Topological models for classical configurations

doi:10.1016/S0378-3758(99)00122-6

Journal of Statistical Planning and Inference

Volume 86, Issue 2, 1 May 2000, Pages 421-434

https://doi.org/10.1016/S0378-3758(99)00122-6 Get rights and content

Abstract

Two theorems from Euclidean plane geometry, due respectively to Pappus and Desargues, each give rise to an interesting finite geometry. Both of these geometries are readily modelled by any configuration in the plane arising from the relevant theorem, but these models have deficiencies. We consider alternative models, in the form of (1) partially balanced incomplete block designs, with association classes determined by a strongly regular graph corresponding to the geometry, and (2) surface imbeddings of that strongly regular graph. This approach is motivated by a third geometry, the Fano plane.

Introduction

Formally, a geometry consists of a point set $P$ , a line set $L$ disjoint from $P$ , and a symmetric incidence relation $I⊆(P×L)∪(L×P)$ . Thus if $(p,l)$ — and hence $(l,p)$ — belongs to $I$ , we regard $p$ as being on $l$ — and $l$ as being on $p$ . This formulation facilitates discussion of duality, where the roles of the undefined terms “point” and “line” are interchanged. In discussions not involving duality, it is convenient to regard each line as being a subset of the point set $P$ and, consistent with our usual intuition for Euclidean geometry, that is our choice here.

To impose more structure on a geometry, we add conditions:

(A1) each point belongs to exactly $r$ lines;

(A2) each line consists of exactly $k$ points;

(A3) each pair of distinct points belong to at most one line.

A finite geometry satisfying these axioms is called a configuration. In this paper, we focus on configurations having $k=3$ . (For a more general discussion of configurations see Coxeter (1950); for additional background material, see Colbourn and Dinitz (1996) and chapter 1 of Smart (1994).) A common example is the geometry of Fano, or the Fano plane, given its usual depiction in Fig. 1. For this labelling, $P={0,1,2,3,4,5,6}$ , and $L={{0,1,3},{1,2,4},{2,3,5},{3,4,6},{4,5,0},{5,6,1},{6,0,2}}$ , where we note that the lines can be cyclically generated, using the group $Z_{7}$ , from the initial line ${0,1,3}$ . The latter is an example of a perfect difference set, in that each element of $Z_{7} −{0}$ occurs exactly once as a difference $a−b$ , where $a$ and $b$ are distinct in ${0,1,3}$ . This means that, for this configuration, (A3) can be strengthened to:

$(A3 ′)$ each pair of distinct points belong to exactly one line.

(This can also be verified directly from Fig. 1. Note also that $r=3$ .) Thus we have a balanced incomplete block design (BIBD), consisting of $v=7$ objects (points) and $b=7$ blocks (lines), now recorded in Table 1. A BIBD has a fifth parameter $λ$ (to go with $v, b, r$ , and $k$ ), with each pair of distinct objects belonging to exactly $λ$ blocks; here $λ=1$ , from $(A 3′)$ . Thus the Fano plane is a $(7,7,3,3,1)$ -BIBD, or Steiner Triple System (since $k=3$ and $λ=1$ ).

So far we have two models for the Fano Plane, given by Table 1 — or the preceding listings of $P$ and $L$ — and by the representation of Fig. 1. But the former lacks all geometric flavor (of points and lines being subsets of some Euclidean space, for example), and the latter has certain deficiencies:

(D1) The line ${1,2,4}$ is differently shaped, yet that line is not distinguishable from the others via the axioms.

(D2) Each other line has two “end” points and one “middle” point, yet there is no axiom for “betweenness” here.

(D3) There are three “crossings” in the figure, that have no significance in the geometry.

(D4) One cannot discern that $r=3$ , by looking at small neighborhoods of points 1, 2, 4, and 0 in the figure.

To overcome these deficiencies, we first associate a graph with the geometry. We follow Coxeter (1950) in calling this the Menger graph. The points of the geometry become the vertices of the graph, and two vertices are adjacent if the points they represent are collinear (belong to a common line). The geometry of Fano has Menger graph $K_{7}$ , the complete graph of order seven. Then, we imbed $K_{7}$ on an orientable surface $S_{k}$ (a sphere with $k$ “handles” attached, where $k⩾0$ ; thus $S_{0}$ is the sphere, $S_{1}$ the torus, and so forth) so that certain regions depict the lines of the geometry. The remaining regions are hyperregions of the imbedding of the geometry. (Every geometry is a hypergraph, with lines as hyperedges, so an imbedded geometry decomposes a surface into hyperregions.) The graph imbedding will have bichromatic dual, reflecting a $K_{3}$ -decomposition of the graph (edges partitioned into 3-cycles). That is, each edge will bound one region depicting a line of the geometry and one hyperregion. Any imbedding of $K_{7}$ on the torus models the Fano plane in this fashion; see Fig. 2a. The unshaded regions model the lines, while the shaded regions are what remains in the torus after the geometry is modelled: the hyperregions. As there is no imbedding of the Fano plane in $S_{0}$ (we cannot do better than having all hyperregions triangular, for $λ=1$ ), we say that this geometry has genus one. Note that the deficiencies (D1)–(D4) of the model of Fig. 1 have all been remedied by the model of Fig. 2a. The reader can also check the following formal axioms for the geometry of Fano against Fig. 2a.

(FA1) There is at least one line.

(FA2) Every line contains exactly three points.

(FA3) Not all points lie on one line.

(FA4) Two distinct points belong to exactly one common line.

(FA5) Two distinct lines contain exactly one common point.

From these axioms, one can deduce theorems, such as

(FT1) $|P|=7$ .

(FT2) $|L|=7$ .

(FT3) Each point belongs to exactly three lines (i.e. $r=3$ ).

The axioms are said to be consistent, as we have found a model realizing them. If the model is unique up to isomorphism, as is the case here, then the axioms are said to be complete. Thus the three theorems above can also be deduced from our model.

Now denote the map, consisting of the $K_{7}$ imbedding of Fig. 2a into $S_{1}$ , by $M$ . We comment briefly on $Aut M$ , the group of map automorphisms (graph automorphisms which preserve oriented region boundaries) which also preserve region colors (so that lines of the geometry are preserved as well). From the figure it is apparent that diagonal translations are in $Aut M$ , so that $Aut M$ contains $Z_{7}$ , which acts regularly as group of collineations. (This is also clear from Table 1.) It can also be checked that $Aut M$ contains 3-fold rotations at every vertex. In fact all of $Aut M$ is generated by these two.

The Fano plane, $PG (2,2)$ , is the simplest case of a finite projective geometry $PG (n,q)$ of dimension $n⩾2$ and order a prime power $q$ . $PG (2,q)$ is a $(q^{2} +q+1, q^{2} +q+1,q+1,q+1,1)$ -BIBD. For extensions of the construction of Fig. 2 to $PG (2,q)$ for $q⩾3$ , see White (1995); for extensions to $n⩾3$ , see Figueroa-Centeno (1999). In both places we find a formal definition of the genus $γ(G)$ of a geometry $G$ . First define the bipartite incidence graph $G(G)$ of $G$ , by taking $P∪L$ as the vertex set, and all pairs ${p,l}$ , where $p∈P$ and $l∈L$ , and where $p∈l$ , as edges. (Coxeter calls this the Levi graph in Coxeter (1950), where he constructs regular maps for it.) Then $γ(G)=γ(G(G))$ , where the latter is the minimum $k$ for which $G(G)$ imbeds on $S_{k}$ . We observe that this parameter is well defined when the geometries considered are complete, as is the case for all geometries we consider in this paper. Then a natural modification of the imbedding of $G(G)$ produces a model of $G$ on $S_{k}$ , and conversely. By condition (A3) for configurations, $G(G)$ is easily seen to have girth at least six, so that a hexagonal imbedding is necessarily minimal. For the $K_{7}$ imbedding of Fig. 2a modelling the Fano plane $F, G(F)$ can be obtained by inserting a new (line) vertex in each white region, joining it by an edge to each (point) vertex in the boundary of that region, and then deleting all the $K_{7}$ edges (see Fig. 2b). The result is a hexagonal imbedding of the Heawood graph in $S_{1}$ . (The seven hexagonal regions are mutually incident, affirming that seven is the map-coloring number of the torus.)

In this paper we study two other configurations, that rival the Fano plane for prominence among finite geometries.

Section snippets

The geometry of Pappus

The geometry of Pappus of Alexandria (ca. 300–350 a.d.) arises from the following Theorem of Pappus, as part of Euclidean plane geometry:

Theorem of Pappus

If $A, B$ , and $C$ are three distinct points on line $L$ and $A′, B′$ , and $C′$ are three different distinct points on line $L′≠L$ , then the points $AB′ ↔ ∩ A′B ↔, AC′ ↔ ∩ A′C ↔$ , and $BC′ ↔ ∩ B′C ↔$ are collinear.

One such situation is illustrated in Fig. 3. Immediately we get a partial block design of nine objects (points) and nine blocks (lines), given in Table 2. The associated Menger

The geometry of Desargues

The geometry of Desargues ( $1593$ – $1662$ ) arises from the following Theorem of Desargues, as part of Euclidean plane geometry:

Theorem of Desargues

If two triangles are perspective from a point $,$ then they are perspective from a line.

In Fig. 6, triangles $BCD$ and $EFG$ are perspective from point $A$ . Let $H= BC ↔ ∩ EF ↔$ , $I= CD ↔ ∩ FG ↔$ , and $J= BD ↔ ∩ EG ↔$ . Then the claim of Desargues’ Theorem is that $H$ , $I$ , and $J$ are collinear.

Now we get a partial block design of 10 objects (points) and 10 blocks (lines), given in Table 3. The Menger graph is $Π$

Conclusions

The geometries of Pappus and of Desargues are interesting in their own right, as nontrivial examples of simple axiom systems and for the properties they possess. They both occur as configurations in the Euclidean plane arising from theorems in the Euclidean plane geometry. And, they both occur similarly within each $PG (2,q)$ , for $q≠2$ . (The Fano plane has too few points.) For example, $PG (2,3)$ is a $(13,13,4,4,1)$ -BIBD, with lines cyclically generated by $Z_{13}$ from initial line ${0,1,3,9}$ , a perfect

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