A Rabbit Hole Between Topology and Geometry

Hindawi Publishing Corporation ISRN Geometry Volume 2013, Article ID 379074, 9 pages http://dx.doi.org/10.1155/2013/379074

Research Article A Rabbit Hole between Topology and Geometry

David G. Glynn

CSEM, Flinders University, P.O. Box 2100, Adelaide, SA 5001, Australia

Correspondence should be addressed to David G. Glynn; [email protected]

Received 10 July 2013; Accepted 13 August 2013

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

Copyright © 2013 David G. Glynn. This is an open access article distributed under the Creative Commons Attribution License, which 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 different aspects, the first with properties preserved under deformations, and the second with more linear or rigid aspects, properties invariant under translations, rotations, or projections. The present paper shows a way to go between them in an unexpected way that uses graphs on orientable surfaces, which already have widespread applications. In this way infinitely many geometrical properties are found, starting with the most basic such as the bundle and Pappus theorems. An interesting philosophical consequence is that the most general geometry over noncommutative skewfields such as Hamilton’s quaternions corresponds to planar graphs, while graphs on surfaces of higher genus are related to geometry over commutative fields such as the real or complex numbers.

1. Introduction 𝐾5 and 𝐾3,3 are minimal nonplanar (toroidal) graphs and both lead to configurational theorems in the same manner. The British/Canadian mathematician H.S.M. Coxeter (1907– In this paper, we explain how virtually all basic linear 2003) was one of most influential geometers of the 20th properties of projective space can be derived from graphs century. He learnt philosophy of mathematics from L. and topology. We show that any map (induced by a graph Wittgenstein at Cambridge, inspired M.C. Escher with his of vertices and edges) on an orientable surface of genus 𝑔, drawings, and influenced the architect R. Buckminster Fuller. having V vertices, 𝑒 edges, and 𝑓 faces, where V −𝑒+𝑓 = See [1].Whenonelooksatthecoverofhisbook“Introduction 2−2𝑔, is equivalent to a linear property of projective space to Geometry”[2], there is the depiction of the complete graph of dimension V −1, coordinatized by a general commutative 𝐾5 on five vertices. It might surprise some people that such field. This property is characterized by a configuration having a discrete object as a graph could be deemed important V+𝑓points and 𝑒 hyperplanes. This leads to the philosophical in geometry. However, Desargues 10-point 10-line theorem deduction that topology and geometry are closely related, via in the projective plane is in fact equivalent to the graph graph theory. If 𝑔=0(and the graph is planar), the linear 𝐾5: in mathematical terms the cycle matroid of 𝐾5 is the property is also valid for the most general projective spaces, Desargues configuration in three-dimensional space, and a which are over skewfields that in general have noncommuta- projection from a general point gives the configurational tive multiplication. This is a powerful connection between the theorem in the plane. Desargues theorem has long been topology of orientable surfaces and discrete configurational recognised (by Hilbert, Coxeter, Russell, and so on) as one of properties of the most general projective spaces. the foundational theorems in projective geometry. However, There are various “fundamental” theorems that pro- there is an unexplained gap left in their philosophies: why vide pathways between different areas of mathematics. For does the graph give a theorem in space? Certainly, the example, the fundamental theorem of projective geometry matroids of almost all graphs are not theorems. The only (FTPG) describes the group of automorphisms of projective other example known to the author of a geometrical theorem geometries over fields or skewfields (all those of dimensions coming directly from a graphic matroid is the complete greater than two) as a group of nonsingular semilinear bipartite graph 𝐾3,3,whichgivesthe9-point 9-plane theorem transformations. This most importantly allows the choice of in three-dimensional space; see [3]. It is interesting that both coordinate systems in well-defined ways. Hence, the FTPG is 2 ISRN Geometry a pathway between projective geometry and algebra, matrix, a multiset containing two vertices. If a vertex is repeated, then and group theory. the edge is a loop.Thegraphis simple if it contains no loop Another example is the fundamental theorem of algebra. and no multiple edges, edges that are repeated. This provides another pathway between polynomials of An orientable surface is a surface in real three- degree 𝑛 over the real number field and multisets of 𝑛 roots, dimensional space that can be constructed from the sphere by which are complex numbers. It explains why the complex appending 𝑔 handles;see[2,Section21.1].Thissurfacehas𝑔 numbers are important for an understanding of the real holes,andwesaythatithas genus 𝑔.Oneclassicaluseforsuch numbers. a surface is to parametrize the points on an algebraic curve in In a similar vein we show here how our “rabbit hole” the complex plane, but we have another application in mind. betweentopologyandgeometrycanbeusedtoobtainthe A skewfield or division ring is an algebraic structure basic properties of the most general projective geometry (𝐹,+,⋅),where𝐹 is a set containing distinct elements 0 and directly from topological considerations. 1,forwhich(𝐹, +) is an abelian (i.e., commutative) group, ∗ Here is an outline of the approach. with identity 0,and(𝐹 := 𝐹 \ {0}, ⋅) is a group (nonabelian if the skewfield is “proper”). The left and right distributive laws (1) Consider the properties of fundamental configura- 𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐 (𝑎 + 𝑏)𝑐 = 𝑎𝑐 +𝑏𝑐 ∀𝑎, 𝑏, 𝑐 ∈𝐹 (V −1) and hold, for . tions in -dimensional projective geometry, The classical example of a proper skewfield is the quaternion which are collections of points and hyperplanes with system of Hamilton (four-dimensional over the reals). If the incidences between them. The most important have 𝐹∗ 𝐹 V multiplicative group is abelian (i.e., commutative), is points on each hyperplane, and these points form a called a field.Thusafieldisaspecialcaseofskewfield. minimal dependent set (a “circuit” in matroid theory). Classical examples of a field are the rational numbers, the (2) In most of these configurations, the algebraic property real numbers, and the complex numbers. It is known (by that corresponds to a configurational theorem is that a Wedderburn’s theorem and elementary field theory) that the set of 𝑒 subdeterminants of size two in a general V ×𝑓 only finite skewfields are the Galois fields GF(𝑞),where𝑞 is a matrix over a field has a linear dependency; that is, power of a prime. the vanishing of any 𝑒−1subdeterminants implies A projective geometry of dimension 𝑛 over a skewfield the vanishing of the remaining subdeterminant. is the set of subspaces of a (left or right) vector space 𝑛+1 (3) The condition for such a set of subdeterminants is of rank over the skewfield. Points are subspaces of topological: the dependency amongst the subdeter- projective dimension zero, while hyperplanes are subspaces 𝑛−1 minants happens if and only if there exists a graph of projective dimension . It is well-known (or by the having V vertices and 𝑒 edges embedded on an FTPG) that every projective space of dimension at least three orientable surface of genus 𝑔 and inducing 𝑓 faces has a coordinatization involving a skewfield and comes from (certain circuits of the graph that can be contracted the relevant vector space. There are some incidence properties to a point on the surface). for geometries over fields that are not valid for those over the more general skewfields. For example, the bundle theorem is (4) A bonus is that when the surface has genus zero (i.e., valid for skewfields (and fields), but Pappus 93 theorem only the graph is planar), the commutative field restriction holds for geometries over fields. for the algebraic coordinates of the space can be It is known that certain of the configurational theorems relaxedtononcommutativeskewfieldsincludingthe are in some sense “equivalent” in that assuming any one of quaternions. This requires a different interpretation them implies the remaining ones. These include the theorems for a 2×2determinant and another proof depending of Pappus, Mobius,¨ and Gallucci. These latter theorems are all upon topological methods. explained by the present topological theory. Desargues theo- (5)Sincethelattermethodofplanargraphsproducesthe remandthebundletheorem(oritsdual,theconfiguration main axiom for projective geometry (the bundle theo- of Pasch) are also in some sense equivalent in the case of the rem or its dual Pasch axiom; see [4,page24])andthe more general geometries over skewfields; see [6]. We show former one for standard determinants over commu- that the bundle theorem comes from the topology of planar tative fields produces the Pappus theorem, we see that graphs. all bases are covered, and a topological explanation An abstract configuration is a set of points and a distin- for standard projective geometry, that is, embeddable guished collection of subsets, called blocks.An embedding into space of dimension greater than two, is obtained. of such a configuration is a way of putting the points into Inthecaseof2-dimensional geometries (planes) there a projective space so that each of the blocks generates a exist non-Desarguesian projective planes so these hyperplane and not the whole space. The point-set as a whole geometries do not appear to be produced topologi- should generate the whole projective space. There are several cally; see [5,page120]and[6,Section23]. ways of thinking about embeddings (e.g., often they may have more incidences than specified by the abstract configuration), 2. Definitions and Concepts and we refer the reader to [7] for a discussion. However, extra incidences do not bother us here. Let us summarize the topological and geometrical concepts Our configurations have blocks with all the same size 𝑘. thatareusedinthispaper.A graph is a collection of vertices We say that such a configuration is a configurational theorem with a certain specified multiset of edges,eachofwhichis if for each embedding of the configuration into space of ISRN Geometry 3

Consider any V ×𝑓matrix 𝑀 over a field 𝐹 (where 2 the multiplication is commutative, i.e. 𝑥𝑦 = 𝑦𝑥 for all B2 𝑥, 𝑦), ∈𝐹 with rows in correspondence with the vertices of 𝐺 A2 𝐴, 𝐵, . A B ( ) and the columns in correspondence with the faces A of 𝐺 (1,2,...,𝑓). We assume that a typical matrix element 1 B1 corresponding to vertex 𝐶 and face 𝑖 has 𝑚𝐶𝑖 =𝑚𝑖𝐶.Thus, 1 the subscripts are treated like unordered sets {𝐶, 𝑖}.Forany “graph fragment” corresponding to an edge 𝐴𝐵 of 𝐺,see 2×2 𝑀 𝐴 Figure 1: Graph fragment. Figure 1,thereisa submatrix of in the rows and 𝐵 andinthecolumns1 and 2.The“angles”𝐴1, 𝐵1, 𝐴2,and 𝐵2 correspond to the four positions in the submatrix, while the determinant of this submatrix is 𝑚𝐴1 ⋅𝑚𝐵2 −𝑚𝐴2 ⋅𝑚𝐵1.Ina 𝑘−1 dimension , the property that all but one of the blocks general embedding of 𝐾 into PG(V−1, 𝐹),wemayassumethat lie in hyperplanes implies the same is true for the remaining the points from 𝑉 formabasisandsoarecoordinatizedbythe block. This might hold only for spaces over fields but not unit vectors. If the remaining points of 𝐾 had no constraints general skewfields, as with Pappus theorem. upon them except for being embedded in PG(V −1,𝐹), they would be coordinatized by completely general (nonzero) 3. Main Results vectors of length V andrealizedbythe𝑓 columns of the matrix 𝑀. Then the vanishing of the subdeterminant corresponding We present two main results. Theorem 1 relatesgraphsor to the edge 𝐴𝐵 is found to be equivalent to the fact that the V maps on orientable surfaces of any genus to configurational points 𝐴𝐵12, as defined above, lie in a hyperplane. theorems in general projective space over any commutative 𝐺 field (such as the rational numbers, real numbers, complex Since the surface of is orientable, we may orient it so numbers, or finite fields). This uses 2×2determinants with that at each vertex there is an anticlockwise direction. The the standard definition. However, for general skewfields this equivalence between cyclic graphs,graphsinwhichthereis definition of determinant does not work, and so we use a cyclic order at each vertex, and embeddings of graphs on Lemma 2 to find an alternate way and find that there is such surfaces has been discussed by many people, starting a restriction to surfaces of genus zero. Thus, Theorem 4 apparently with Heffter [9] and later clarified by Edmonds investigates the graphs or maps on a surface of genus zero and [10]. They have been given many names, such as graphs relates them to configurational theorems over skewfields. with rotation systems, ribbon graphs, combinatorial premaps, and fatgraphs:see[11–13]. Consider Figure 1 again. Small Theorem 1. Any graph 𝐺 embeddedonanorientablesurface anticlockwise-oriented circles around 𝐴 and 𝐵 induce a larger of genus 𝑔≥0,havingV vertices, 𝑒 edges, and 𝑓 faces, where clockwise-oriented circle going from 𝐴→2→𝐵→1→ by Euler’s formula V −𝑒+𝑓=2−2𝑔,isequivalenttoacertain 𝐴. Thus, given any edge of 𝐺 containing a vertex 𝐶 and being configurational theorem (explained in the proof) in projective the boundary of a face 𝑖, this orients the angle from vertex 𝑃𝐺(V −1,𝐹) 𝐹 space ,where is any commutative field. 𝐶 to face 𝑖 or from 𝑖 to 𝐶.Denotethesepossibilitiesby𝐶𝑖 or 𝑖𝐶 𝐺 𝐴, 𝐵, 𝐶, . , respectively. However, such an angle occurs with precisely Proof. Letuslabeltheverticesof with the letters two edges, and one edge gives 𝐶𝑖 and the other 𝑖𝐶. in a set 𝑉 of cardinality V and label the faces (which are The 2×2subdeterminant, with rows 𝐴 and 𝐵 and columns certaincircuitsonthesurface)withthenaturalnumbers 1 and 2,maybewritten𝑚1𝐴 ⋅𝑚2𝐵 −𝑚𝐴2 ⋅𝑚𝐵1, according to 1, 2, 3, . . . , 𝑓.Then,eachofthe𝑒 edges of the graph joins the clockwise orientation. (We purposely forget for a while precisely two vertices, for example, 𝐴 and 𝐵,anditformspart 1 that 𝑚𝑖𝐶 =𝑚𝐶𝑖.) Now the vanishing of this determinant is of the boundaries of precisely two faces, for example, and 𝑚 ⋅𝑚 =𝑚 ⋅𝑚 2. (For simplicity we are assuming that there are no loops in equivalent to 1𝐴 2𝐵 𝐴2 𝐵1 (wecouldcallthetwosides both the graph and its dual, but these can easily be accounted of this equation the “diagonals” of the determinant), and if all 𝐺𝑑 the determinants corresponding to the edges of 𝐺 vanish, we for in a more expansive theory.) Note that the dual graph 𝑒 is the graph embeddable on the same surface where we switch can take the product over all edges on both sides to obtain Π 𝑚 ⋅𝑚 =Π 𝑚 ⋅𝑚 := 𝑝 the roles of vertex and face, joining two faces if they have a 𝐴𝐵∈𝐺 1𝐴 2𝐵 𝐴𝐵∈𝐺 𝐴2 𝐵1 . This is clearly a trivial 𝑖𝐶 common edge. This dual graph depends strongly upon the identity since any angle, for example, ,occursonceonthe 𝐶𝑖 embedding, so that a graph may have different dual graphs left and once (as ) on the right. Now we can assume that the 𝑚 on other surfaces: see [8] for recent research on this topic. “angle variables” 𝑖𝐶 are all nonzero, as otherwise there will We define an abstract configuration 𝐾 having V+𝑓points be an unwanted hyperplane in 𝐾 which would not be in the and 𝑒 blocks, which are subsets of V points as follows. The most general position. Then the vanishing of any 𝑒−1of the points are identified with 𝑉 ∪ {1,...,𝑓},thatis,theunion subdeterminants implies the vanishing of the remaining one, of the set of points and the set of faces of 𝐺. Additionally, for since we can divide 𝑝 by 𝑒−1“diagonals” 𝑚1𝐴 ⋅𝑚2𝐵 on the each edge 𝐴𝐵 bounded by the two faces 1 and 2,thereisthe left and by the corresponding 𝑒−1“diagonals” 𝑚𝐴2 ⋅𝑚𝐵1 on corresponding set of V points which is 𝐴𝐵12 := 𝑉 \ {𝐴, 𝐵}∪ the right, and we obtain the vanishing of the last determinant. {1, 2}; that is, we replace 𝐴 and 𝐵 in 𝑉 by 1 and 2,andwecall This shows the theorem in the general case where 𝐹 is a field this a block of 𝐾. with commutative multiplication. 4 ISRN Geometry

The converse construction holds: a configurational the- itisthesameascreatinganewgeometricaltheoremby orem in space that relies on 2×2matrices as above must identifying points or hyperplanes. However, these examples come from a graph on an orientable surface. The problem is to can then be expanded out again by splitting the rows or determine the cyclic graph 𝐺 from the set of 𝑒2×2subdeter- columns into bigger collections of rows or columns as above, minants of a matrix having the property that the vanishing of and the pattern of subdeterminants in the largest matrix is any 𝑒−1of them implies that the remaining subdeterminant canonical up to permutations of rows and columns. So we see vanishes. Around the edges of each vertex of 𝐺 there should how to get around this minor problem in the proof. be an anticlockwise cyclic orientation or “cyclic order.” If we 𝐾 start with a vertex 𝐴 and an edge {𝐴, 𝐵} containing it, proceed What kind of configurational theorems corresponds to {𝐴, 𝐶} graphs on orientable surfaces? One obvious condition is that to the next edge in the cyclic order, and using the cyclic V +𝑓 (V−1, 𝐹) order at 𝐶, find the next edge {𝐶, 𝐷},andsoon,weshould the configuration must have points in PG .There are 𝑒 hyperplanes or blocks in 𝐾, each containing V points. follow around all the edges of a face of the embedding in a 𝑉 V 𝐾 clockwisewayandreturntothefirstvertex𝐴 and edge {𝐴, 𝐵}. More importantly, there should be a subset of points in such that each hyperplane of 𝐾 contains precisely V−2points Wewillshowhowthisisachieved.Now,asbefore,wecan 𝑉 assume that the entries, where the subdeterminants occur, of and two others. are all nonzero. If the subdeterminants have the assumed Now we explain the noncommutative case which is property, they can be ordered so that one “diagonal” of each related to planar graphs. is selected, and the product of all these selected diagonals is Lemma 2. Let 𝐹 be a skewfield with perhaps noncommutative thesameastheproductofthenonselectedones(asinthefirst multiplication. The condition that a set of V points of 𝑃𝐺(V − part of the proof above). As before we may write the selected 1, 𝐹), consisting of 𝐴,,andtheunitvectors 𝐵 𝑒3,...,𝑒V,is diagonals in the form 𝑚1𝐴 ⋅𝑚2𝐵 and the nonselected ones −1 −1 𝑚 ⋅𝑚 contained in a hyperplane is a “cyclic identity” 𝑎 𝑏𝑐 𝑑=1, in the form 𝐴2 𝐵1.Tofindthegraphwemustassociate ( 𝑎𝑏) 2×2 𝐹 𝑀 where 𝑑𝑐 is a certain matrix over .(Hereweare the rows of the matrix with the vertices, the columns with 𝑎, 𝑏, 𝑐,𝑑 faces, and the subdeterminants with the edges. Consider a assuming a “generic” case where all the are nonzero.) particular vertex 𝐴 of 𝐺 (a row of 𝑀). We obtain a cyclic Proof. A point of PG(V −1,𝐹)is a nonzero column vector (anticlockwise) chain of 𝑛 subdeterminants using that row with V coefficients from 𝐹 that are not all zero. Two of these (equivalently, edges of 𝐺 containing 𝐴) as follows: 𝑑:=𝑚1𝐴 ⋅ column vectors y and z give the same point if one can find a 𝑚2𝐵−𝑚𝐴2⋅𝑚𝐵1, 𝑒:=𝑚2𝐴⋅𝑚3𝐶−𝑚𝐴3⋅𝑚𝐶2, 𝑓:=𝑚3𝐴⋅𝑚4𝐷−𝑚𝐴4⋅ nonzero element 𝑓∈𝐹such that y = z𝑓. The hyperplanes of 𝑚𝐷3,..., 𝑔:=𝑚𝑛𝐴 ⋅𝑚1𝐸 −𝑚𝐴1 ⋅𝑚𝐸𝑛.Nowwecancheckthat PG(V −1,𝐹)can be coordinatized by row vectors of length the faces of 𝐺 also arise from this construction. Starting with V over 𝐹, in a similar way to the points. Then a point y is the vertex 𝐴 and edge containing it 𝑑=𝑚1𝐴 ⋅𝑚2𝐵 −𝑚𝐴2 ⋅𝑚𝐵1, contained in a hyperplane h if and only if hy =0,(h is a row the next edge determinant in 𝐴’s anticlockwise order from 𝑑 and y is a column vector). Notice that here we are multiplying is 𝑒:=𝑚2𝐴 ⋅𝑚3𝐶 −𝑚𝐴3 ⋅𝑚𝐶2 which contains the vertex- points on the left (and hyperplanes on the right). Thus we row 𝐶. The cyclic ordering at 𝐶 makes 𝑚2𝐶 ⋅𝑚𝑘𝐹 −𝑚𝐶𝑘 ⋅𝑚𝐹2 must restrict ourselves to operations on the points of PG(V − the next edge (for some vertex-row 𝐹 and column-face 𝑘). 1, 𝐹) that act on the left. A square V × V matrix is “singular” Following this sequence of subdeterminants (edges) around (and its column points are in a hyperplane) if and only if it we see that the edges surround the column-face 2,andwe cannot be row-reduced (by multiplying on the left by a square can say that the cyclic ordering induced on the edges of the matrix) to the identity matrix, or equivalently, it can be row face in this way is clockwise. So it works out similarly given reduced so that a zero row appears. In our situation we have any vertex and edge containing that vertex. However, one a V × V matrix that consists of V −2different unit vectors and might see a minor problem with this argument. In a standard 𝑎𝑏 a 2×2two submatrix 𝑋=( ) (with 𝑎, 𝑏, 𝑐, 𝑑 all nonzero) (cyclic) graph 𝐺 thereshouldbeonecycle(ofedges)ateach 𝑑𝑐 in the remaining part row disjoint from the ones of the unit vertex: if there are 𝑥𝑟 cycles determined by a row 𝐴 of 𝑀, vectors. We can then restrict our row reductions to the two we “split” that row into 𝑥𝑟 distinct rows, one for each disjoint rows of 𝑋,andweseethatthewholematrixissingularifand cycle of subdeterminants with 𝐴.Similarlywelookateach only if 𝑋 is singular. It is still not possible to use the ordinary column 𝑐,andtherewillbe𝑦𝑐 disjoint cycles on the rows determinant to work out if 𝑋 is singular. But assuming that induced by the subdeterminants with that column. Splitting 𝑎 𝑑 𝑦 both and arenonzerowemaymultiplythefirstrowby that column into 𝑐 distinct columns will enable us to look at 𝑎−1 𝑑−1 a larger matrix with the same number of subdeterminants, and the second by . This leaves us with the matrix butwitheachrowandcolumncorrespondingtoaunique 1𝑎−1𝑏 ( ), cycle. Subdeterminants in different cycles will not have rows 1𝑑−1𝑐 (1) or columns in common. Then the graph on the orientable Σ 𝑥 Σ 𝑦 surface has 𝑟 𝑟 vertices and 𝑐 𝑐 faces. The other way and the condition for singularity of this matrix is clearly 2×2 −1 −1 around, given a set of determinants with our special 𝑎 𝑏=𝑑 𝑐,asthenwecanfurtherrow-reducetoobtaina −1 −1 property, if we collapse the matrix by identifying certain rows zero row. This gives the “cyclic condition” 𝑎 𝑏𝑐 𝑑=1(= −1 −1 −1 −1 −1 −1 or columns, then the property is retained, as long as we do 𝑑𝑎 𝑏𝑐 =𝑐 𝑑𝑎 𝑏=𝑏𝑐 𝑑𝑎 ),if𝑐 is also nonzero. notidentifytworowsorcolumnsbelongingtothesame −1 −1 −1 −1 subdeterminant. By this process cycles of subdeterminants Note that 𝑎 𝑏=𝑑 𝑐 does not imply that 𝑎𝑏 𝑐𝑑 = canbecreatedwiththesameroworcolumn.Geometrically, 1: equivalently, transposing a general 2×2matrix over ISRN Geometry 5 a skewfield does not always preserve its singularity. There is B quite a lot of theory about determinants for skewfields, see for example, [14, 15], but we can have a more elementary approach here since we only deal with 2×2subdeterminants. This leads us to consider a special type of planar graph that has cyclic identities at each vertex. It is well known that 3 1 any planar graph with an even number of edges on each face is bipartite; see, for example, [8]. By dualizing this statement we D also know that any planar graph which is Eulerian, that is, has an even valency at each vertex, has a bipartite dual. What this 2 means is that the edges of such a planar Eulerian graph may A 4 C be oriented so that the edges on each face go in a clockwise or Figure 2: The tetrahedron (graph of the bundle theorem) in the in an anticlockwise direction. Then, if we travel around any plane. vertex in a clockwise direction, the edges alternate, going out andintothevertex.Wecallsuchanorientation Eulerian. −1 −1 In general, an Eulerian orientation of a graph having even 𝐴 is 𝑥1 ⋅𝑥2 ⋅⋅⋅𝑥2𝑑−1⋅𝑥2𝑑 =1, with the odd edges directed from valencyateachvertexisanorientationofeachedge(put 𝐴 to 𝐵 and the even edges from 𝐵 to 𝐴, then the clockwise an arrow on the edge) such that there are equal numbers of order at 𝐵 will be the reverse of that at 𝐴,andsothecyclic −1 −1 edges going out or into each vertex. For the above embedding identity at 𝐵 will be 𝑥2𝑑 ⋅𝑥2𝑑−1 ⋅⋅⋅𝑥2 ⋅𝑥1 =1which is the in the plane we find a natural Eulerian orientation that is inverse identity to that at 𝐴 andsoequivalenttoit.Hence determined by the faces. the dependency among all the cyclic identities of the original graph is established. Lemma 3. Consider a planar graph 𝐻 with a bipartite dual having its Eulerian orientation of the edges. Then there is non- Theorem 4. Any graph 𝐺 embeddedonanorientablesurface commutative cyclic identity with variables over any skewfield of genus 𝑔=0,havingV vertices, 𝑒 edges, and 𝑓 faces, at each vertex, and any one of these cyclic identities is implied where by Euler’s formula V −𝑒+𝑓,isequivalenttoa =2 by the remaining cyclic identities. configurational theorem in projective space 𝑃𝐺(V−1,𝐹),where 𝐹 is any skewfield or field. Proof. Consider the list of edges 𝐸, and for each 𝑒∈𝐸let 𝑒= (𝐴, 𝐵), where the Eulerian orientation goes from vertex 𝐴 on 𝑒 Proof. First we construct the configuration 𝐾 from the graph to vertex 𝐵 on 𝑒. The “cyclic identity” at vertex 𝐴 is of the form 𝐺 in precisely the same manner as Theorem 1. 𝑥−1 ⋅𝑥 ⋅⋅⋅𝑥−1 ⋅𝑥 =1 𝐴 𝐺 𝑒1 𝑒2 𝑒2𝑑−1 𝑒2𝑑 , where the edges of the graph on When the graph is embedded in any orientable surface, are (in the clockwise ordering around 𝐴) 𝑒1, 𝑒2,...,𝑒2𝑑,where which in the present case is now the plane (or the sphere), 𝑒1 = (𝐴, 𝐵),2 𝑒 =(𝐶,𝐴),𝑒3 = (𝐴,𝐷),...,𝑒2𝑑 =(𝑋,𝐴).Note there is a natural cyclic structure at each vertex. We now go to 𝑑 that if we had have started with any other edge, for example, a graph that is intermediate between 𝐺 and its dual 𝐺 . This is 󸀠 𝑒3,goingoutfrom𝐴, we would have obtained an equivalent called the “medial” graph 𝑀(𝐺),andithasV =𝑒vertices and 𝑥−1𝑥 𝑓󸀠 = V +𝑓 4 identity, since by multiplying both sides on the left by 𝑒2 𝑒1 faces. It is -regular, in that every vertex is joined 𝑥−1𝑥 to four others. Since each edge has two vertices, it is easy to andthenbothsidesontherightby 𝑒 𝑒 we obtain 󸀠 󸀠 1 2 see that the medial graph has 𝑒 =2V edges. Notice that since 𝑥−1 ⋅𝑥 ⋅⋅⋅𝑥−1 ⋅𝑥 =1, V−𝑒+𝑓 = 2−2𝑔(Euler’sformula) we have in the medial graph 𝑒 𝑒2 𝑒 𝑒2𝑑 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 1 2𝑑−1 with V −𝑒 +𝑓 = V −2V +𝑓 =𝑓−V = V+𝑓−𝑒 = 2−2𝑔:itis −1 −1 −1 󳨐⇒ 𝑥 𝑒 ⋅𝑥𝑒 ⋅⋅⋅𝑥𝑒 ⋅𝑥𝑒 =𝑥𝑒 𝑥𝑒 (2) clear the medial graph is also embedded on the same surface 3 4 2𝑑−1 2𝑑 2 1 as 𝐺. −1 −1 −1 𝐺 󳨐⇒ 𝑥 𝑒 ⋅𝑥𝑒 ⋅⋅⋅𝑥𝑒 ⋅𝑥𝑒 ⋅𝑥𝑒 ⋅𝑥𝑒 =1. For example, if is the planar tetrahedral graph of 3 4 2𝑑−1 2𝑑 1 2 Figure 2,then𝑀(𝐺) is the planar octahedral graph, having Nowconsideranyfaceofthegraphwithitsclockwiseor six vertices and eight faces. anticlockwise orientation. If it has 𝑛 vertices (in the cyclic In detail, the set of vertices of 𝑀(𝐺) is {V𝐴𝐵 | order labelled 𝐴1,...,𝐴𝑛), then there are 𝑛 cyclic identities 𝐴𝐵 edge of 𝐺},andV𝐴𝐵 is joined with V𝐵𝐶 in 𝑀(𝐺) when 𝐴𝐵 attached. Consider the operation of collapsing the face down and 𝐵𝐶 are adjacent to the same face 𝑓 of 𝐺 on the surface: to a single vertex and erasing all the edges of the face. The they are also adjacent in the cyclic order at 𝐵 andinthatof𝑓. cyclic identities can be multiplied in the cyclic order so that a The dual of this medial graph is always bipartite so that there new cyclic identity is obtained. If a loop having adjacent ins are two types of faces, corresponding to the vertices and to the and outs at a vertex appears, then it may be safely purged from faces of the original graph 𝐺.(Conversely,a4-regular graph thegraph,sincetherecanbenoholesinthesurfaceandsince on an orientable surface, for which the dual graph is bipartite, in the cyclic identity at the vertex the edge variable will cancel is easily seen to be the medial graph of a unique graph on that with itself. The new collapsed graph has cyclic identities that surface.) derive from the larger graph. By continuing this process we Consider Figure 1,andadjoin𝐶 and 𝐷,whicharethe obtain eventually a planar graph with two vertices 𝐴 and 𝐵 vertices in 𝐺 adjacent to 𝐴 on the boundaries of faces 1 and 2, joined by an even number 2𝑑 of edges. If the cyclic identity at respectively, and adjoin 𝐸 and 𝐹 which are the vertices 6 ISRN Geometry

Table 1: A table of five geometrical theorems.

Name Graph: V,𝑒,𝑓 Dual Surface: 𝑔 P’s H’s Space

Bundle Thm 𝐾4: 4, 6, 4 𝐾4 Plane: 0 86PG(3, 𝐻)

Pappus 93 Thm 3𝐾3: 3, 9, 6 𝐾3,3 Torus: 199PG (2, 𝐹)

Mobius¨ 84 Thm 2𝐶4: 4, 8, 4 2𝐶4 Torus: 188PG (3, 𝐹)

Other 84 Thm 𝐾4 +2𝑒: 4, 8, 4 𝐾4 +2𝑒 Torus: 188PG (3, 𝐹)

Gallucci’s Thm 2𝐾4: 4, 12, 8 Cube Torus: 11212PG (3, 𝐹)

The bundle theorem states that if four lines are such that five of the unordered pairs of the lines are coplanar, then sois the final unordered pair. Translating this to a theorem about points and planes, we can define a line as the span of a pair of AD14 distinct points. Thus the lines correspond to pairs of points, 1 4 andthetheoremisabouteightpointsandsixplanes.Itturns out that the configuration is in three-dimensional space, and AC13 A 23 D CD34 the four lines must be concurrent. The dual in terms of points and lines is that if four lines AB12 BC23 in space have five intersections in points, then so is the sixth B C BD24 intersection.Thenallthelinesarecoplanar.Thisisthe“Axiom (a) (b) of Pasch”; see for example, [4],anditisoneofthefundamen- tal axioms from which all the other basic properties derive. Figure 3: The bundle theorem in 3dspaceanditsdualPaschaxiom. Comparing Figure 2 with Figure 3 the bundle theorem is seen to be the configurational theorem that arises from 𝐾4, 𝐵 2 1 the tetrahedral graph or equivalently the complete graph adjacent to on the boundaries of faces and .Weseethat embedded in the plane. V 𝑀(𝐺) 𝐴𝐵 is joined in the medial graph with the four vertices Relating this to the proof of Theorem 4,themedialgraph V𝐴𝐶, V𝐴𝐷, V𝐵𝐸,andV𝐵𝐹 in the clockwise direction. Notice that of 𝐾4 is the octahedral graph having six vertices and eight 𝑀(𝐺) these edges of are in bijective correspondence with the faces. Thus the theorem shows that the bundle theorem 𝐴1 𝐴2 𝐵2 𝐵1 “angles” , , , ,respectively.Also,asintheproofof is valid for all projective geometries of dimension at least Theorem 1 the selection of “diagonals” of the determinants three. This leads to the philosophic conclusion that projective 𝑚1𝐴 ⋅𝑚2𝐵 −𝑚𝐴2 ⋅𝑚𝐵1 at each edge implies that we can −1 geometry and our perceptions of linear geometry may have orientate the edge (V𝐴𝐵, V𝐴𝐶) in 𝑀(𝐺) and label it with 𝑚1𝐴; −1 topological origins. similarly the directed edge (V𝐴𝐵, V𝐴𝐸) is labelled 𝑚2𝐵.Thenthe It is noted that the dual graph of the octahedral graph (in remaining unselected diagonal of the determinant gives two theplane)isthecube,whichhaseightsquarefacesandsix edges of 𝑀(𝐺) directed the other way: (V𝐴𝐷, V𝐴𝐵) is labelled vertices. 𝑚𝐴2 and (V𝐵𝐹, V𝐴𝐵) is labelled 𝑚𝐵1.Repeatingthisforalledges Thesixblocksoffourpointsobtainedfromtheedgesof of 𝐺 we obtain an Eulerian orientation, and each vertex of the graph are 𝑀(𝐺) corresponds to a cyclic identity with four variables which is equivalent to the determinant condition. For the 𝐴𝐵34 = 𝐶𝐷34, −1 −1 edge 𝐴𝐵 above the “cyclic” identity is 𝑚1𝐴 ⋅𝑚𝐴2 ⋅𝑚2𝐵 ⋅𝑚𝐵1 =1. Applying Lemma 3 to the medial graph 𝑀(𝐺) we see that 𝐴𝐶24 = 𝐵𝐷24, the final cyclic identity is dependent upon the others, and so we have proved that 𝐾 is a configurational theorem for every 𝐴𝐷23 = 𝐵𝐶23, skewfield and therefore also for every field. (3) 𝐵𝐶14 = 𝐴𝐷14,

4. Examples of Configurational Theorems 𝐵𝐷13 = 𝐴𝐶13,

If a graph on an orientable surface 𝑆 gives a configurational 𝐶𝐷12 = 𝐴𝐵12. theorem 𝐾, then the dual graph on 𝑆 gives a configurational 𝐾 theorem that is the matroid dual of . It corresponds to the The eight points of this “bundle” theorem in 3d space V ×𝑓 𝑀 simple process of transposing the matrix containing are members of the set {𝐴,𝐵,𝐶,𝐷,1,2,3,4},whilethesix the subdeterminants in the construction. blocks (contained in planes) are in correspondence with the Table 1 summarizes the five examples of this section. six edges of the 𝐾4 graph (the tetrahedron); see Figure 2. In the Pasch configuration on the right of Figure 3,there 4.1. The Bundle Theorem. The bundle theorem in three- are again four lines which we could label 𝐴1, 𝐵2, 𝐶3, 𝐷4. Each dimensional projective space is a theorem of eight points and pair of lines intersect in a point, for example, 𝐴1 and 𝐵2 six planes. See Figure 3. intersect in the point labelled 𝐴1𝐵2.Theintersectionofthe ISRN Geometry 7

43 B 36 A A

1 C 6 2 6 4 C

5 B 1 1 A 25

A A Figure 5: The Pappus theorem derived from the toric map. 4 3 B (a) (b) A A

Figure 4: The toroidal Pappus graph 3𝐶3 and its dual 𝐾3,3. 12 C D D 4 3 final pair of lines 𝐵2 and 𝐶3 is a consequence of the other intersections. So we verify that the geometric dual of the bundle theorem is the Pasch configuration. A A B

(a) (b) 4.2. The Pappus Theorem. The nine points of the Pappus 93 configurationaltheoremintheplanearemembersoftheset Figure 6: The toroidal Mobius¨ graph 2𝐶4 and its dual 2𝐶4. {𝐴,𝐵,𝐶,1,2,3,4,5,6}, while the nine blocks (contained in lines when the configuration is embedded in the plane) are in correspondence with the nine edges of the 3𝐶3 graph; see The eight blocks obtained from the edges of the graph are Figure 4. 𝐴𝐵41 = 𝐶𝐷41, The nine blocks obtained from the edges of the graph are 𝐴𝐵23 = 𝐶𝐷23, 𝐴𝐵14 = 𝐶14, 𝐵𝐶12 = 𝐴𝐷12,

𝐴𝐵26 = 𝐶26, 𝐵𝐶34 = 𝐴𝐷34,

𝐴𝐵35 = 𝐶35, 𝐶𝐷23 = 𝐴𝐵23, (5)

𝐵𝐶16 = 𝐴16, 𝐶𝐷41 = 𝐴𝐵41,

𝐵𝐶25 = 𝐴25, (4) 𝐷𝐴34 = 𝐵𝐶34, 𝐵𝐶34 = 𝐴34, 𝐷𝐴12 = 𝐵𝐶12.

𝐶𝐴15 = 𝐵15, There are many references for this configuration; see 𝐶𝐴24 = 𝐵24, [2, 3, 5, 16–20]. Perhaps the easiest way to construct this configuration in space is to first construct a 4×4grid of eight 𝐶𝐴36 = 𝐵36. lines; see Figure 7.Theeight“Mobius”¨ points can be eight points grouped in two lots of four as in the figure. The planes There are many references for this configuration which then correspond to the remaining eight points on the grid. dates back to Pappus of Alexandria circa 330 CE; see [2, 3, A recent observation by the author [21] is that one can find 16 5, 16–18]. Perhaps the easiest way to construct it in the plane three four by four matrices with the same variables such is first to draw any two lines. Put three points on each and that their determinants sum to zero, and it is closely related to connect them up with six lines in the required manner; see the fact that there are certain three quadratic surfaces in space Figure 5. associated with this configuration. See16 [ ]foradiscussionof the three quadrics.

4.3. The Mobius¨ Theorem. The eight points of the Mobius¨ 4.4. The Non-Mobius¨ 84 Configurational Theorem. The eight 84 configurationaltheoremin3dspacearemembersofthe 8 {𝐴,𝐵,𝐶,𝐷,1,2,3,4} points of the “other” 4 configurational theorem in 3d space set , while the eight blocks (contained in can be abstractly considered to be the members of the set planes when the configuration is in 3d space) are in correspondence with the eight edges of the 2𝐶4 graph; see Figure 6. {0=𝐴,2=𝐵,4=𝐶,6=𝐷,1,3,5,7} , (6) 8 ISRN Geometry

A B 0 5

D C 7 2

12 6 3

4 3 1 4

Figure 7: The Mobius¨ 84 configuration on eight lines. Figure 9: The other 84 configuration on eight lines.

6 B

0 1

7 A C 2 2 5 D 4 3

A 6 C

(a) (b) B Figure 8: The toroidal graph 𝐾4 +2𝑒and its dual 𝐾4 +2𝑒. (a) 8 4 while the eight blocks (contained in planes when the configuration is embedded in 3d space) are in correspondence with 35 the eight edges of the 𝐾4 +2𝑒graph which has four vertices: 6 2 it can be constructed as the complete graph on four vertices 4 8 plus two other nonadjacent edges. 1 7 The eight blocks obtained from the edges of the graph are

𝐶𝐷14 = 0215 = 0125, 8 4

𝐴𝐶13 = 2613 = 1236, (b) 2𝐾 𝐴𝐷37 = 2437 = 2347, Figure 10: The toroidal Gallucci graph 4 and its dual, cube graph.

𝐵𝐷35 = 0435 = 3450, (7) 𝐴𝐵15 = 4615 = 4561, are 𝐴,𝐵,𝐶,𝐷,1,...,8, while the twelve blocks (contained in planes when the configuration is in 3d space) are in 𝐴𝐶57 = 2657 = 5672, correspondencewiththetwelvepointsonthe4×4grid other than 𝐴, 𝐵, 𝐶,𝐷.Notethatwearerepresentingthetorus 𝐵𝐶37 = 0637 = 6703, as a hexagon with opposite sides identified. This is just an alternative to the more common representation of the torus 𝐵𝐷17 = 0417 = 7014. as a rectangle with opposite sides identified. The arrows on the outside of the hexagons show the directions for which the The standard cyclic representation of this configuration is identifications are applied. (The hexagons’ boundaries are not that the points are the integers modulo eight, while the blocks graph edges.) {0,1,2,5} + 𝑖( 8) are the subsets mod ;seeGlynn[3]and Another thing to note is that the only place the author Figure 8.AswiththeMobius¨ configuration, the configuration has seen the name “Gallucci” attached to this configuration 4×4 can always be constructed on a grid of lines; see Figure 9. is in the works of Coxeter; see [2, Section 14.8]. The theorem The planes then correspond to the remaining eight points on appears in Baker’s book [5,page49],whichappearedinits the grid. first edition in 1921, well before Gallucci’s major work of 1928; see [18]. Due to its fairly basic nature it was obviously known 4.5. The Gallucci Theorem. Consider Figures 10 and 11.The to geometers of the 19th century. However, in deference to twelve points of the Gallucci configuration in 3d space Coxeter, we are calling it “Gallucci’s theorem.” ISRN Geometry 9

5 A [7] D. G. Glynn, “A note on 𝑁𝐾 configurations and theorems in projective space,” Bulletin of the Australian Mathematical Society,vol.76,no.1,pp.15–31,2007. B 6 [8] S. Huggett and I. Moffatt, “Bipartite partial duals and circuits in medial graphs,” Combinatorica,vol.33,no.2,pp.231–252,2013. C 7 [9] L. Heffter, “Ueber das Problem der Nachbargebiete,” Mathema- tische Annalen,vol.38,no.4,pp.477–508,1891. [10] J. R. Edmonds, “A combinatorial representation for polyhedral D 8 surfaces,” Notices of the American Mathematical Society,vol.7, 1234 article A646, 1960. [11] B. Bollobas´ and O. Riordan, “A polynomial invariant of graphs Figure 11: The Gallucci theorem of eight lines in 3d space. on orientable surfaces,” Proceedings of the London Mathematical Society,vol.83,no.3,pp.513–531,2001. [12] B. Bollobas´ and O. Riordan, “A polynomial of graphs on surfaces,” Mathematische Annalen,vol.323,no.1,pp.81–96, The Gallucci configuration is normally thought of asa 2002. collection of eight lines, but here we are obtaining it from [13]G.A.JonesandD.Singerman,“Theoryofmapsonorientable certain subsets of points and planes related to it. One set of surfaces,” Proceedings of the London Mathematical Society,vol. fourmutuallyskewlinesisgeneratedbythepairsofpoints 37, no. 2, pp. 273–307, 1978. 𝐴1,𝐵2,𝐶3,𝐷4, and the other set of four lines by the four pairs 𝐴5,𝐵6,𝐶7,𝐷8 [14] J. Dieudonne,´ “Les determinants´ sur un corps non commutatif,” . Bulletin de la Societ´ eMath´ ematique´ de France,vol.71,pp.27–45, The twelve blocks obtained from the edges of the graph 1943. are [15]I.Gelfand,S.Gelfand,V.Retakh,andR.L.Wilson,“Quasi- 𝐶𝐷25 = 𝐴𝐵25, 𝐵𝐷35 = 𝐴𝐶35, 𝐵𝐶45 = 𝐴𝐷45, determinants,” Advances in Mathematics,vol.193,no.1,pp.56– 141, 2005. 𝐴𝐷36 = 𝐵𝐶36, 𝐴𝐶46 = 𝐵𝐷46, 𝐴𝐵47 = 𝐶𝐷47, [16] W. Blaschke, Projektive Geometrie,Birkhauser,¨ Basel, Switzer- land, 3rd edition, 1954. 𝐶𝐷16 = 𝐴𝐵16, 𝐵𝐷17 = 𝐴𝐶17, 𝐵𝐶18 = 𝐴𝐷18, [17] H. S. M. Coxeter, “Self-dual configurations and regular graphs,” Bulletin of the American Mathematical Society,vol.56,pp.413– 𝐴𝐷27 = 𝐵𝐶27, 𝐴𝐶28 = 𝐵𝐷28, 𝐴𝐵38 = 𝐶𝐷38. 455, 1950. (8) [18] G. Gallucci, Complementi di geometria proiettiva. Contributo allageometriadeltetraedroedallostudiodelleconfigurazioni, Universita` degli Studi di Napoli, Napoli, Italy, 1928. Some practical considerations remain: small graphs may determine relatively trivial properties of space, but we have [19] A. F. Moebius, “Kann von zwei dreiseitigen Pyramiden eine seen in our examples that many graphs correspond to jede in Bezug auf die andere um- und eingeschrieben zugleich heissen?” Crelle’s Journal fur¨ die reine und angewandte Mathe- fundamental and nontrivial properties. We also obtain an matik,vol.3,pp.273–278,1828. automatic proof for these properties just from the embedding [20] A. F. Moebius, “Kann von zwei dreiseitigen Pyramiden eine onto the surface. For some graphs on orientable surfaces jede in Bezug auf die andere um- und eingeschrieben zugleich the constructed geometrical configuration must collapse into heissen?” Gesammelte Werke,vol.1,pp.439–446,1886. smaller dimensions upon embedding into space or have [21] D. G. Glynn, “A slant on the twisted determinants theorem,” points or hyperplanes that merge. This is a subject for further Submitted to Bulletin of the Institute of Combinatorics and Its investigation. Applications.

References

[1] S. Lavietes, New York Times obituary, 2003, http://www.ny- times.com/2003/04/07/world/harold-coxeter-96-who-found- profound-beauty-in-geometry.html. [2]H.S.M.Coxeter,Introduction to Geometry,JohnWiley&Sons, New York, NY, USA, 1961. [3] D. G. Glynn, “Theorems of points and planes in three-dimensional projective space,” Journal of the Australian Mathematical Society,vol.88,no.1,pp.75–92,2010. [4] P. Dembowski, Finite Geometries,vol.44ofErgebnisse der Mathematik und ihrer Grenzgebiete, Springer, New York, NY, USA, 1968. [5] H. F. Baker, Principles of Geometry, vol. 1, Cambridge University Press, London, UK, 2nd edition, 1928. [6] D. Hilbert, Grundlagen der Geometrie,Gottingen,¨ 1899. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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