Motivation and definitions INCIDENCE GEOMETRIES Motivation and definitions

Plan In this chapter we will cover the following: motivation incidence geometries incidence structures combinatorial configurations

Incidence structure An incidence structure C is a triple C = (P,L,I) where P is the set of points, L is the set of blocks or lines I  P  L is an incidence relation. Elements from I are called flags. The bipartite incidence graph G(C) with black vertices P, white vertices L and edges I is known as the Levi graph of the structure C.

Examples 1. Each graph G = (V,E) is an incidence structure: P = V, L = E, (x,e) 2 I if and only if x is an endvertex of e. 2. Any family of sets F µ P(X) is an incidence structure. P = X, L = F, I = 2. 3. A line arrangement L = {l1, l2, ..., ln} consisting of a finite number of n distinct lines in Euclidean plane E2 defines an incidence structure. Let V denote the set of points from E2 that are contained in at least two lines from L. Then: P = V, L = L and I is the point-line incidence in E2.

Exercises N1. Draw the Levi graph of the incidence structure defined by the complete bipartite graph K3,3. N2. Draw the Levi graph of the incidence structure defined by the powerset P({a,b,c}). N3. Determine the Levi graph of the incidence structure, defined by an arrangemnet of three lines forming a triangle in E2.

Incidence geometry Incidence geometry (G,c)of rank k is a graph G with a proper vertrex coloring c, where k colors are used. Sometimes we denote the geometry by (G,c,I,~). Here c:VG ! I is the coloring and |I| = k is the number of colors, also known as the rank of G. Also ~ is the incidence. I is the set of types. Note that only object of different types may be incident.

Examples 1. Each incidence structure is a rank 2 geometry. (Actualy, look at its Levi graph.) 2. Each 3 dimensional polyhedron is a rank 3 geometry. There are three types of objects: vertices, edges and faces with obvious geometric incidence. 3. Each (abstract) simplicial complex is an incidence geometry. 4. Any complete multipartite graph is a geometry. Take for instance K2,2,2, K2,2,2,2, K2,2, ..., 2.

Pasini Geometry Pasini defines incidence geometry (that we call Pasini geometry) in more restrictive way. For k=1, the graph must contain at least two vertices: |V(G)|>1. For k>1: G has to be connected, For each x  V(G) the (k-1)-colored graph (Gx,c), called residuum, induced on the neigbors of x is a Pasini geometry of rank (k-1).

Incidence geometries of rank 2 Incidence geometries of rank 2 are simply bipartite graphs with a given black and white vertex coloring. Rank 2 Pasini geometries are in addition connected and the valence of each vertex is at least 2: d(G) >1.

Example of Rank 2 Geometry Graph H on the left is known as the Heawood graph. H is connected H is trivalent: d(H) = D(H) = 3. H je bipartite. H is a Pasini geometry.

Another View Geometry of the Heawood graph H has another interpretation. Rank = 2. There are two types of objects in Euclidean plane, say, points and curves. There are 7 points, 7 curves, 3 points on a curve, 3 curves through a point. The corresponding Levi graph is H!

In other words ... The Heawood graph (with a given black and white coloring) is the same thing as the Fano plane (73), the smallest finite projective plane. Any incidence geometry can be interpeted in terms of abstract points, lines. If we want to distinguish geometry (interpretation) from the associated graph we refer to the latter the Levi graph of the corresponding geometry.

Simplest Rank 2 Pasini Geometries Cycle (Levi Graph) “Simplest” geometries of rank 2 in the sense of Pasini are even cycles. For instance the Levi graph C6 corresponds to the triangle. Triangle (Geometry)

Rank 3 Incidence geometries of rank 3 are exactly 3-colored graphs. Pasini geometries of rank 3 are much more restricted. Currently we are interested in those geometries whose residua are even cycles. Such geometries correspond to Eulerian surface triangulations with a given 3-vertex coloring.

Flag System as Geometries Any flag system  µ V £ E £ F defines a rank 3 geometry on X = V t E t F. There are three types of elements and two distinct elements of X are incicent if and only if they belong to the same flag of .

Exercises N1. Prove that the Petrie dual of a self-avoiding map is self-avoiding. N2. Prove that any operation Du,Tr,Me,Su1, ... of a self-avoiding map is self-avoiding. N3. Prove that BS of any map is self-avoiding. N4. Show that any self-avoding map may be considered as a geometry of rank 4 (add the fourth involution).

Self-avoiding maps as Geometries of rank 4 Consider a generalized flag system  µ V £ E £ F £ P that defines a rank 4 geometry on X = V t E t F t P. There are four types of elements and two distinct elements of X are incident if and only if they belong to the same flag of . We may take any self-avoiding map M and the four involutions 0,1,2 and 3 and define the above geometry.

Homework H1 Describe the rank 4 geometry of the projective planar map on the left.

Geometries from Groups Let G be a group and let {G1,G2,...,Gk} be a family of subgroups of G. Form the cosets xGt, t 2 {1,2, ..., k}. An incidence geometry of rank k is obtained as follows: Elements of type t 2 {1,2,...,k} are the cosets xGt. Two cosets are incident: xGt ~ yGs if and only if xGt Å yGs ¹ ;.

Q – The Quaternion Units 1 -1 i -i j -j k -k

Geometry from Quaternions Example: Q = {+1,-1,+i,-i,+j,-j,+k,-k}. Gi = {+1,-1,+i,-i}, Gj = {+1,-1,+j,-j}, Gk ={+1,-1,+k,-k}.

Quaternions - Continiuation j,k Levi graph is an octahedron. Labels on the left: i = {+1,-1,+i,-i} j,k = {+j,-j,+k,-k}, etc. k j i i,j i,k

Quaternions– Examle of Rank 4 Geometry. j,k Levi graph was an octahedron. Notation: i = {+1,-1,+i,-i} j,k = {+j,-j,+k,-k}, itd. If we add the sugroup G0 = {+1,-1}, four more cosets are obtained: Additional notation: 1 = {+1,-1},i’={+i,-i}, etc. k’ j’ k 1 j i i’ i,k i,j