academic Editors A Ferrandez J Keesling E Previato M Przanowski and H J Van Maldeghem

Copyright copy 2013 David G Glynn This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Topology and geometry should be very closely related mathematical subjects dealing with space However they deal with differentaspects the first with properties preserved under deformations and the second with more linear or rigid aspects propertiesinvariant under translations rotations or projectionsThe present paper shows away to go between them in an unexpected way thatuses graphs on orientable surfaces which already have widespread applications In this way infinitely many geometrical propertiesare found starting with the most basic such as the bundle and Pappus theorems An interesting philosophical consequence is thatthe most general geometry over noncommutative skewfields such as Hamiltonrsquos quaternions corresponds to planar graphs whilegraphs on surfaces of higher genus are related to geometry over commutative fields such as the real or complex numbers

1 Introduction

The BritishCanadian mathematician HSM Coxeter (1907ndash2003) was one of most influential geometers of the 20thcentury He learnt philosophy of mathematics from LWittgenstein at Cambridge inspired MC Escher with hisdrawings and influenced the architect R Buckminster FullerSee [1]When one looks at the cover of his book ldquoIntroductionto Geometryrdquo [2] there is the depiction of the complete graph1198705on five vertices It might surprise some people that such

a discrete object as a graph could be deemed importantin geometry However Desargues 10-point 10-line theoremin the projective plane is in fact equivalent to the graph1198705 in mathematical terms the cycle matroid of 119870

5is the

Desargues configuration in three-dimensional space and aprojection from a general point gives the configurationaltheorem in the plane Desargues theorem has long beenrecognised (by Hilbert Coxeter Russell and so on) as one ofthe foundational theorems in projective geometry Howeverthere is an unexplained gap left in their philosophies whydoes the graph give a theorem in space Certainly thematroids of almost all graphs are not theorems The onlyother example known to the author of a geometrical theoremcoming directly from a graphic matroid is the completebipartite graph119870

33 which gives the 9-point 9-plane theorem

in three-dimensional space see [3] It is interesting that both

1198705and 119870

33are minimal nonplanar (toroidal) graphs and

both lead to configurational theorems in the same mannerIn this paper we explain how virtually all basic linear

properties of projective space can be derived from graphsand topology We show that any map (induced by a graphof vertices and edges) on an orientable surface of genus 119892having V vertices 119890 edges and 119891 faces where V minus 119890 + 119891 =

2 minus 2119892 is equivalent to a linear property of projective spaceof dimension V minus 1 coordinatized by a general commutativefieldThis property is characterized by a configuration havingV+119891 points and 119890 hyperplanesThis leads to the philosophicaldeduction that topology and geometry are closely related viagraph theory If 119892 = 0 (and the graph is planar) the linearproperty is also valid for the most general projective spaceswhich are over skewfields that in general have noncommuta-tivemultiplicationThis is a powerful connection between thetopology of orientable surfaces and discrete configurationalproperties of the most general projective spaces

There are various ldquofundamentalrdquo theorems that pro-vide pathways between different areas of mathematics Forexample the fundamental theorem of projective geometry(FTPG) describes the group of automorphisms of projectivegeometries over fields or skewfields (all those of dimensionsgreater than two) as a group of nonsingular semilineartransformations This most importantly allows the choice ofcoordinate systems in well-defined ways Hence the FTPG is

2 ISRN Geometry

a pathway between projective geometry and algebra matrixand group theory

Another example is the fundamental theorem of algebraThis provides another pathway between polynomials ofdegree 119899 over the real number field and multisets of 119899 rootswhich are complex numbers It explains why the complexnumbers are important for an understanding of the realnumbers

In a similar vein we show here how our ldquorabbit holerdquobetween topology and geometry can be used to obtain thebasic properties of the most general projective geometrydirectly from topological considerations

Here is an outline of the approach

(1) Consider the properties of fundamental configura-tions in (V minus 1)-dimensional projective geometrywhich are collections of points and hyperplanes withincidences between them The most important haveV points on each hyperplane and these points form aminimal dependent set (a ldquocircuitrdquo inmatroid theory)

(2) Inmost of these configurations the algebraic propertythat corresponds to a configurational theorem is that aset of 119890 subdeterminants of size two in a general V times119891matrix over a field has a linear dependency that isthe vanishing of any 119890 minus 1 subdeterminants impliesthe vanishing of the remaining subdeterminant

(3) The condition for such a set of subdeterminants istopological the dependency amongst the subdeter-minants happens if and only if there exists a graphhaving V vertices and 119890 edges embedded on anorientable surface of genus 119892 and inducing 119891 faces(certain circuits of the graph that can be contractedto a point on the surface)

(4) A bonus is that when the surface has genus zero (iethe graph is planar) the commutative field restrictionfor the algebraic coordinates of the space can berelaxed to noncommutative skewfields including thequaternions This requires a different interpretationfor a 2 times 2 determinant and another proof dependingupon topological methods

(5) Since the latter method of planar graphs produces themain axiom for projective geometry (the bundle theo-rem or its dual Pasch axiom see [4 page 24]) and theformer one for standard determinants over commu-tative fields produces the Pappus theorem we see thatall bases are covered and a topological explanationfor standard projective geometry that is embeddableinto space of dimension greater than two is obtainedIn the case of 2-dimensional geometries (planes) thereexist non-Desarguesian projective planes so thesegeometries do not appear to be produced topologi-cally see [5 page 120] and [6 Section 23]

2 Definitions and Concepts

Let us summarize the topological and geometrical conceptsthat are used in this paper A graph is a collection of verticeswith a certain specified multiset of edges each of which is

amultiset containing two vertices If a vertex is repeated thenthe edge is a loop The graph is simple if it contains no loopand nomultiple edges edges that are repeated

An orientable surface is a surface in real three-dimensional space that can be constructed from the sphere byappending 119892 handles see [2 Section 211] This surface has 119892holes andwe say that it has genus119892 One classical use for sucha surface is to parametrize the points on an algebraic curve inthe complex plane but we have another application in mind

A skewfield or division ring is an algebraic structure(119865 + sdot) where 119865 is a set containing distinct elements 0 and1 for which (119865 +) is an abelian (ie commutative) groupwith identity 0 and (119865lowast = 119865 0 sdot) is a group (nonabelian ifthe skewfield is ldquoproperrdquo) The left and right distributive laws119886(119887 + 119888) = 119886119887+ 119886119888 and (119886 + 119887)119888 = 119886119888 + 119887119888 hold for forall119886 119887 119888 isin 119865The classical example of a proper skewfield is the quaternionsystem of Hamilton (four-dimensional over the reals) If themultiplicative group 119865lowast is abelian (ie commutative) 119865 iscalled a field Thus a field is a special case of skewfieldClassical examples of a field are the rational numbers thereal numbers and the complex numbers It is known (byWedderburnrsquos theorem and elementary field theory) that theonly finite skewfields are the Galois fields GF(119902) where 119902 is apower of a prime

A projective geometry of dimension 119899 over a skewfieldis the set of subspaces of a (left or right) vector spaceof rank 119899 + 1 over the skewfield Points are subspaces ofprojective dimension zero while hyperplanes are subspacesof projective dimension 119899 minus 1 It is well-known (or by theFTPG) that every projective space of dimension at least threehas a coordinatization involving a skewfield and comes fromthe relevant vector spaceThere are some incidence propertiesfor geometries over fields that are not valid for those over themore general skewfields For example the bundle theorem isvalid for skewfields (and fields) but Pappus 9

3theorem only

holds for geometries over fieldsIt is known that certain of the configurational theorems

are in some sense ldquoequivalentrdquo in that assuming any one ofthem implies the remaining onesThese include the theoremsof PappusMobius andGallucciThese latter theorems are allexplained by the present topological theory Desargues theo-rem and the bundle theorem (or its dual the configurationof Pasch) are also in some sense equivalent in the case of themore general geometries over skewfields see [6] We showthat the bundle theorem comes from the topology of planargraphs

An abstract configuration is a set of points and a distin-guished collection of subsets called blocks An embeddingof such a configuration is a way of putting the points intoa projective space so that each of the blocks generates ahyperplane and not the whole spaceThe point-set as a wholeshould generate the whole projective space There are severalways of thinking about embeddings (eg often theymay havemore incidences than specified by the abstract configuration)and we refer the reader to [7] for a discussion However extraincidences do not bother us here

Our configurations have blocks with all the same size 119896We say that such a configuration is a configurational theoremif for each embedding of the configuration into space of

ISRN Geometry 3

2

1

A B

A1 B

1

A2

B2

Figure 1 Graph fragment

dimension 119896 minus 1 the property that all but one of the blockslie in hyperplanes implies the same is true for the remainingblock This might hold only for spaces over fields but notgeneral skewfields as with Pappus theorem

3 Main Results

We present two main results Theorem 1 relates graphs ormaps on orientable surfaces of any genus to configurationaltheorems in general projective space over any commutativefield (such as the rational numbers real numbers complexnumbers or finite fields) This uses 2 times 2 determinants withthe standard definition However for general skewfields thisdefinition of determinant does not work and so we useLemma 2 to find an alternate way and find that there isa restriction to surfaces of genus zero Thus Theorem 4investigates the graphs ormaps on a surface of genus zero andrelates them to configurational theorems over skewfields

Theorem 1 Any graph 119866 embedded on an orientable surfaceof genus 119892 ge 0 having V vertices 119890 edges and 119891 faces whereby Eulerrsquos formula V minus 119890 +119891 = 2 minus 2119892 is equivalent to a certainconfigurational theorem (explained in the proof) in projectivespace 119875119866(V minus 1 119865) where 119865 is any commutative field

Proof Let us label the vertices of119866with the letters119860 119861 119862 in a set 119881 of cardinality V and label the faces (which arecertain circuits on the surface) with the natural numbers1 2 3 119891 Then each of the 119890 edges of the graph joinsprecisely two vertices for example119860 and 119861 and it forms partof the boundaries of precisely two faces for example 1 and2 (For simplicity we are assuming that there are no loops inboth the graph and its dual but these can easily be accountedfor in a more expansive theory) Note that the dual graph 119866119889is the graph embeddable on the same surface where we switchthe roles of vertex and face joining two faces if they have acommon edge This dual graph depends strongly upon theembedding so that a graph may have different dual graphson other surfaces see [8] for recent research on this topic

We define an abstract configuration119870 having V+119891 pointsand 119890 blocks which are subsets of V points as follows Thepoints are identified with 119881 cup 1 119891 that is the unionof the set of points and the set of faces of 119866 Additionally foreach edge 119860119861 bounded by the two faces 1 and 2 there is thecorresponding set of V points which is 11986011986112 = 119881 119860 119861 cup1 2 that is we replace 119860 and 119861 in 119881 by 1 and 2 and we callthis a block of119870

Consider any V times 119891 matrix 119872 over a field 119865 (wherethe multiplication is commutative ie 119909119910 = 119910119909 for all119909 119910 isin 119865) with rows in correspondence with the vertices of119866(119860 119861 ) and the columns in correspondence with the facesof 119866 (1 2 119891) We assume that a typical matrix elementcorresponding to vertex 119862 and face 119894 has 119898

119862119894= 119898119894119862 Thus

the subscripts are treated like unordered sets 119862 119894 For anyldquograph fragmentrdquo corresponding to an edge 119860119861 of 119866 seeFigure 1 there is a 2 times 2 submatrix of119872 in the rows 119860 and119861 and in the columns 1 and 2 The ldquoanglesrdquo 1198601 1198611 1198602 and1198612 correspond to the four positions in the submatrix whilethe determinant of this submatrix is119898

1198601sdot1198981198612minus1198981198602sdot1198981198611 In a

general embedding of119870 into PG(Vminus1 119865) wemay assume thatthe points from119881 form a basis and so are coordinatized by theunit vectors If the remaining points of 119870 had no constraintsupon them except for being embedded in PG(V minus 1 119865)they would be coordinatized by completely general (nonzero)vectors of length V and realized by the119891 columns of thematrix119872 Then the vanishing of the subdeterminant correspondingto the edge 119860119861 is found to be equivalent to the fact that the Vpoints 11986011986112 as defined above lie in a hyperplane

Since the surface of 119866 is orientable we may orient it sothat at each vertex there is an anticlockwise direction Theequivalence between cyclic graphs graphs in which there isa cyclic order at each vertex and embeddings of graphs onsuch surfaces has been discussed by many people startingapparently with Heffter [9] and later clarified by Edmonds[10] They have been given many names such as graphswith rotation systems ribbon graphs combinatorial premapsand fatgraphs see [11ndash13] Consider Figure 1 again Smallanticlockwise-oriented circles around119860 and119861 induce a largerclockwise-oriented circle going from 119860 rarr 2 rarr 119861 rarr 1 rarr

119860 Thus given any edge of 119866 containing a vertex 119862 and beingthe boundary of a face 119894 this orients the angle from vertex119862 to face 119894 or from 119894 to 119862 Denote these possibilities by 119862119894 or119894119862 respectively However such an angle occurs with preciselytwo edges and one edge gives 119862119894 and the other 119894119862

The 2times2 subdeterminant with rows119860 and119861 and columns1 and 2 may be written 119898

1119860sdot 1198982119861minus 1198981198602sdot 1198981198611 according to

the clockwise orientation (We purposely forget for a whilethat 119898

119894119862= 119898119862119894) Now the vanishing of this determinant is

equivalent to1198981119860sdot1198982119861= 1198981198602sdot1198981198611(we could call the two sides

of this equation the ldquodiagonalsrdquo of the determinant) and if allthe determinants corresponding to the edges of 119866 vanish wecan take the product over all 119890 edges on both sides to obtainΠ119860119861isin119866

1198981119860sdot1198982119861= Π119860119861isin119866

1198981198602sdot1198981198611= 119901This is clearly a trivial

identity since any angle for example 119894119862 occurs once on theleft and once (as119862119894) on the right Nowwe can assume that theldquoangle variablesrdquo 119898

119894119862are all nonzero as otherwise there will

be an unwanted hyperplane in 119870 which would not be in themost general position Then the vanishing of any 119890 minus 1 of thesubdeterminants implies the vanishing of the remaining onesince we can divide 119901 by 119890 minus 1 ldquodiagonalsrdquo 119898

1119860sdot 1198982119861

on theleft and by the corresponding 119890 minus 1 ldquodiagonalsrdquo119898

1198602sdot 1198981198611

onthe right andwe obtain the vanishing of the last determinantThis shows the theorem in the general case where 119865 is a fieldwith commutative multiplication

4 ISRN Geometry

The converse construction holds a configurational the-orem in space that relies on 2 times 2 matrices as above mustcome from a graph on an orientable surfaceTheproblem is todetermine the cyclic graph 119866 from the set of 119890 2 times 2 subdeter-minants of a matrix having the property that the vanishing ofany 119890 minus 1 of them implies that the remaining subdeterminantvanishes Around the edges of each vertex of 119866 there shouldbe an anticlockwise cyclic orientation or ldquocyclic orderrdquo If westart with a vertex119860 and an edge 119860 119861 containing it proceedto the next edge 119860 119862 in the cyclic order and using the cyclicorder at 119862 find the next edge 119862119863 and so on we shouldfollow around all the edges of a face of the embedding in aclockwise way and return to the first vertex119860 and edge 119860 119861We will show how this is achieved Now as before we canassume that the entries where the subdeterminants occurare all nonzero If the subdeterminants have the assumedproperty they can be ordered so that one ldquodiagonalrdquo of eachis selected and the product of all these selected diagonals isthe same as the product of the nonselected ones (as in the firstpart of the proof above) As before we may write the selecteddiagonals in the form 119898

1119860sdot 1198982119861

and the nonselected onesin the form 119898

1198602sdot 1198981198611 To find the graph we must associate

the rows of the matrix119872 with the vertices the columns withfaces and the subdeterminants with the edges Consider aparticular vertex 119860 of 119866 (a row of 119872) We obtain a cyclic(anticlockwise) chain of 119899 subdeterminants using that row(equivalently edges of 119866 containing 119860) as follows 119889 = 119898

1119860sdot

1198982119861minus1198981198602sdot1198981198611 119890 = 119898

2119860sdot1198983119862minus1198981198603sdot1198981198622119891 = 119898

3119860sdot1198984119863minus1198981198604sdot

1198981198633 119892 = 119898

119899119860sdot 1198981119864minus 1198981198601sdot 119898119864119899 Now we can check that

the faces of 119866 also arise from this construction Starting withthe vertex119860 and edge containing it 119889 = 119898

1119860sdot1198982119861minus1198981198602sdot1198981198611

the next edge determinant in 119860rsquos anticlockwise order from 119889is 119890 = 119898

2119860sdot 1198983119862minus 1198981198603sdot 1198981198622

which contains the vertex-row 119862 The cyclic ordering at 119862makes119898

2119862sdot 119898119896119865minus119898119862119896sdot 1198981198652

the next edge (for some vertex-row 119865 and column-face 119896)Following this sequence of subdeterminants (edges) aroundwe see that the edges surround the column-face 2 and wecan say that the cyclic ordering induced on the edges of theface in this way is clockwise So it works out similarly givenany vertex and edge containing that vertex However onemight see a minor problem with this argument In a standard(cyclic) graph 119866 there should be one cycle (of edges) at eachvertex if there are 119909

119903cycles determined by a row 119860 of 119872

we ldquosplitrdquo that row into 119909119903distinct rows one for each disjoint

cycle of subdeterminants with 119860 Similarly we look at eachcolumn 119888 and there will be 119910

119888disjoint cycles on the rows

induced by the subdeterminants with that column Splittingthat column into 119910

119888distinct columns will enable us to look at

a larger matrix with the same number of subdeterminantsbut with each row and column corresponding to a uniquecycle Subdeterminants in different cycles will not have rowsor columns in common Then the graph on the orientablesurface has Σ

119903119909119903vertices and Σ

119888119910119888faces The other way

around given a set of 2 times 2 determinants with our specialproperty if we collapse thematrix by identifying certain rowsor columns then the property is retained as long as we donot identify two rows or columns belonging to the samesubdeterminant By this process cycles of subdeterminantscan be created with the same row or column Geometrically

it is the same as creating a new geometrical theorem byidentifying points or hyperplanes However these examplescan then be expanded out again by splitting the rows orcolumns into bigger collections of rows or columns as aboveand the pattern of subdeterminants in the largest matrix iscanonical up to permutations of rows and columns So we seehow to get around this minor problem in the proof

What kind of configurational theorems119870 corresponds tographs on orientable surfaces One obvious condition is thatthe configurationmust have V +119891 points in PG(Vminus1 119865)Thereare 119890 hyperplanes or blocks in 119870 each containing V pointsMore importantly there should be a subset119881 of V points in119870such that each hyperplane of119870 contains precisely Vminus2 pointsof 119881 and two others

Now we explain the noncommutative case which isrelated to planar graphs

Lemma 2 Let 119865 be a skewfield with perhaps noncommutativemultiplication The condition that a set of V points of 119875119866(V minus1 119865) consisting of 119860 119861 and the unit vectors 119890

3 119890V is

contained in a hyperplane is a ldquocyclic identityrdquo 119886minus1119887119888minus1119889 = 1where ( 119886 119887

119889 119888) is a certain 2 times 2 matrix over 119865 (Here we are

assuming a ldquogenericrdquo case where all the 119886 119887 119888 119889 are nonzero)

Proof A point of PG(V minus 1 119865) is a nonzero column vectorwith V coefficients from 119865 that are not all zero Two of thesecolumn vectors y and z give the same point if one can find anonzero element 119891 isin 119865 such that y = z119891 The hyperplanes ofPG(V minus 1 119865) can be coordinatized by row vectors of lengthV over 119865 in a similar way to the points Then a point y iscontained in a hyperplane h if and only if hy = 0 (h is a rowand y is a column vector) Notice that here we aremultiplyingpoints on the left (and hyperplanes on the right) Thus wemust restrict ourselves to operations on the points of PG(V minus1 119865) that act on the left A square V times V matrix is ldquosingularrdquo(and its column points are in a hyperplane) if and only if itcannot be row-reduced (bymultiplying on the left by a squarematrix) to the identity matrix or equivalently it can be rowreduced so that a zero row appears In our situation we havea V times Vmatrix that consists of V minus 2 different unit vectors anda 2 times 2 two submatrix 119883 = ( 119886 119887

119889 119888) (with 119886 119887 119888 119889 all nonzero)

in the remaining part row disjoint from the ones of the unitvectors We can then restrict our row reductions to the tworows of119883 and we see that the whole matrix is singular if andonly if119883 is singular It is still not possible to use the ordinarydeterminant to work out if 119883 is singular But assuming thatboth 119886 and 119889 are nonzero we may multiply the first row by119886minus1 and the second by 119889minus1 This leaves us with the matrix

(1 119886minus1119887

1 119889minus1119888) (1)

and the condition for singularity of this matrix is clearly119886minus1119887 = 119889

minus1119888 as then we can further row-reduce to obtain a

zero row This gives the ldquocyclic conditionrdquo 119886minus1119887119888minus1119889 = 1 (=119889119886minus1119887119888minus1= 119888minus1119889119886minus1119887 = 119887119888

minus1119889119886minus1) if 119888 is also nonzero

Note that 119886minus1119887 = 119889minus1119888 does not imply that 119886119887minus1119888119889minus1 =1 equivalently transposing a general 2 times 2 matrix over

ISRN Geometry 5

a skewfield does not always preserve its singularity There isquite a lot of theory about determinants for skewfields seefor example [14 15] but we can have a more elementaryapproach here since we only deal with 2times2 subdeterminants

This leads us to consider a special type of planar graphthat has cyclic identities at each vertex It is well known thatany planar graphwith an even number of edges on each face isbipartite see for example [8] By dualizing this statement wealso know that any planar graphwhich is Eulerian that is hasan even valency at each vertex has a bipartite dual What thismeans is that the edges of such a planar Eulerian graph maybe oriented so that the edges on each face go in a clockwise orin an anticlockwise direction Then if we travel around anyvertex in a clockwise direction the edges alternate going outand into the vertex We call such an orientation Eulerian

In general an Eulerian orientation of a graph having evenvalency at each vertex is an orientation of each edge (putan arrow on the edge) such that there are equal numbers ofedges going out or into each vertex For the above embeddingin the plane we find a natural Eulerian orientation that isdetermined by the faces

Lemma 3 Consider a planar graph 119867 with a bipartite dualhaving its Eulerian orientation of the edges Then there is non-commutative cyclic identity with variables over any skewfieldat each vertex and any one of these cyclic identities is impliedby the remaining cyclic identities

Proof Consider the list of edges 119864 and for each 119890 isin 119864 let 119890 =(119860 119861) where the Eulerian orientation goes fromvertex119860 on 119890to vertex119861 on 119890The ldquocyclic identityrdquo at vertex119860 is of the form119909minus1

1198901

sdot 1199091198902

sdot sdot sdot 119909minus1

1198902119889minus1

sdot 1199091198902119889

= 1 where the edges of the graph on119860are (in the clockwise ordering around119860) 119890

1 1198902 119890

2119889 where

1198901= (119860 119861) 119890

2= (119862 119860) 119890

3= (119860119863) 119890

2119889= (119883119860) Note

that if we had have started with any other edge for example1198903 going out from 119860 we would have obtained an equivalent

identity since by multiplying both sides on the left by 119909minus11198902

1199091198901

and then both sides on the right by 119909minus11198901

1199091198902

we obtain

119909minus1

1198901

sdot 1199091198902


1198902119889minus1

sdot 1199091198902119889

= 1

997904rArr 119909minus1

1198903

sdot 1199091198904


1198902119889minus1

sdot 1199091198902119889

= 119909minus1

1198902

1199091198901

997904rArr 119909minus1

1198903

sdot 1199091198904


1198902119889minus1

sdot 1199091198902119889

sdot 119909minus1

1198901

sdot 1199091198902

= 1

(2)

Now consider any face of the graph with its clockwise oranticlockwise orientation If it has 119899 vertices (in the cyclicorder labelled 119860

1 119860

119899) then there are 119899 cyclic identities

attached Consider the operation of collapsing the face downto a single vertex and erasing all the edges of the face Thecyclic identities can be multiplied in the cyclic order so that anew cyclic identity is obtained If a loop having adjacent insand outs at a vertex appears then itmay be safely purged fromthe graph since there can be no holes in the surface and sincein the cyclic identity at the vertex the edge variable will cancelwith itself The new collapsed graph has cyclic identities thatderive from the larger graph By continuing this process weobtain eventually a planar graph with two vertices 119860 and 119861joined by an even number 2119889 of edges If the cyclic identity at

1

24

3

A

B

C

D

Figure 2 The tetrahedron (graph of the bundle theorem) in theplane

119860 is119909minus11sdot1199092sdot sdot sdot 119909minus1

2119889minus1sdot1199092119889= 1 with the odd edges directed from

119860 to 119861 and the even edges from 119861 to 119860 then the clockwiseorder at 119861 will be the reverse of that at 119860 and so the cyclicidentity at 119861 will be 119909minus1

2119889sdot 1199092119889minus1


2sdot 1199091= 1 which is the

inverse identity to that at 119860 and so equivalent to it Hencethe dependency among all the cyclic identities of the originalgraph is established

Theorem 4 Any graph 119866 embedded on an orientable surfaceof genus 119892 = 0 having V vertices 119890 edges and 119891 faceswhere by Eulerrsquos formula V minus 119890 + 119891 = 2 is equivalent to aconfigurational theorem in projective space 119875119866(Vminus1 119865) where119865 is any skewfield or field

Proof First we construct the configuration119870 from the graph119866 in precisely the same manner as Theorem 1

When the graph119866 is embedded in any orientable surfacewhich in the present case is now the plane (or the sphere)there is a natural cyclic structure at each vertexWe now go toa graph that is intermediate between119866 and its dual119866119889This iscalled the ldquomedialrdquo graph119872(119866) and it has V1015840 = 119890 vertices and1198911015840= V + 119891 faces It is 4-regular in that every vertex is joined

to four others Since each edge has two vertices it is easy tosee that the medial graph has 1198901015840 = 2V1015840 edges Notice that sinceVminus119890+119891 = 2minus2119892 (Eulerrsquos formula) we have in themedial graphwith V1015840minus1198901015840+1198911015840 = V1015840minus2V1015840+1198911015840 = 1198911015840minusV1015840 = V+119891minus119890 = 2minus2119892 it isclear the medial graph is also embedded on the same surfaceas 119866

For example if 119866 is the planar tetrahedral graph ofFigure 2 then 119872(119866) is the planar octahedral graph havingsix vertices and eight faces

In detail the set of vertices of 119872(119866) is V119860119861

|

119860119861 edge of 119866 and V119860119861

is joined with V119861119862

in119872(119866)when119860119861and 119861119862 are adjacent to the same face 119891 of 119866 on the surfacethey are also adjacent in the cyclic order at 119861 and in that of 119891The dual of this medial graph is always bipartite so that thereare two types of faces corresponding to the vertices and to thefaces of the original graph 119866 (Conversely a 4-regular graphon an orientable surface for which the dual graph is bipartiteis easily seen to be themedial graph of a unique graph on thatsurface)

Consider Figure 1 and adjoin 119862 and 119863 which are thevertices in119866 adjacent to119860 on the boundaries of faces 1 and 2respectively and adjoin 119864 and 119865 which are the vertices

6 ISRN Geometry

Table 1 A table of five geometrical theorems

Name Graph V 119890 119891 Dual Surface 119892 Prsquos Hrsquos SpaceBundleThm 119870

4 4 6 4 119870

4Plane 0 8 6 PG (3119867)

Pappus 93Thm 3119870

3 3 9 6 119870

33Torus 1 9 9 PG (2 119865)

Mobius 84Thm 2119862

4 4 8 4 2119862

4Torus 1 8 8 PG (3 119865)

Other 84Thm 119870

4+ 2119890 4 8 4 119870

4+ 2119890 Torus 1 8 8 PG (3 119865)

GalluccirsquosThm 21198704 4 12 8 Cube Torus 1 12 12 PG (3 119865)

1

2 3

4

A

B C

D

(a)

AB12

AC13

AD14

BD24

BC23

CD34

(b)

Figure 3The bundle theorem in 3d space and its dual Pasch axiom

adjacent to 119861 on the boundaries of faces 2 and 1 We see thatV119860119861

is joined in the medial graph119872(119866)with the four verticesV119860119862 V119860119863 V119861119864 and V

119861119865in the clockwise direction Notice that

these edges of119872(119866) are in bijective correspondence with theldquoanglesrdquo 1198601 1198602 1198612 1198611 respectively Also as in the proof ofTheorem 1 the selection of ldquodiagonalsrdquo of the determinants1198981119860sdot 1198982119861minus 1198981198602sdot 1198981198611

at each edge implies that we canorientate the edge (V

119860119861 V119860119862) in119872(119866) and label it with 119898minus1

1119860

similarly the directed edge (V119860119861 V119860119864) is labelled119898minus1

2119861Then the

remaining unselected diagonal of the determinant gives twoedges of119872(119866) directed the other way (V

119860119863 V119860119861) is labelled

1198981198602

and (V119861119865 V119860119861) is labelled119898

1198611 Repeating this for all edges

of 119866 we obtain an Eulerian orientation and each vertex of119872(119866) corresponds to a cyclic identity with four variableswhich is equivalent to the determinant condition For theedge119860119861 above the ldquocyclicrdquo identity is119898minus1

1119860sdot1198981198602sdot119898minus1

2119861sdot1198981198611= 1

Applying Lemma 3 to the medial graph119872(119866)we see thatthe final cyclic identity is dependent upon the others and sowe have proved that119870 is a configurational theorem for everyskewfield and therefore also for every field

4 Examples of Configurational Theorems

If a graph on an orientable surface 119878 gives a configurationaltheorem 119870 then the dual graph on 119878 gives a configurationaltheorem that is the matroid dual of 119870 It corresponds to thesimple process of transposing the V times 119891matrix119872 containingthe subdeterminants in the construction

Table 1 summarizes the five examples of this section

41 The Bundle Theorem The bundle theorem in three-dimensional projective space is a theorem of eight points andsix planes See Figure 3

The bundle theorem states that if four lines are such thatfive of the unordered pairs of the lines are coplanar then so isthe final unordered pair Translating this to a theorem aboutpoints and planes we can define a line as the span of a pair ofdistinct points Thus the lines correspond to pairs of pointsand the theorem is about eight points and six planes It turnsout that the configuration is in three-dimensional space andthe four lines must be concurrent

The dual in terms of points and lines is that if four linesin space have five intersections in points then so is the sixthintersectionThen all the lines are coplanarThis is the ldquoAxiomof Paschrdquo see for example [4] and it is one of the fundamen-tal axioms from which all the other basic properties derive

Comparing Figure 2 with Figure 3 the bundle theoremis seen to be the configurational theorem that arises fromthe tetrahedral graph or equivalently the complete graph 119870

4

embedded in the planeRelating this to the proof ofTheorem 4 the medial graph

of 1198704is the octahedral graph having six vertices and eight

faces Thus the theorem shows that the bundle theoremis valid for all projective geometries of dimension at leastthreeThis leads to the philosophic conclusion that projectivegeometry and our perceptions of linear geometry may havetopological origins

It is noted that the dual graph of the octahedral graph (inthe plane) is the cube which has eight square faces and sixvertices

The six blocks of four points obtained from the edges ofthe graph are

11986011986134 = 11986211986334

11986011986224 = 11986111986324

11986011986323 = 11986111986223

11986111986214 = 11986011986314

11986111986313 = 11986011986213

11986211986312 = 11986011986112

(3)

The eight points of this ldquobundlerdquo theorem in 3d spaceare members of the set 119860 119861 119862119863 1 2 3 4 while the sixblocks (contained in planes) are in correspondence with thesix edges of the 119870

4graph (the tetrahedron) see Figure 2

In the Pasch configuration on the right of Figure 3 thereare again four lines whichwe could label1198601 1198612 11986231198634 Eachpair of lines intersect in a point for example 1198601 and 1198612intersect in the point labelled 11986011198612 The intersection of the

ISRN Geometry 7

A

A

A

A

B

C

(a)

1

6 2

4 3

6

15

34

(b)

Figure 4 The toroidal Pappus graph 31198623and its dual 119870

33

final pair of lines 1198612 and 1198623 is a consequence of the otherintersections So we verify that the geometric dual of thebundle theorem is the Pasch configuration

42 The Pappus Theorem The nine points of the Pappus 93

configurational theorem in the plane are members of the set119860 119861 119862 1 2 3 4 5 6 while the nine blocks (contained inlines when the configuration is embedded in the plane) arein correspondence with the nine edges of the 3119862

3graph see

Figure 4The nine blocks obtained from the edges of the graph are

11986011986114 = 11986214

11986011986126 = 11986226

11986011986135 = 11986235

11986111986216 = 11986016

11986111986225 = 11986025

11986111986234 = 11986034

11986211986015 = 11986115

11986211986024 = 11986124

11986211986036 = 11986136

(4)

There are many references for this configuration whichdates back to Pappus of Alexandria circa 330 CE see [2 35 16ndash18] Perhaps the easiest way to construct it in the planeis first to draw any two lines Put three points on each andconnect them up with six lines in the required manner seeFigure 5

43 The Mobius Theorem The eight points of the Mobius84configurational theorem in 3d space are members of the

set 119860 119861 119862119863 1 2 3 4 while the eight blocks (contained inplanes when the configuration is in 3d space) are in corre-spondence with the eight edges of the 2119862

4graph see Figure 6

3 6

41

2 5A

B

C

Figure 5 The Pappus theorem derived from the toric map

A

A

A

A

B

B

CDD

(a)

1 2

34

(b)

Figure 6 The toroidal Mobius graph 21198624and its dual 2119862

4

The eight blocks obtained from the edges of the graph are

11986011986141 = 11986211986341

11986011986123 = 11986211986323

11986111986212 = 11986011986312

11986111986234 = 11986011986334

11986211986323 = 11986011986123

11986211986341 = 11986011986141

11986311986034 = 11986111986234

11986311986012 = 11986111986212

(5)

There are many references for this configuration see[2 3 5 16ndash20] Perhaps the easiest way to construct thisconfiguration in space is to first construct a 4times4 grid of eightlines see Figure 7 The eight ldquoMobiusrdquo points can be eightpoints grouped in two lots of four as in the figure The planesthen correspond to the remaining eight points on the gridA recent observation by the author [21] is that one can findthree four by four matrices with the same 16 variables suchthat their determinants sum to zero and it is closely related tothe fact that there are certain three quadratic surfaces in spaceassociated with this configuration See [16] for a discussion ofthe three quadrics

44 The Non-Mobius 84Configurational Theorem The eight

points of the ldquootherrdquo 84configurational theorem in 3d space

can be abstractly considered to be the members of the set

0 = 119860 2 = 119861 4 = 119862 6 = 119863 1 3 5 7 (6)

8 ISRN Geometry

1 2

34

A B

CD

Figure 7 The Mobius 84configuration on eight lines

0

2 2

6

6

4

(a)

3

57

1

(b)

Figure 8 The toroidal graph 1198704+ 2119890 and its dual 119870

4+ 2119890

while the eight blocks (contained in planes when the config-uration is embedded in 3d space) are in correspondence withthe eight edges of the 119870

4+ 2119890 graph which has four vertices

it can be constructed as the complete graph on four verticesplus two other nonadjacent edges


11986211986314 = 0215 = 0125

11986011986213 = 2613 = 1236

11986011986337 = 2437 = 2347

11986111986335 = 0435 = 3450

11986011986115 = 4615 = 4561

11986011986257 = 2657 = 5672

11986111986237 = 0637 = 6703

11986111986317 = 0417 = 7014

(7)

The standard cyclic representation of this configuration isthat the points are the integersmodulo eight while the blocksare the subsets 0 1 2 5 + 119894 (mod 8) see Glynn [3] andFigure 8 Aswith theMobius configuration the configurationcan always be constructed on a 4times4 grid of lines see Figure 9The planes then correspond to the remaining eight points onthe grid

45 The Gallucci Theorem Consider Figures 10 and 11 Thetwelve points of the Gallucci configuration in 3d space

0

2

4

6

1

3

5

7

Figure 9 The other 84configuration on eight lines

A

A

B

B

C

C

D

(a)

1

2

3 5

6

7

4

4

4

8

8

8

(b)

Figure 10The toroidal Gallucci graph 21198704and its dual cube graph

are 119860 119861 119862119863 1 8 while the twelve blocks (containedin planes when the configuration is in 3d space) are incorrespondence with the twelve points on the 4 times 4 gridother than119860 119861 119862119863 Note that we are representing the torusas a hexagon with opposite sides identified This is just analternative to the more common representation of the torusas a rectangle with opposite sides identified The arrows onthe outside of the hexagons show the directions for which theidentifications are applied (The hexagonsrsquo boundaries are notgraph edges)

Another thing to note is that the only place the authorhas seen the name ldquoGalluccirdquo attached to this configurationis in the works of Coxeter see [2 Section 148] The theoremappears in Bakerrsquos book [5 page 49] which appeared in itsfirst edition in 1921 well before Galluccirsquos major work of 1928see [18] Due to its fairly basic nature it was obviously knownto geometers of the 19th century However in deference toCoxeter we are calling it ldquoGalluccirsquos theoremrdquo

ISRN Geometry 9

1 2 3 4

5

6

7

8

A

B

C

D

Figure 11 The Gallucci theorem of eight lines in 3d space

The Gallucci configuration is normally thought of as acollection of eight lines but here we are obtaining it fromcertain subsets of points and planes related to it One set offour mutually skew lines is generated by the pairs of points1198601 1198612 11986231198634 and the other set of four lines by the four pairs1198605 1198616 11986271198638

The twelve blocks obtained from the edges of the graphare

11986211986325 = 11986011986125 11986111986335 = 11986011986235 11986111986245 = 11986011986345

11986011986336 = 11986111986236 11986011986246 = 11986111986346 11986011986147 = 11986211986347

11986211986316 = 11986011986116 11986111986317 = 11986011986217 11986111986218 = 11986011986318

11986011986327 = 11986111986227 11986011986228 = 11986111986328 11986011986138 = 11986211986338

(8)

Some practical considerations remain small graphs maydetermine relatively trivial properties of space but we haveseen in our examples that many graphs correspond tofundamental and nontrivial properties We also obtain anautomatic proof for these properties just from the embeddingonto the surface For some graphs on orientable surfacesthe constructed geometrical configuration must collapse intosmaller dimensions upon embedding into space or havepoints or hyperplanes that mergeThis is a subject for furtherinvestigation

References

[1] S Lavietes New York Times obituary 2003 httpwwwny-timescom20030407worldharold-coxeter-96-who-found-profound-beauty-in-geometryhtml

[2] H S M Coxeter Introduction to Geometry JohnWiley amp SonsNew York NY USA 1961

[3] D G Glynn ldquoTheorems of points and planes in three-dimens-ional projective spacerdquo Journal of the Australian MathematicalSociety vol 88 no 1 pp 75ndash92 2010

[4] P Dembowski Finite Geometries vol 44 of Ergebnisse derMathematik und ihrer Grenzgebiete Springer New York NYUSA 1968

[5] H F Baker Principles of Geometry vol 1 Cambridge UniversityPress London UK 2nd edition 1928

[6] D Hilbert Grundlagen der Geometrie Gottingen 1899

[7] D G Glynn ldquoA note on 119873119870

configurations and theoremsin projective spacerdquo Bulletin of the Australian MathematicalSociety vol 76 no 1 pp 15ndash31 2007

[8] S Huggett and I Moffatt ldquoBipartite partial duals and circuits inmedial graphsrdquo Combinatorica vol 33 no 2 pp 231ndash252 2013

[9] L Heffter ldquoUeber das Problem der NachbargebieterdquoMathema-tische Annalen vol 38 no 4 pp 477ndash508 1891

[10] J R Edmonds ldquoA combinatorial representation for polyhedralsurfacesrdquo Notices of the American Mathematical Society vol 7article A646 1960

[11] B Bollobas and O Riordan ldquoA polynomial invariant of graphson orientable surfacesrdquo Proceedings of the LondonMathematicalSociety vol 83 no 3 pp 513ndash531 2001

[12] B Bollobas and O Riordan ldquoA polynomial of graphs onsurfacesrdquo Mathematische Annalen vol 323 no 1 pp 81ndash962002

[13] G A Jones and D Singerman ldquoTheory of maps on orientablesurfacesrdquo Proceedings of the London Mathematical Society vol37 no 2 pp 273ndash307 1978

[14] J Dieudonne ldquoLes determinants sur un corps non commutatifrdquoBulletin de la Societe Mathematique de France vol 71 pp 27ndash451943

[15] I Gelfand S Gelfand V Retakh and R L Wilson ldquoQuasi-determinantsrdquo Advances in Mathematics vol 193 no 1 pp 56ndash141 2005

[16] W Blaschke Projektive Geometrie Birkhauser Basel Switzer-land 3rd edition 1954

[17] H S M Coxeter ldquoSelf-dual configurations and regular graphsrdquoBulletin of the American Mathematical Society vol 56 pp 413ndash455 1950

[18] G Gallucci Complementi di geometria proiettiva Contributoalla geometria del tetraedro ed allo studio delle configurazioniUniversita degli Studi di Napoli Napoli Italy 1928

[19] A F Moebius ldquoKann von zwei dreiseitigen Pyramiden einejede in Bezug auf die andere um- und eingeschrieben zugleichheissenrdquo Crellersquos Journal fur die reine und angewandte Mathe-matik vol 3 pp 273ndash278 1828

[20] A F Moebius ldquoKann von zwei dreiseitigen Pyramiden einejede in Bezug auf die andere um- und eingeschrieben zugleichheissenrdquo Gesammelte Werke vol 1 pp 439ndash446 1886

[21] D G Glynn ldquoA slant on the twisted determinants theoremrdquoSubmitted to Bulletin of the Institute of Combinatorics and ItsApplications

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 ISRN Geometry

a pathway between projective geometry and algebra matrixand group theory

Another example is the fundamental theorem of algebraThis provides another pathway between polynomials ofdegree 119899 over the real number field and multisets of 119899 rootswhich are complex numbers It explains why the complexnumbers are important for an understanding of the realnumbers

In a similar vein we show here how our ldquorabbit holerdquobetween topology and geometry can be used to obtain thebasic properties of the most general projective geometrydirectly from topological considerations

Here is an outline of the approach

(1) Consider the properties of fundamental configura-tions in (V minus 1)-dimensional projective geometrywhich are collections of points and hyperplanes withincidences between them The most important haveV points on each hyperplane and these points form aminimal dependent set (a ldquocircuitrdquo inmatroid theory)

(2) Inmost of these configurations the algebraic propertythat corresponds to a configurational theorem is that aset of 119890 subdeterminants of size two in a general V times119891matrix over a field has a linear dependency that isthe vanishing of any 119890 minus 1 subdeterminants impliesthe vanishing of the remaining subdeterminant

(3) The condition for such a set of subdeterminants istopological the dependency amongst the subdeter-minants happens if and only if there exists a graphhaving V vertices and 119890 edges embedded on anorientable surface of genus 119892 and inducing 119891 faces(certain circuits of the graph that can be contractedto a point on the surface)

(4) A bonus is that when the surface has genus zero (iethe graph is planar) the commutative field restrictionfor the algebraic coordinates of the space can berelaxed to noncommutative skewfields including thequaternions This requires a different interpretationfor a 2 times 2 determinant and another proof dependingupon topological methods

(5) Since the latter method of planar graphs produces themain axiom for projective geometry (the bundle theo-rem or its dual Pasch axiom see [4 page 24]) and theformer one for standard determinants over commu-tative fields produces the Pappus theorem we see thatall bases are covered and a topological explanationfor standard projective geometry that is embeddableinto space of dimension greater than two is obtainedIn the case of 2-dimensional geometries (planes) thereexist non-Desarguesian projective planes so thesegeometries do not appear to be produced topologi-cally see [5 page 120] and [6 Section 23]

2 Definitions and Concepts

Let us summarize the topological and geometrical conceptsthat are used in this paper A graph is a collection of verticeswith a certain specified multiset of edges each of which is

amultiset containing two vertices If a vertex is repeated thenthe edge is a loop The graph is simple if it contains no loopand nomultiple edges edges that are repeated

An orientable surface is a surface in real three-dimensional space that can be constructed from the sphere byappending 119892 handles see [2 Section 211] This surface has 119892holes andwe say that it has genus119892 One classical use for sucha surface is to parametrize the points on an algebraic curve inthe complex plane but we have another application in mind

A skewfield or division ring is an algebraic structure(119865 + sdot) where 119865 is a set containing distinct elements 0 and1 for which (119865 +) is an abelian (ie commutative) groupwith identity 0 and (119865lowast = 119865 0 sdot) is a group (nonabelian ifthe skewfield is ldquoproperrdquo) The left and right distributive laws119886(119887 + 119888) = 119886119887+ 119886119888 and (119886 + 119887)119888 = 119886119888 + 119887119888 hold for forall119886 119887 119888 isin 119865The classical example of a proper skewfield is the quaternionsystem of Hamilton (four-dimensional over the reals) If themultiplicative group 119865lowast is abelian (ie commutative) 119865 iscalled a field Thus a field is a special case of skewfieldClassical examples of a field are the rational numbers thereal numbers and the complex numbers It is known (byWedderburnrsquos theorem and elementary field theory) that theonly finite skewfields are the Galois fields GF(119902) where 119902 is apower of a prime

A projective geometry of dimension 119899 over a skewfieldis the set of subspaces of a (left or right) vector spaceof rank 119899 + 1 over the skewfield Points are subspaces ofprojective dimension zero while hyperplanes are subspacesof projective dimension 119899 minus 1 It is well-known (or by theFTPG) that every projective space of dimension at least threehas a coordinatization involving a skewfield and comes fromthe relevant vector spaceThere are some incidence propertiesfor geometries over fields that are not valid for those over themore general skewfields For example the bundle theorem isvalid for skewfields (and fields) but Pappus 9

3theorem only

holds for geometries over fieldsIt is known that certain of the configurational theorems

are in some sense ldquoequivalentrdquo in that assuming any one ofthem implies the remaining onesThese include the theoremsof PappusMobius andGallucciThese latter theorems are allexplained by the present topological theory Desargues theo-rem and the bundle theorem (or its dual the configurationof Pasch) are also in some sense equivalent in the case of themore general geometries over skewfields see [6] We showthat the bundle theorem comes from the topology of planargraphs

An abstract configuration is a set of points and a distin-guished collection of subsets called blocks An embeddingof such a configuration is a way of putting the points intoa projective space so that each of the blocks generates ahyperplane and not the whole spaceThe point-set as a wholeshould generate the whole projective space There are severalways of thinking about embeddings (eg often theymay havemore incidences than specified by the abstract configuration)and we refer the reader to [7] for a discussion However extraincidences do not bother us here

Our configurations have blocks with all the same size 119896We say that such a configuration is a configurational theoremif for each embedding of the configuration into space of

ISRN Geometry 3

2

1

A B

A1 B

1

A2

B2



3 Main Results






119862119894= 119898119894119862 Thus












1198981119860sdot1198982119861= Π119860119861isin119866





1119860sdot 1198982119861


1198602sdot 1198981198611


4 ISRN Geometry


1119860sdot 1198982119861




1119860sdot

1198982119861minus1198981198602sdot1198981198611 119890 = 119898

2119860sdot1198983119862minus1198981198603sdot1198981198622119891 = 119898

3119860sdot1198984119863minus1198981198604sdot

1198981198633 119892 = 119898



1119860sdot1198982119861minus1198981198602sdot1198981198611


2119860sdot 1198983119862minus 1198981198603sdot 1198981198622


2119862sdot 119898119896119865minus119898119862119896sdot 1198981198652









119903119909119903vertices and Σ







3 119890V is





119889 119888) (with 119886 119887 119888 119889 all nonzero)


(1 119886minus1119887

1 119889minus1119888) (1)






ISRN Geometry 5






1198901

sdot 1199091198902


1198902119889minus1

sdot 1199091198902119889


1 1198902 119890

2119889 where

1198901= (119860 119861) 119890

2= (119862 119860) 119890

3= (119860119863) 119890

2119889= (119883119860) Note



1199091198901


1199091198902

we obtain

119909minus1

1198901

sdot 1199091198902


1198902119889minus1

sdot 1199091198902119889

= 1

997904rArr 119909minus1

1198903

sdot 1199091198904


1198902119889minus1

sdot 1199091198902119889

= 119909minus1

1198902

1199091198901

997904rArr 119909minus1

1198903

sdot 1199091198904


1198902119889minus1

sdot 1199091198902119889

sdot 119909minus1

1198901

sdot 1199091198902

= 1

(2)


1 119860



1

24

3

A

B

C

D





2119889sdot 1199092119889minus1










|

119860119861 edge of 119866 and V119860119861




6 ISRN Geometry



4 4 6 4 119870

4Plane 0 8 6 PG (3119867)


3 3 9 6 119870

33Torus 1 9 9 PG (2 119865)


4 4 8 4 2119862

4Torus 1 8 8 PG (3 119865)

Other 84Thm 119870

4+ 2119890 4 8 4 119870

4+ 2119890 Torus 1 8 8 PG (3 119865)


1

2 3

4

A

B C

D

(a)

AB12

AC13

AD14

BD24

BC23

CD34

(b)








1119860


2119861Then the


119860119863 V119860119861) is labelled

1198981198602




1119860sdot1198981198602sdot119898minus1

2119861sdot1198981198611= 1









4






11986011986134 = 11986211986334

11986011986224 = 11986111986324

11986011986323 = 11986111986223

11986111986214 = 11986011986314

11986111986313 = 11986011986213

11986211986312 = 11986011986112

(3)




ISRN Geometry 7

A

A

A

A

B

C

(a)

1

6 2

4 3

6

15

34

(b)


33




3graph see


11986011986114 = 11986214

11986011986126 = 11986226

11986011986135 = 11986235

11986111986216 = 11986016

11986111986225 = 11986025

11986111986234 = 11986034

11986211986015 = 11986115

11986211986024 = 11986124

11986211986036 = 11986136

(4)




4graph see Figure 6

3 6

41

2 5A

B

C


A

A

A

A

B

B

CDD

(a)

1 2

34

(b)


4


11986011986141 = 11986211986341

11986011986123 = 11986211986323

11986111986212 = 11986011986312

11986111986234 = 11986011986334

11986211986323 = 11986011986123

11986211986341 = 11986011986141

11986311986034 = 11986111986234

11986311986012 = 11986111986212

(5)





0 = 119860 2 = 119861 4 = 119862 6 = 119863 1 3 5 7 (6)

8 ISRN Geometry

1 2

34

A B

CD


0

2 2

6

6

4

(a)

3

57

1

(b)


4+ 2119890





11986211986314 = 0215 = 0125

11986011986213 = 2613 = 1236

11986011986337 = 2437 = 2347

11986111986335 = 0435 = 3450

11986011986115 = 4615 = 4561

11986011986257 = 2657 = 5672

11986111986237 = 0637 = 6703

11986111986317 = 0417 = 7014

(7)



0

2

4

6

1

3

5

7


A

A

B

B

C

C

D

(a)

1

2

3 5

6

7

4

4

4

8

8

8

(b)




ISRN Geometry 9

1 2 3 4

5

6

7

8

A

B

C

D




11986211986325 = 11986011986125 11986111986335 = 11986011986235 11986111986245 = 11986011986345

11986011986336 = 11986111986236 11986011986246 = 11986111986346 11986011986147 = 11986211986347

11986211986316 = 11986011986116 11986111986317 = 11986011986217 11986111986218 = 11986011986318

11986011986327 = 11986111986227 11986011986228 = 11986111986328 11986011986138 = 11986211986338

(8)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









ISRN Geometry 3

2

1

A B

A1 B

1

A2

B2



3 Main Results






119862119894= 119898119894119862 Thus












1198981119860sdot1198982119861= Π119860119861isin119866





1119860sdot 1198982119861


1198602sdot 1198981198611


4 ISRN Geometry


1119860sdot 1198982119861




1119860sdot

1198982119861minus1198981198602sdot1198981198611 119890 = 119898

2119860sdot1198983119862minus1198981198603sdot1198981198622119891 = 119898

3119860sdot1198984119863minus1198981198604sdot

1198981198633 119892 = 119898



1119860sdot1198982119861minus1198981198602sdot1198981198611


2119860sdot 1198983119862minus 1198981198603sdot 1198981198622


2119862sdot 119898119896119865minus119898119862119896sdot 1198981198652









119903119909119903vertices and Σ







3 119890V is





119889 119888) (with 119886 119887 119888 119889 all nonzero)


(1 119886minus1119887

1 119889minus1119888) (1)






ISRN Geometry 5






1198901

sdot 1199091198902


1198902119889minus1

sdot 1199091198902119889


1 1198902 119890

2119889 where

1198901= (119860 119861) 119890

2= (119862 119860) 119890

3= (119860119863) 119890

2119889= (119883119860) Note



1199091198901


1199091198902

we obtain

119909minus1

1198901

sdot 1199091198902


1198902119889minus1

sdot 1199091198902119889

= 1

997904rArr 119909minus1

1198903

sdot 1199091198904


1198902119889minus1

sdot 1199091198902119889

= 119909minus1

1198902

1199091198901

997904rArr 119909minus1

1198903

sdot 1199091198904


1198902119889minus1

sdot 1199091198902119889

sdot 119909minus1

1198901

sdot 1199091198902

= 1

(2)


1 119860



1

24

3

A

B

C

D





2119889sdot 1199092119889minus1










|

119860119861 edge of 119866 and V119860119861




6 ISRN Geometry



4 4 6 4 119870

4Plane 0 8 6 PG (3119867)


3 3 9 6 119870

33Torus 1 9 9 PG (2 119865)


4 4 8 4 2119862

4Torus 1 8 8 PG (3 119865)

Other 84Thm 119870

4+ 2119890 4 8 4 119870

4+ 2119890 Torus 1 8 8 PG (3 119865)


1

2 3

4

A

B C

D

(a)

AB12

AC13

AD14

BD24

BC23

CD34

(b)








1119860


2119861Then the


119860119863 V119860119861) is labelled

1198981198602




1119860sdot1198981198602sdot119898minus1

2119861sdot1198981198611= 1









4






11986011986134 = 11986211986334

11986011986224 = 11986111986324

11986011986323 = 11986111986223

11986111986214 = 11986011986314

11986111986313 = 11986011986213

11986211986312 = 11986011986112

(3)




ISRN Geometry 7

A

A

A

A

B

C

(a)

1

6 2

4 3

6

15

34

(b)


33




3graph see


11986011986114 = 11986214

11986011986126 = 11986226

11986011986135 = 11986235

11986111986216 = 11986016

11986111986225 = 11986025

11986111986234 = 11986034

11986211986015 = 11986115

11986211986024 = 11986124

11986211986036 = 11986136

(4)




4graph see Figure 6

3 6

41

2 5A

B

C


A

A

A

A

B

B

CDD

(a)

1 2

34

(b)


4


11986011986141 = 11986211986341

11986011986123 = 11986211986323

11986111986212 = 11986011986312

11986111986234 = 11986011986334

11986211986323 = 11986011986123

11986211986341 = 11986011986141

11986311986034 = 11986111986234

11986311986012 = 11986111986212

(5)





0 = 119860 2 = 119861 4 = 119862 6 = 119863 1 3 5 7 (6)

8 ISRN Geometry

1 2

34

A B

CD


0

2 2

6

6

4

(a)

3

57

1

(b)


4+ 2119890





11986211986314 = 0215 = 0125

11986011986213 = 2613 = 1236

11986011986337 = 2437 = 2347

11986111986335 = 0435 = 3450

11986011986115 = 4615 = 4561

11986011986257 = 2657 = 5672

11986111986237 = 0637 = 6703

11986111986317 = 0417 = 7014

(7)



0

2

4

6

1

3

5

7


A

A

B

B

C

C

D

(a)

1

2

3 5

6

7

4

4

4

8

8

8

(b)




ISRN Geometry 9

1 2 3 4

5

6

7

8

A

B

C

D




11986211986325 = 11986011986125 11986111986335 = 11986011986235 11986111986245 = 11986011986345

11986011986336 = 11986111986236 11986011986246 = 11986111986346 11986011986147 = 11986211986347

11986211986316 = 11986011986116 11986111986317 = 11986011986217 11986111986218 = 11986011986318

11986011986327 = 11986111986227 11986011986228 = 11986111986328 11986011986138 = 11986211986338

(8)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 ISRN Geometry


1119860sdot 1198982119861




1119860sdot

1198982119861minus1198981198602sdot1198981198611 119890 = 119898

2119860sdot1198983119862minus1198981198603sdot1198981198622119891 = 119898

3119860sdot1198984119863minus1198981198604sdot

1198981198633 119892 = 119898



1119860sdot1198982119861minus1198981198602sdot1198981198611


2119860sdot 1198983119862minus 1198981198603sdot 1198981198622


2119862sdot 119898119896119865minus119898119862119896sdot 1198981198652









119903119909119903vertices and Σ







3 119890V is





119889 119888) (with 119886 119887 119888 119889 all nonzero)


(1 119886minus1119887

1 119889minus1119888) (1)






ISRN Geometry 5






1198901

sdot 1199091198902


1198902119889minus1

sdot 1199091198902119889


1 1198902 119890

2119889 where

1198901= (119860 119861) 119890

2= (119862 119860) 119890

3= (119860119863) 119890

2119889= (119883119860) Note



1199091198901


1199091198902

we obtain

119909minus1

1198901

sdot 1199091198902


1198902119889minus1

sdot 1199091198902119889

= 1

997904rArr 119909minus1

1198903

sdot 1199091198904


1198902119889minus1

sdot 1199091198902119889

= 119909minus1

1198902

1199091198901

997904rArr 119909minus1

1198903

sdot 1199091198904


1198902119889minus1

sdot 1199091198902119889

sdot 119909minus1

1198901

sdot 1199091198902

= 1

(2)


1 119860



1

24

3

A

B

C

D





2119889sdot 1199092119889minus1










|

119860119861 edge of 119866 and V119860119861




6 ISRN Geometry



4 4 6 4 119870

4Plane 0 8 6 PG (3119867)


3 3 9 6 119870

33Torus 1 9 9 PG (2 119865)


4 4 8 4 2119862

4Torus 1 8 8 PG (3 119865)

Other 84Thm 119870

4+ 2119890 4 8 4 119870

4+ 2119890 Torus 1 8 8 PG (3 119865)


1

2 3

4

A

B C

D

(a)

AB12

AC13

AD14

BD24

BC23

CD34

(b)








1119860


2119861Then the


119860119863 V119860119861) is labelled

1198981198602




1119860sdot1198981198602sdot119898minus1

2119861sdot1198981198611= 1









4






11986011986134 = 11986211986334

11986011986224 = 11986111986324

11986011986323 = 11986111986223

11986111986214 = 11986011986314

11986111986313 = 11986011986213

11986211986312 = 11986011986112

(3)




ISRN Geometry 7

A

A

A

A

B

C

(a)

1

6 2

4 3

6

15

34

(b)


33




3graph see


11986011986114 = 11986214

11986011986126 = 11986226

11986011986135 = 11986235

11986111986216 = 11986016

11986111986225 = 11986025

11986111986234 = 11986034

11986211986015 = 11986115

11986211986024 = 11986124

11986211986036 = 11986136

(4)




4graph see Figure 6

3 6

41

2 5A

B

C


A

A

A

A

B

B

CDD

(a)

1 2

34

(b)


4


11986011986141 = 11986211986341

11986011986123 = 11986211986323

11986111986212 = 11986011986312

11986111986234 = 11986011986334

11986211986323 = 11986011986123

11986211986341 = 11986011986141

11986311986034 = 11986111986234

11986311986012 = 11986111986212

(5)





0 = 119860 2 = 119861 4 = 119862 6 = 119863 1 3 5 7 (6)

8 ISRN Geometry

1 2

34

A B

CD


0

2 2

6

6

4

(a)

3

57

1

(b)


4+ 2119890





11986211986314 = 0215 = 0125

11986011986213 = 2613 = 1236

11986011986337 = 2437 = 2347

11986111986335 = 0435 = 3450

11986011986115 = 4615 = 4561

11986011986257 = 2657 = 5672

11986111986237 = 0637 = 6703

11986111986317 = 0417 = 7014

(7)



0

2

4

6

1

3

5

7


A

A

B

B

C

C

D

(a)

1

2

3 5

6

7

4

4

4

8

8

8

(b)




ISRN Geometry 9

1 2 3 4

5

6

7

8

A

B

C

D




11986211986325 = 11986011986125 11986111986335 = 11986011986235 11986111986245 = 11986011986345

11986011986336 = 11986111986236 11986011986246 = 11986111986346 11986011986147 = 11986211986347

11986211986316 = 11986011986116 11986111986317 = 11986011986217 11986111986218 = 11986011986318

11986011986327 = 11986111986227 11986011986228 = 11986111986328 11986011986138 = 11986211986338

(8)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









ISRN Geometry 5






1198901

sdot 1199091198902


1198902119889minus1

sdot 1199091198902119889


1 1198902 119890

2119889 where

1198901= (119860 119861) 119890

2= (119862 119860) 119890

3= (119860119863) 119890

2119889= (119883119860) Note



1199091198901


1199091198902

we obtain

119909minus1

1198901

sdot 1199091198902


1198902119889minus1

sdot 1199091198902119889

= 1

997904rArr 119909minus1

1198903

sdot 1199091198904


1198902119889minus1

sdot 1199091198902119889

= 119909minus1

1198902

1199091198901

997904rArr 119909minus1

1198903

sdot 1199091198904


1198902119889minus1

sdot 1199091198902119889

sdot 119909minus1

1198901

sdot 1199091198902

= 1

(2)


1 119860



1

24

3

A

B

C

D





2119889sdot 1199092119889minus1










|

119860119861 edge of 119866 and V119860119861




6 ISRN Geometry



4 4 6 4 119870

4Plane 0 8 6 PG (3119867)


3 3 9 6 119870

33Torus 1 9 9 PG (2 119865)


4 4 8 4 2119862

4Torus 1 8 8 PG (3 119865)

Other 84Thm 119870

4+ 2119890 4 8 4 119870

4+ 2119890 Torus 1 8 8 PG (3 119865)


1

2 3

4

A

B C

D

(a)

AB12

AC13

AD14

BD24

BC23

CD34

(b)








1119860


2119861Then the


119860119863 V119860119861) is labelled

1198981198602




1119860sdot1198981198602sdot119898minus1

2119861sdot1198981198611= 1









4






11986011986134 = 11986211986334

11986011986224 = 11986111986324

11986011986323 = 11986111986223

11986111986214 = 11986011986314

11986111986313 = 11986011986213

11986211986312 = 11986011986112

(3)




ISRN Geometry 7

A

A

A

A

B

C

(a)

1

6 2

4 3

6

15

34

(b)


33




3graph see


11986011986114 = 11986214

11986011986126 = 11986226

11986011986135 = 11986235

11986111986216 = 11986016

11986111986225 = 11986025

11986111986234 = 11986034

11986211986015 = 11986115

11986211986024 = 11986124

11986211986036 = 11986136

(4)




4graph see Figure 6

3 6

41

2 5A

B

C


A

A

A

A

B

B

CDD

(a)

1 2

34

(b)


4


11986011986141 = 11986211986341

11986011986123 = 11986211986323

11986111986212 = 11986011986312

11986111986234 = 11986011986334

11986211986323 = 11986011986123

11986211986341 = 11986011986141

11986311986034 = 11986111986234

11986311986012 = 11986111986212

(5)





0 = 119860 2 = 119861 4 = 119862 6 = 119863 1 3 5 7 (6)

8 ISRN Geometry

1 2

34

A B

CD


0

2 2

6

6

4

(a)

3

57

1

(b)


4+ 2119890





11986211986314 = 0215 = 0125

11986011986213 = 2613 = 1236

11986011986337 = 2437 = 2347

11986111986335 = 0435 = 3450

11986011986115 = 4615 = 4561

11986011986257 = 2657 = 5672

11986111986237 = 0637 = 6703

11986111986317 = 0417 = 7014

(7)



0

2

4

6

1

3

5

7


A

A

B

B

C

C

D

(a)

1

2

3 5

6

7

4

4

4

8

8

8

(b)




ISRN Geometry 9

1 2 3 4

5

6

7

8

A

B

C

D




11986211986325 = 11986011986125 11986111986335 = 11986011986235 11986111986245 = 11986011986345

11986011986336 = 11986111986236 11986011986246 = 11986111986346 11986011986147 = 11986211986347

11986211986316 = 11986011986116 11986111986317 = 11986011986217 11986111986218 = 11986011986318

11986011986327 = 11986111986227 11986011986228 = 11986111986328 11986011986138 = 11986211986338

(8)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









6 ISRN Geometry



4 4 6 4 119870

4Plane 0 8 6 PG (3119867)


3 3 9 6 119870

33Torus 1 9 9 PG (2 119865)


4 4 8 4 2119862

4Torus 1 8 8 PG (3 119865)

Other 84Thm 119870

4+ 2119890 4 8 4 119870

4+ 2119890 Torus 1 8 8 PG (3 119865)


1

2 3

4

A

B C

D

(a)

AB12

AC13

AD14

BD24

BC23

CD34

(b)








1119860


2119861Then the


119860119863 V119860119861) is labelled

1198981198602




1119860sdot1198981198602sdot119898minus1

2119861sdot1198981198611= 1









4






11986011986134 = 11986211986334

11986011986224 = 11986111986324

11986011986323 = 11986111986223

11986111986214 = 11986011986314

11986111986313 = 11986011986213

11986211986312 = 11986011986112

(3)




ISRN Geometry 7

A

A

A

A

B

C

(a)

1

6 2

4 3

6

15

34

(b)


33




3graph see


11986011986114 = 11986214

11986011986126 = 11986226

11986011986135 = 11986235

11986111986216 = 11986016

11986111986225 = 11986025

11986111986234 = 11986034

11986211986015 = 11986115

11986211986024 = 11986124

11986211986036 = 11986136

(4)




4graph see Figure 6

3 6

41

2 5A

B

C


A

A

A

A

B

B

CDD

(a)

1 2

34

(b)


4


11986011986141 = 11986211986341

11986011986123 = 11986211986323

11986111986212 = 11986011986312

11986111986234 = 11986011986334

11986211986323 = 11986011986123

11986211986341 = 11986011986141

11986311986034 = 11986111986234

11986311986012 = 11986111986212

(5)





0 = 119860 2 = 119861 4 = 119862 6 = 119863 1 3 5 7 (6)

8 ISRN Geometry

1 2

34

A B

CD


0

2 2

6

6

4

(a)

3

57

1

(b)


4+ 2119890





11986211986314 = 0215 = 0125

11986011986213 = 2613 = 1236

11986011986337 = 2437 = 2347

11986111986335 = 0435 = 3450

11986011986115 = 4615 = 4561

11986011986257 = 2657 = 5672

11986111986237 = 0637 = 6703

11986111986317 = 0417 = 7014

(7)



0

2

4

6

1

3

5

7


A

A

B

B

C

C

D

(a)

1

2

3 5

6

7

4

4

4

8

8

8

(b)




ISRN Geometry 9

1 2 3 4

5

6

7

8

A

B

C

D




11986211986325 = 11986011986125 11986111986335 = 11986011986235 11986111986245 = 11986011986345

11986011986336 = 11986111986236 11986011986246 = 11986111986346 11986011986147 = 11986211986347

11986211986316 = 11986011986116 11986111986317 = 11986011986217 11986111986218 = 11986011986318

11986011986327 = 11986111986227 11986011986228 = 11986111986328 11986011986138 = 11986211986338

(8)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









ISRN Geometry 7

A

A

A

A

B

C

(a)

1

6 2

4 3

6

15

34

(b)


33




3graph see


11986011986114 = 11986214

11986011986126 = 11986226

11986011986135 = 11986235

11986111986216 = 11986016

11986111986225 = 11986025

11986111986234 = 11986034

11986211986015 = 11986115

11986211986024 = 11986124

11986211986036 = 11986136

(4)




4graph see Figure 6

3 6

41

2 5A

B

C


A

A

A

A

B

B

CDD

(a)

1 2

34

(b)


4


11986011986141 = 11986211986341

11986011986123 = 11986211986323

11986111986212 = 11986011986312

11986111986234 = 11986011986334

11986211986323 = 11986011986123

11986211986341 = 11986011986141

11986311986034 = 11986111986234

11986311986012 = 11986111986212

(5)





0 = 119860 2 = 119861 4 = 119862 6 = 119863 1 3 5 7 (6)

8 ISRN Geometry

1 2

34

A B

CD


0

2 2

6

6

4

(a)

3

57

1

(b)


4+ 2119890





11986211986314 = 0215 = 0125

11986011986213 = 2613 = 1236

11986011986337 = 2437 = 2347

11986111986335 = 0435 = 3450

11986011986115 = 4615 = 4561

11986011986257 = 2657 = 5672

11986111986237 = 0637 = 6703

11986111986317 = 0417 = 7014

(7)



0

2

4

6

1

3

5

7


A

A

B

B

C

C

D

(a)

1

2

3 5

6

7

4

4

4

8

8

8

(b)




ISRN Geometry 9

1 2 3 4

5

6

7

8

A

B

C

D




11986211986325 = 11986011986125 11986111986335 = 11986011986235 11986111986245 = 11986011986345

11986011986336 = 11986111986236 11986011986246 = 11986111986346 11986011986147 = 11986211986347

11986211986316 = 11986011986116 11986111986317 = 11986011986217 11986111986218 = 11986011986318

11986011986327 = 11986111986227 11986011986228 = 11986111986328 11986011986138 = 11986211986338

(8)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









8 ISRN Geometry

1 2

34

A B

CD


0

2 2

6

6

4

(a)

3

57

1

(b)


4+ 2119890





11986211986314 = 0215 = 0125

11986011986213 = 2613 = 1236

11986011986337 = 2437 = 2347

11986111986335 = 0435 = 3450

11986011986115 = 4615 = 4561

11986011986257 = 2657 = 5672

11986111986237 = 0637 = 6703

11986111986317 = 0417 = 7014

(7)



0

2

4

6

1

3

5

7


A

A

B

B

C

C

D

(a)

1

2

3 5

6

7

4

4

4

8

8

8

(b)




ISRN Geometry 9

1 2 3 4

5

6

7

8

A

B

C

D




11986211986325 = 11986011986125 11986111986335 = 11986011986235 11986111986245 = 11986011986345

11986011986336 = 11986111986236 11986011986246 = 11986111986346 11986011986147 = 11986211986347

11986211986316 = 11986011986116 11986111986317 = 11986011986217 11986111986218 = 11986011986318

11986011986327 = 11986111986227 11986011986228 = 11986111986328 11986011986138 = 11986211986338

(8)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









ISRN Geometry 9

1 2 3 4

5

6

7

8

A

B

C

D




11986211986325 = 11986011986125 11986111986335 = 11986011986235 11986111986245 = 11986011986345

11986011986336 = 11986111986236 11986011986246 = 11986111986346 11986011986147 = 11986211986347

11986211986316 = 11986011986116 11986111986317 = 11986011986217 11986111986218 = 11986011986318

11986011986327 = 11986111986227 11986011986228 = 11986111986328 11986011986138 = 11986211986338

(8)


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







