INCIDENCE GEOMETRIES

1
INCIDENCE GEOMETRIES

• CHAPTER 4

2
Contents

1. Motivation
2. Incidence Geometries
3. Incidence Geometry Constructions
4. Residuals, Truncations - Sections, Shadow Spaces
5. Incidence Structures and Combinatorial
   Configurations
6. Substructures, Symmetry and Duality
7. Haar Graphs and Cyclic Configurations
8. Algebraic Structures
9. Euclidean Plane, Affine Plane, Projective Plane
10. Point Configurations, Line Arrangements and
    Polarity

• Pappus and Desarguers Theorem
• Existence and Countnig
• Coordinatization
• Combinatorial Configurations on Surfaces
• Generalized Polygons
• Cages and Combinatorial Configurations
• A Case Study The Gray Graph
• Another Case Study - Tennis Doubles
• Martinetti-Boben Theorem

3
1. Motivation
4
Motivation

• When Slovenia joined the European Union it
  obtained 7 seats in the Parliament of the
  European Union. In 2004 the first elections to
  the European Parliament in Slovenia were held.
• There were 13 political parties (7 parliamentary
  parties 1, 2, 3, 4, 5, 6, 7, and 6
  non-parliamentary parties A, B, C, D, E, F)
  competing for these seats. TV Slovenia decided to
  cover the campain by hosting political parties in
  6 TV shows a,b,c,d,e,f.
• TV asked mathematicians to help them select the
  guests in a fair way.

5
Motivation

• With a little help from mathematicians TV came up
  with the following schedule.

a b c d e f
A B C D E F
1 2 3 1 2 3
4 6 4 7 5 6
5 7
6
Example TV coverage of EU parliamentary
elections in Slovenia
TV Shows Parties Parties Parties Parties
a A 1 4 5
b B 2 6
c C 3 4
d D 1 7
e E 2 5
f F 3 6 7
7
Model

• We can model the above schedule as follows
• Let P 1,2,3,4,5,6,7,A,B,C,D,E,F
• Let L a,b,c,d,e,f
• Let I ½ P L such that
• (p,L) 2 I if and only if political party p
  appears in the show L.
• I (A,a), (1,a), (4,a), (5,a), ...

8
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.

9
Levi Graph

• 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.

10
Levi graph for the Election structure
A
a

• On the left there is the Levi graph for the
  incidence structure of the media coverage of the
  European Union Parliament elections in Slovenia.
• Each parliamentary party appears twice and each
  non-parliamentary party appears once. (check
  valence!)

1
5
4
B
e
D
C
c
2
d
E
b
6
7
3
F
f
11
Menger graph

• Given an incidence structure C (P,L,I) we say
  that two points p and q are collinear, if there
  is a line L that contains both of them.
• Menger graph M(C).
• Vertices P
• p q if and only if p and q are collinear.

12
Menger Graph from Levi Graph

• There is a simple procedure for computing M from
  L. Take the pure graph power L(2). It is obtained
  from L by taking the same vertex set and making
  two vertices adjacent in L(2) if and only if they
  are at distance two in L. Since L is bipartite
  L(2). has (at least) two components. The one
  defined on the black vertices (corresponding to
  points of the incidence structure) is Menger
  graph M. The other one is called dual Menger
  graph.

13
Menger graph for the Election structure

• On the left there is the Menger graph for the
  incidence structure of the media coverage of the
  European Union Parliament elections in Slovenia.

A
1
4
5
B
C
D
2
E
6
3
F
7
14
Configuration Graph

• The configuration graph K is the complement of
  the Menger graph. The dual configuration graph
  is the complement of the dual Menger graph.

15
Dual Configuration graph for the Election
structure

• On the left there is the dual configuration graph
  for the incidence structure of the media coverage
  of the European Union Parliament elections in
  Slovenia.

f
e
c
d
a
b
16
Dual Configuration graph for the Election
structure

• The Hamilton path abcdef in the dual
  configuration graph guarantees that no political
  party appears in two consecutive TV shows.

f
e
c
d
a
b
17
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 the 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.

18
Exercises 1

• 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 power set P(a,b,c).
• N3. Determine the Levi graph of the incidence
  structure, defined by an arrangement of three
  lines forming a triangle in E2.
• N4 Determine the Levi graph of the line
  arrangement on the left.

19
2. Incidence Geometry
20
Incidence geometry

• An 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,,T,c).
  Here cV(G) ! T is the coloring and T k is
  the number of colors, also known as the rank of
  G. The relation is called the incidence.
• T is the set of types. Note that only objects of
  different types may be incident.

21
Morphisms or representations

• Given two incidence geometries (G,,c,T) and
  (G',,c',T') a pair (f,g) of mappings
• f G ! G' and
• g T ! T' is called a morphism of geometries (or
  representation) if the following is true
• for any v 2 V(G) c'(f(v)) g(c(v)).
• for any u,v 2 V(G) if u v then either f(u)
  f(v) or f(u) f(v).

22
Special morphisms

• Some morphisms have nice properties and deserve
  special attention.
• We call a representation dimension-preserving if
• for any u,v 2 V(G) if u v then f(u) f(v).
• We call a representation faithful or strong if
• for any u,v 2 V(G) u v if and only if f(u)
  f(v).
• A faithful representation in which both f and g
  are injective is called realization.
• A morphism is an isomorphism if both f and g are
  bijections and the inverse pair (f-1,g-1) is a
  morphism too.
• The image of a representation is geometry. The
  image of a realization is isomorphic to the
  original.
• For stirng geometries we seek representations and
  realizations in sets. Vertices are mapped to the
  elements (or singletons) of S and the faces to
  subsets of S. The incidence ui uj, i lt j is
  represented by inclusions S(ui) µ S(uj).

23
Automorphisms

• There are two types of automorphisms in a
  geometry (G,,c,T). Aut0 G contains
  type-preserving automorphisms. (g id). Aut G
  contains all (extended) automorphisms.
• In the case of string geometries we want the
  linear order on T to be respected (or reversed).
  In the case of extended automorphisms we speak of
  dualities that map faces of rank r to rank n-r.

24
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. Incidence is defined by
  inclusion of simplices.
• 4. Any complete multipartite graph is a geometry.
  Take for instance K2,2,2, K2,2,2,2, K2,2, ..., 2.
  The vertex coloring defining the geometry in each
  case is obvious.

25
Pasini Geometry

• Pasini defines incidence geometry (that we call
  Pasini geometry) in a more restrictive way.
• For k1, the graph must contain at least two
  vertices V(G)gt1.
• For kgt1
• 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).

26
String geometries

• A geometry G over the set of types T -1,0,1,
  ..., n is called a string geometry if the
  following (1-2) is true (the elements of G are
  called faces, faces of type 0 are called vertices
  (or points), faces of type 1 are called edges (or
  lines), faces of type n-1 are called facets.). It
  is called pure string geometry if (1-3) is true.
• There are exactly two improper faces u-1 2 V(G)
  of type -1 and one element un 2 V(G) of type n
  (both incident with every other face). The rest
  are called proper faces.
• If ui, uj, uk are elements of respective types
  i lt j lt k and ui uj, uj uk, then ui uk.
• Every collection of mutually incident faces U can
  be extended to a sequence of (n2) mutually
  incident faces. (In other words all chambers
  have rank n2.)

27
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) gt1.

28
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 is bipartite.
• H is a Pasini geometry.

29
Another View

• The 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!

30
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 the geometry
  (interpretation) from the associated graph we
  refer to the latter as the Levi graph of the
  corresponding geometry.

31
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)
32
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 vertex 3-coloring.

33
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 ?.

34
Self-avoiding maps

• Recall that a map is self-avoiding if and only if
  neither the skeleton of the map nor the skeleton
  of its dual has a loop.

35
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 a geometry
  as above.

36
Exercises 2

• 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-avoiding map may be
  considered as a geometry of rank 4 (add the
  fourth involution).

37
Homework 2

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

38
3. Incidence Geometry Constructions
39
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 .

40
Q The Quaternion Units
Q 1 -1 i -i j -j k -k
1 1 -1 i -i j -j k -k
-1 -1 1 -i i -j j -k k
i i -i -1 1 k -k -j j
-i -i i 1 -1 -k k j -j
j j -j k -k -1 1 i -i
-j -j j -k k 1 -1 -i i
k k -k j -j -i i -1 1
-k -k k -j j i -i 1 -1
41
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.

42
Quaternions - Continiuation
j,k

• The Levi graph is an octahedron.
• Labels on the left
• i 1,-1,i,-i
• j,k j,-j,k,-k, etc.

j
k
i
i,j
i,k
43
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, etc.
• If we add the sugroup G0 1,-1, four more
  cosets are obtained
• Additional notation
• 1 1,-1,ii,-i, etc.

k
j
k
1
j
i
i
i,k
i,j
44
Reyes Configuration

• Reyes configuration of points, lines and planes
  in 3-dimensional projective space consists of
• 8 1 3 12 points (3 at infinity)
• 12 4 16 lines
• 6 6 12 planes.

P12 L16 S12
P12 - 4 6
L16 3 - 3
S12 6 4 -
45
Theodor Reye

• Theodor Reye (1838 - 1919), German Geometer.
• Known for his book
  Geometrie der Lage (1866 and 1868).
• Published his famous configuration in 1878.
• Posed the problem of configurations.

46
Centers of Similitude

• We are interested in tangents common to two
  circles in the plane.
• The two intersections are called the centers of
  similitudes of the two circles. The blue center
  is called the internal, the red one is the
  external center.
• If the radii are the same, the external center is
  at infinity.

47
Reyes Configuration -Revisited

• Reyes configuration can be obtained from centers
  of similitudes of four spheres in three space
  (see Hilbert ...)
• Each plane contains a complete quadrangle.
• There are 2 C(4,2) 2 4
  3/2 12 points.

48
Exercises 3-1

• N1. Consider the geometry defined by Z3 and Z5 in
  Z15. Draw its Levi graph.
• N2. Draw the Levi graph of the geometry defined
  by all non-trivial subgroups of the symmetric
  group S3.
• N3. Draw the Levi graph of the geometry defined
  by all non-trivial subgroups of the group Z23.

49
Exercises 3-2

• N4. Let there be three circles in a plane giving
  rise to 3 internal and 3 external centers of
  similitude. Prove that the three external centers
  of similitude are colinear.

50
4. Residuals, Truncations - Sections, Shadow
Spaces
51
Residual geometry

• Each incidence geometry
• G (G, , T,c)
• (G,) a simple graph
• c, proper vertex coloring,
• T collection of colors.
• c V(G) ! T
• Each element x 2 V(G) determines a residual
  geometry Gx. defined by an induced graph defined
  on the neighborhood of x in G.

G
Gx
x
52
Flags and Residuals

• In an incidence geometry G a clique on m vertices
  (complete subgraph) is called a flag of rank m.
• Residuum can be definied for each flag F ½ V(G).
  G(F) ÅG(x) Gx x 2 F.

53
Chambers and Walls

• A maximal flag (flag of rank T is called a
  chamber. A flag of rank T-1 is called a wall.
• To each geometry G we can associate the chamber
  graph
• Vertices chambers
• Two chambers are adjacent if and only if they
  share a common wall.
• (See Egon Shulte, ..., Tits systems)

54
The 4-Dimensional Cube Q4.
0010
0001
0000
0100
1000
55
Hypercube

• The graph with one vertex for each n-digit binary
  sequence and an edge joining vertices that
  correspond to sequences that differ in just one
  position is called an n-dimensional cube or
  hypercube.
• v 2n
• e n 2n-1

56
4-dimensional Cube.
0110
0010
0111
1110
0011
1010
1111
1011
0001
1101
1001
0000
0100
1100
1000
57
4-dimensional Cube and a Famous Painting by
Salvador Dali

• Salvador Dali (1904 1998) produced, in 1954,
  the Crucifixion (Metropolitan Museum of Art, New
  York) in which the cross is a 3-dimensional net
  of a 4-dimensional hypercube.

58
4-dimensional Cube and a Famous Painting by
Salvador Dali

• Salvador Dali (1904 1998) produced in 1954, the
  Crucifixion (Metropolitan Museum of Art, New
  York) in which the cross is a 3-dimensional net
  of a 4-dimensional hypercube.

59
The Geometry of Q4.

• Vertices (Q0) of Q4 16
• Edges (Q1)of Q4 32
• Squares (Q2) of Q4 24
• Cubes (Q3) of Q4 8
• Total 80
• The Levi graph of Q4 has 80 vertices and is
  colored with 4 colors.

60
Residual geometries of Q4.
V E S Q3.
G(V) - 4 6 4
G(E) 2 - 3 3
G(S) 4 4 - 2
G(Q3) 8 12 6 -
61
Truncations or Sections

• Given a geometry G (V,,T,c) and a subset of
  types J µ T, define a J-section G/J of G as the
  geometry H (U,,J,c), where U v 2 V c(v) 2
  J and H is the induced subgraph of G.

62
Quaternions Example of Rank 4 Geometry - Section
j,k
j
k
i
i,j
i,k
Rank 3 section
Rank 4 geometry
63
Shadow Spaces

• Given a geometry G (V,,T,c) and J µ T we may
  define an incidence structure Spa(G,J) whose
  points are J-flags and the blocks are composed of
  those sets of J-flags that belong to the residual
  geometry G(F) for some flag F from the original
  geometry G.

64
Shadow Spaces - An Example
3
4

• Let us denote the types
• I g,r,b.
• Let J r,b. There are three J-flags 26, 45
  and 56. The set system for the shadow space
• 45,26,45,56,26,56.
• For J g,b we get three flags
• 16,14,34
• The set system for the shadow space
• 16,34,14,16,14,34

5
6
1
2
65
Shadow spaces of Maps

• For maps as rank 3 geometries the notion of
  shadow spaces gives rise to an interesting
  interpretation. There are three types of objects
  v,e,f.
• Hence, there are 7 types of shadow spaces

• v - primal id
• e - medial Me
• f - dual Du
• v,e - truncation Tr
• v,f - Me Me
• e,f - leapfrog Le
• v,e,f- Co

66
Shadows - Example

• Our map is a prism. All flags (structured by
  type)
• ,
• 1,2,3,4,5,6
• a,b,c,d,e,f,g,h,i
• A,B,C,D,E
• 1a,1d,1e,2a,2b,2f,3e,3f,3i,4c,4d,4h,5b,5c,5g,6g,6h
  ,6i
• 1A,1B,1C,2A,2B,2D,3B,3C,3D,4A,4C,4E,5A,5D,5E,6C,6D
  ,6E
• aA,aB,bA,bD,cA,cE,dA,dC,eB,eC,fB,fD,gD,gE,hC,hE,iC
  ,iD
• 1aA,1aB,1dA,1dC,1eB,1eC,2aA,2aB,2bA,2bD,2fB,2fD,3e
  B,3eC,3fB,3fD,3iC,3iD,4cA,3cE,4dA,4dC,4hC,4hE,5bA,
  5bD,5cA,5cE,5gD,5gE,6gD,6gE,6hC,6hE,6iC,8iD

5
4
c
E
g
h
6
d
b
C
D
i
3
f
e
B
2
a
1
A
67
Shadows - Example - Primal

• Our map is a prism. T v,e,f
• J v
• J-flags 1, 2, 3, 4, 5, 6
• Sets 12, 13, 14, 23, 25, 36, 45, 46, 56, 123,
  456, 1245, 1346, 2356.

5
4
c
E
g
h
6
d
b
C
D
i
3
f
e
B
2
a
1
A
68
Shadows - Example - Dual

• Our map is a prism. T v,e,f
• J f
• Flags A,B,C,D,E
• Sets AB, AC, AD, AE, BC, BD, CD, CE, DE, ABC,
  ABD, BCD, CDE, ACE, ADE.

5
4
c
E
g
h
6
d
b
C
D
i
3
f
e
B
2
a
1
A
69
Shadows - Example - Medial

• Our map is a prism. T v,e,f
• J e
• Flags a,b,c,d,e,f,g,h,i
• Sets ae,ab,ad,af,bc,bf,bg,cd,cg,ch,de,dh,ef,ei,f
  i,gh,gi,hi, aef, bfgi, dehi, abcd,cgh.

5
4
c
E
g
h
6
d
b
C
D
i
3
f
e
B
2
a
1
A
70
Shadows - Example - Truncation

• Our map is a prism. T v,e,f
• J v,e
• Flags 1a,1d,1e,2a,2b,2f,3e,3f,3i,4c,4d,4h,5b,5c,5
  g,6g,6h,6i
• Sets 1a,1d,1e,2a,2b,2f,3e,3f,3i,4c,4d,4h,5b,5c,5g
  ,6g,6h,6i
• ...

5
4
c
E
g
h
6
d
b
C
D
i
3
f
e
B
2
a
1
A
71
Posets

• Let (P,) be a poset. We assume that we add two
  special (called trivial) elements, 0, and 1, such
  that for each x 2 P, we have 0 x 1.

72
Ranked Posets

• Note that a ranked poset (P,) of rank n has the
  property that there exists a rank function rP !
  -1,0,1,...,n, r(0) -1, r(1) n and if y
  covers x then r(y) r(x) 1. (All linear chains
  have the same length)
• If we are given a poset (P, ) with rank function
  r, then such a poset defines a natural incidence
  geometry.
• V(G) P.
• x y if and only if x lt y.
• c(x) r(x). Vertex color is just the rank.
• The corresponding geometry is a string geometry.
  There is a one-to-one correspondence between the
  two concepts.

73
Intervals in Posets

• Let (P,) be a poset.
• Then I(x,z) y x y z is called the
  interval between x and z.
• Note that I(x,z) is empty if and only if x z.
• I(x,z) is also a ranked poset with 0 and 1.

74
Connected Posets.

• A ranked poset (P,) wih 0 and 1 is called
  connected, if either rank(P) 1 or for any two
  non-trivial elements x and y there exists a
  sequence x z0, z1, ..., zm y, such that there
  is a path avoiding 0 and 1 in the Levi graph from
  x to y and the rank function is changed by 1 at
  each step of the path.

75
Abstract Polytopes

• Peter McMullen and Egon Schulte define abstract
  polytopes as special ranked posets.
• Their definition is equivalent to the following
• (P,) is a ranked poset with 0 and 1 (minimal and
  maximal element)
• For any two elements x and z, such that r(z)
  r(x)2, x lt z there exist exactly two elements
  y1, y2 such that x lt y1 lt z, x lt y2 lt z.
• Each section is connected.
• Note that abstract poytopes are a special case of
  posets but they form also a generalization of the
  convex polytopes.

76
Convex vs abstract polytopes

• To each convex polytope we may associate an
  abstract polytope. For instance, the tetrahedron
  
• 0
• 4 vertices v1, v2, v3, v4.
• 6 edges e1, e2, ..., e6,
• 4 faces t1,t2,t3, t4
• 1
• e1 v1v2, e2 v1v3, e3 v1v4, e4 v2v3, e5
  v2v4, e6 v3v4.
• t4 v1v2v3, t1 v2v3v4, t3 v1v2v4, t2
  v1v3v4.

77
The Poset
1

• In the Hasse diagram we have the following local
  picture

t2
t1
t3
t4
e2
e3
e4
e5
e1
e6
v2
v1
v3
v4
0
78
Diagram geometries

• For any incidence geometry G(V,,T,c) we usually
  study for each pair i,j 2 T the section
  (truncation) of rank two G/(i,j). We
  deliberatly make distinction between G/(i,j) and
  its dual G/(j,i). Sometimes each connected
  component of G/(i,j) has the same structure. This
  is indicated by a diagram. A diagram in an
  edge-labeled graph on the vertex set T, where the
  lables indicate the structure of each section.

79
String diagram geometries

• The edge between i an j is omitted if and only if
  G/(i,j) is a generalized digon. This means that
  each connected component is a complete bipartite
  graph.
• G is called a string diagram geometry if the
  corresponding diagram has a shape of a path (or
  union of paths).
• Example Each abstract polytope is a string
  diagram geometry.

80
The Grassmann graph

• Let G(V,,T,c) be an incidence geometry and let i
  2 T be a type. Then we define the Grassmann graph
  G(i) to on the vertex set V(i) v 2 V c(v)
  i and two vertices u and v are adjacent in G(i)
  if an only if for each j ? i there exists an w 2
  V(j) such that u w and w v (in the original
  geometry.)
• Example For instance, in the case of rank two
  geometries, the Grassmann graphs are exactly the
  Menger graph and the dual Menger graph.

81
Exercises 4-1

• N1. Our map is a prism. I v,e,f
• For each set of type
• J v,f
• J e,f
• J v,e,f
• determine the shadow space.

5
4
c
E
g
h
6
d
b
C
D
i
3
f
e
B
2
a
1
A
82
Exercises 4-2

• N2. Repeat the analysis of previous two slides
  for the simplex K5.
• N3. Repeat the analysis of the previous two
  slides for the generalized octahedron K2,2,2,2.

83
Exercises 4-3

• N4 Determine all residual geometries of Reyes
  configuration
• N5 Determine all residual geometries of Q4.
• N6 Determine all residual geometries of the
  Platonic solids.
• N7 Determine the Levi graph of the geometry for
  the group Z2 Z2 Z2, with three cyclic
  subgroups, generated by 100, 010, 001,
  respectively.

84
Exercises 4-4

• N18 Determine the posets and Levi graphs of
  each of the polytopes on the left.
• Solution for the haxagonal pyramid
• 0
• 7 vertices v0, v1, v2, ..., v6.
• 12 edges e1, e2, ..., e6, f1, f2, ..., f6
• 7 faces h,t1,t2,t3,.., t6
• 1
• e1 v1v2, e2 v2v3, e3 v3v4, e4 v4v5, e5
  v5v6, e6 v6v1, f1 v1v0, f2 v2v0,f3 v3v0,
  f4 v4v0, f5v5v0, f6 v6v0.
• h v1v2v3v4v5v6,
• t1 v1v2v0, t2 v2v3v0, t3 v3v4v0, t4
  v4v5v0, t5 v5v6v0, t6 v6v1v0,

85
5. Incidence Structures
86
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.

87
(Combinatorial) Configuration

• A (vr,bk) configuration is an incidence structure
  C (P,L,I) of points and lines, such that
• v P
• b L
• Each point lies on r lines.
• Each line contains k points.
• Two lines intersect in at most one point.
• Warning Levi graph is semiregular of girth ? 6

88
Symmetric configurations

• A (vr,bk) configuration is symmetric, if
• v b (this is equivalent to r k).
• A (vk,vk) configuration is usually denoted by
  (vk).

89
Small Configurations

• Triangle, the only (32) configuration.
• Pasch configuration (62,43) and its dual Perfect
  Quadrangle (43,62) have the same Levi graph.

90
6. Substructures, Symmetry and Duality
91
Substructures

• An incidence structure C (P, L,I) is a
  substructure of an incidence structure C (P,
  L,I), C µ C, if P µ P, L µ L and I µ I.

92
Duality

• Each incidence structure C (P,L,I) gives rise
  to a dual structure Cd (L,P,Id) with the role
  of points and lines reversed and keeping the
  incidence.
• The structures C and Cd share the same Levi graph
  with the roles of black and white vertices
  reversed.

93
Self-Duality and Automorphisms

• If C is isomorphic to its dual Cd , it is said
  that C is selfdual, the corresponding
  isomorphism is called a (combinatorial) duality.
• A duality of order 2 is called (combinatorial)
  polarity.
• An isomorphism mapping C to itself is called an
  automorphism or (combinatorial) collinearity.

94
Automorphisms and Antiautomorphisms

• Automorphisms of the incidence structure C form a
  grup that is called the group of automorphisms
  and is denoted by Aut0C.
• If automorphisms and dualities (antiautomorphisms)
  are considered together as permutations, acting
  on the disjoint union P ? L, we obtain the
  extended group of automorphism Aut C.
• Warning If C is disconnected there may be
  mixed automorphisms.

95
Graphs and Configurations

• The Levi graph of a configuration is bipartite
  and carries complete information about the
  configuration.
• Assume that C is connected. The extended group of
  automorphisms AutC coincides with the group of
  automorphisms of the Levi graph L ignoring the
  vertex coloring, while Aut0C stabilises both
  colors.

96
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 the 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.

97
Exercises 6

• 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.

98
7. Haar Graphs and Cyclic Configurations
99
Haar graph of a natural number

• Let us write n in binary
• n bk-12k-1 bk-2 2k-2 ... b12 b0
• where B(n) (bk-1, bk-2, ..., b1, b0), bk-1
  1are binary digits of n. Graph H(n) H(k n),
  called the Haar graph of the natural number n,
  has vertex set ui, vi, i0,1,...,k-1. Vertex ui
  is adjacent to vij, if and only if bj 1
  (arithmetic is mod k).

100
Remark

• When defininig H(n) we assumed that k is the
  number of binary digits of n. In general, for
  H(kn) one can take k to be greater than the
  number of binary digits. In such a case a
  different graph is obtained!

101
Example

• Determine H(37).
• Binary digits
• B(37) 1,0,0,1,0,1
• k 6.
• H(37) H(637) is depicted on the left!

102
Dipoles qn

• The dipole qn has two vertices, joined by n
  parallel edges. If we want to distinguish the two
  vertices, we call one black, the other one white.
  On the left we see q5.
• Each dipole is a bipartite graph. Therefore each
  of its covering graphs is a bipartite graph.
• In particular q3 is a cubic graph also known as
  the theta graph q.

103
Cyclic covers over a dipole

• Each Haar graph is a cyclic cover over a dipole.
  One can use the following recipe
• H(37) is determined by a natural number 37, or,
  equivalently by a binary sequence(1 0 0 1 0 1).
• The length is k6, therefore the group Z6.
• The indices are written below
• (1 0 0 1 0 1)
• (0 1 2 3 4 5)
• The 1s appear in positions 0, 3 in 5. These
  numbers are used as voltages for H(37).

0
3
5
Z6
104
Connected Haar graphs

• Graph G is connected if there is a path between
  any two of its vertices.
• There exist disconnected Haar graphs, for
  instance H(10).
• Define n to be connected, if the corresponding
  Haar graph H(n) is connected.
• Disconnected numbers 2,4,8,10,16,32,34,36,40,42,6
  4...

105
The Mark Watkins Graph

• The cubic Haar graph H(536870930) has an
  interesting property. 536870930 is the smallest
  connected number that is cyclically equivalent to
  no odd number.
• Recall that two sets S,T µ Zn are cyclically
  equivalent if there exists a 2 Zn and b 2 Zn
  such that S aT b (mod n).

106
Girth of Connected Haar graphs

• K2 is the only connected 1-valent Haar graph.
• Even cycles C2n are connected 2-valent Haar
  graphs.
• Theorem Let H be a connected Haar graph of
  valence d gt 2. Then either girth(H) 4 or
  girth(H) 6.

107
Cyclic Configurations

• A symmetric (vr) configuration determined by its
  first column s of the configuration table where
  each additional column is obtained from s by
  addition (mod m) is called a cyclic
  configuration Cyc(ms).
• The left figure depicts a cyclic Fano
  configuration Cyc(71,2,4) Cyc(70,1,3).

a b c d e f g
k 1 2 3 4 5 6 0
k1 2 3 4 5 6 0 1
k3 4 5 6 0 1 2 3
108
Connection to Haar graphs

• Theorem A symmetric configuration (vr), r 1
  is cyclic, if and only if its Levi graph is a
  Haar graph with girth ¹ 4.
• Corollary Each cyclic configuration is point-
  and line-transitive and combinatorially
  self-dual.
• Corollary Each cyclic configuration (vr), r gt 2
  contains a triangle.
• Question Does there exist a cyclic configuration
  that is not combinatorially self-polar?

109
Problem

• Study cyclic configurations with respect to flag
  orbits.
• Example On the left we see the smallest
  0-symmetric graph Haar(261) on 18 vertices. It is
  the Levi graph of the cyclic (93) configuration
  having 3 flag orbits.

110
Exercises 7-1

• The graph on the left is the so-called Heawood
  graph H. Prove
• N1 H is bipartite
• N2 H is a Haar graph. (Find n!)
• N3 Determine H as a cyclic cover over q3..
• N4 Prove that H has no cycle of length lt 6.
• N5 Prove that H is the smallest cubic graph of
  girth 6.
• N6 Find a hexagonal torus embedding of H .
• N7 Determine the dual of the embedded H.

111
Exercises 7-2

• N8 Prove that each 2m is a disconnected number.
• N9 Show that the Möbius-Kantor graph G(8,3) is a
  Haar graph of some number. Which number is that?
• N10 () Determine all generalized Petersen
  graphs that are Haar graphs of some natural
  number.
• N11 Show that some Haar graphs are circulants.
• N12 Show that some Haar graphs are
  non-circulants.

112
Exercises 7-3

• N13 Prove that each Haar graph is vertex
  transitive.
• N14 Prove that each Haar graph is a Cayley graph
  for a dihedral group.
• N15 Prove that there exist bipartite Cayley
  graphs of dihedral groups that are not Haar
  graphs (such as the graph on the left).

113
Exercises 7-4

• N16 The numbers n and m are cyclically
  equivalent, if the binary string of the first
  number can be cyclically transformed to the
  binary string of the second number. This means
  that the string can be cyclically permuted,
  mirrored or multiplied by a number relatively
  prime with the string length.
• N17 The numbers n and m are Haar equivalent, if
  their Haar graphs are isomorphic H(n) H(m).
• N18 Prove that cyclic equivalence implies Haar
  equivalence.
• N19 Determine all numbers that are cyclically
  equivalent to 69.
• N20 Use a computer to show that 137331 and
  143559 are Haar equivalent, but are not
  cyclically equivalent.

114
Exercises 7-5

• N21 Show that each Haar graph of an odd number
  H(2n1) is hamiltonian and therefore connected.

115
Homework 7

• Use Vega to explore the edge-orbits of cyclic
  Haar graphs.
• H1. Find an example of a cubic Haar graph that
  has 1,2, or 3 edge orbits.
• H2. Find an example of a quartic Haar graph that
  has 1, 2, 3, or 4 edge orbits. Study the graphs
  with 2 edge orbits.

116
8. Algebraic Structures
117
Real Numbers R.

• Let us review the structure of the set of real
  numbers (real line) R.
• In particular, consider addition and
  multiplication .
• (R,) forms an abelian group.
• (R,) does not form a group. Why?
• (R,,) forms a (commutative) field.

118
Real Numbers R. - Exercises

• N43 Write down the axioms for a group, abelian
  group, a ring and a field.
• N44 What algebraic structure is associated with
  the integers (Z,,)?
• N45 Draw a line and represent the numbers R.
  Mark 0, 1, 2, -1, ½, p.

119
A Skew Field K

• A skew field is a set K endowed with two
  constants 0 and 1, two unary operations
• - K ! K,
• K ! K,
• and with two binary operations
• K K ! K,
• K K ! K,
• satisfying the following axioms
• (x y) z x (y z) associativity
• x 0 0 x x neutral element
• x (-x) 0 inverse
• x y y x commutativity
• (x y) z x (y z). associativity
• (x 1) (1 x) x unit
• (x x) (x x) 1, for x ¹ 0. inverse
• (x y) z x z y z. left
  distributivity
• x (y z) x y y z. right
  distributivity
• A (commutative) field satisfies also
• x y y x.

120
Examples of fields and skew fields

• Reals R
• Rational numbers Q
• Complex numbers C
• Quaterions H (non-commutative!! Will consider
  briefly later!)
• Residues mod prime p Fp
• Residues mod prime power q pk Fq (more
  complicated, need irreducible poynomials!!Will
  consider briefly later!)

121
Complex numbers C

• a a bi 2 C
• a a bi
• b c di 2 C
• ab (ac bd) (bc ad)i
• b ¹ 0, a/b (ac bd) (bc ad)i/c2 d2
• a-1 (a bi)/(a2 b2)

122
Quaternions H.

• Quaternions form a non-commutative field.
• General form
• q x y i z j w k., x,y,z,w 2 R.
• i 2 j 2 k 2 -1.
• q x y i z j w k.
• q x y i z j w k.
• q q (x x) (y y) i (z z) j (w
  w) k.
• How to define q .q ?
• i.j k, j.k i, k.i j, j.i -k, k.j -i,
  i.k -j.
• q.q (x y i z j w k)(x y i z j
  w k)

123
Quaternions H. - Exercises

• N46 There is only one way to complete the
  definition of multiplication and respect
  distributivity!
• N47 Represent quaternions by complex matrices
  (matrix addition and matrix multiplication)!
  Hint q a b -b a. (We are using Matlab
  notation).

a b
-b a
124
Residues mod n Zn.

• Two views
• Zn 0,1,..,n-1
• Define on Z
• x y x y cn
• Zn Z/
• (Zn,) is an abelian group, namely a cyclic
  group. Here is taken mod n!!!

125
Example (Z6, ).
0 1 2 3 4 5
0 0 1 2 3 4 5
1 1 2 3 4 5 0
2 2 3 4 5 0 1
3 3 4 5 0 1 2
4 4 5 0 1 2 3
5 5 0 1 2 3 4
126
Example (Z6, ).
0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 1 2 3 4 5
2 0 2 4 0 2 4
3 0 3 0 3 0 3
4 0 4 2 0 4 2
5 0 5 4 3 2 4
127
Example (Z6\0, ).
It is not a group!!! For p prime, (Zp\0, )
forms a group (Zp, ,) Fp.
1 2 3 4 5
1 1 2 3 4 5
2 2 4 0 2 4
3 3 0 3 0 3
4 4 2 0 4 2
5 5 4 3 2 4
128
Vector space V over a field K

• V V ! V (vector addition)
• . K V ! V (scalar multiple)
• (V,) abelian group
• (l m)x l x m x
• 1.x x
• (l m).x l(m x)
• l.(x y) l.x l.y

129
9. Euclidean Plane, Affine Plane, Projective Plane
130
Euclidean plane E2 and real plane R2

• R2 (x,y) x,y 2 R
• R2 is a vector space over R. The elements of R2
  are ordered pairs of reals.
• (x,y) (x,y) (xx,yy)
• l(x,y) (l x,l y)
• We may visualize R2 as an Euclidean plane (with
  the origin O).

131
Subspaces

• One-dimensional (vector) subspaces are lines
  through the origin. (y ax)
• One-dimensional affine subspaces are lines. (y
  ax b)

y ax b
y ax
o
132
Three important results

• Thm1 Through any pair of distinct points passes
  exactly one affine line.
• Thm2 Through any point P there is exactly one
  affine line l that is parallel to a given affine
  line l.
• Thm3 There are at least three points not on the
  same affine line.
• Note parallel not intersecting or identical!

133
Affine Plane

• Axioms
• A1 Through any pair of distinct points passes
  exactly one line.
• A2 Through any point P there is exactly one line
  l that is parallel to a given line l.
• A3 There are at least three points not on the
  same line.
• Note parallel not intersecting or identical!

134
Examples

• Each affine plane is an incidence structure C
  (P,L,I) of points and lines.
• Let K be a field, then K2 has a structure of an
  affine plane.
• K Fp.
• Determine the number of points and lines in the
  affine plane A2(p) Fp2.

135
Parallel Lines

• Parallel lines l m define an equivalence
  relation on the set of lines.
• l l
• l m ) m l
• l m, m n ) l n.

136
A pencil of parallel lines

• An equivalence class of parallel lines is called
  a pencil of parallel lines.
• Thm. Each pencil of parallel lines defines an
  equivalence relation on the set of lines.

137
Ideal points and Ideal line

• Each pencil of parallel lines defines a new
  point, called an ideal point (or a point at
  inifinity.) New point is incident with each line
  of the pencil.
• In addition we add a new ideal line (or line at
  infinity)

138
Extended Plane

• Let A be an arbitrary affine plane. The incidence
  structure obtained from A by adding ideal points
  and ideal lines is called the extended plane and
  is denoted by P(A).

• Theorem. Let C be an extended plane obtained from
  any affine plane. The following holds
• T1. For any two distinct points P and Q there
  exists a unique line l connecting them.
• T2. For any two distinct lines l and m there
  exists a unique point P in their intersection.
• T3. There exist at least four points P,Q,R,S such
  that no three of them are colinear.

139
Projective Plane

• Axioms for the Projective Plane. Let C be an
  incidence structure of points and lines that
  satisfies the following axioms
• P1. For any two distinct points P and Q there
  exists a unique line l connecting them.
• P2. For any two distinct lines l and m there
  exists a unique point P in their intersesction.
• P3. There exist at least four points P,Q,R,S such
  that no three of them are colinear.

140
Linear Transformations

• In a vector space the important mappings are
  linear transformations
• L(l x m y) l L(x) m L(y). L-1 exists.
• L can be represented by a nonsingular square
  matrix.

141
Semi Linear Transformations

• A semi linear transformation is more general
• L(lx m y) f(l) L(x) f(m) L(y). L-1 exists,
  f K ! K is an automorphism of K.

142
Affine Transformations

• In an affine plane the important mappings are
  affine transformations (affinities).
• An affine transformation maps sets of collinear
  points to collinear points.
• Each affine transformation is of the form A(x)
  c, where A is a semilinear transformation.

143
Projective plane from R3

• Consider the incidence structure defined by
  1-dimensional and 2-dimensional subspaces of R3
  where the incidence is defined by inclusion.
• Call 1-dimensional subspaces points and
  2-dimensional subspaces lines.

144
Homogeneous Coordinates

• Let (a,b,c) ¹ (0,0,0) be a point in R3. There is
  exactly one line through the origin passing
  through (a,b,c). Hence a projective point can be
  represented by (a,b,c). However, for any l ¹ 0
  the same projective point can be represented by
  (l a, l b, l c).
• That is why (a,b,c) are called homogeneous
  coordinates.

145
Unit sphere model

• Take a unit sphere in R3.
• Let pairs of antipodal points be projective
  points.
• Let big circles be projective lines.
• Prove that this system is a model for a
  projective plane.

146
Exercises 9-1

• N1. Conditions 1. and 2. are true for any
  incidence structure. (Prove it!)
• N2 Prove condition 3 for affine planes and find
  a counter-example for general incidence
  structure.
• N3. Prove that this structure satisfies all three
  axioms for the projective plane.
• N4 Prove that in R, Q, Fp, (p- prime) there are
  no nontrivial automorphisms.

147
Exercises 9-2

• N5 Prove that z a z (conjugate) is an
  automorphism of C.
• N6 Go to the library or the internet and find a
  reference to the group of authomorphisms of the
  complex numbers C and the quaternions H.
• N7 Determine the size of the group of
  automorphisms of Fq, for q pk, a power of a
  prime.

148
10. Point Configurations, Line Arrangements,
Polarity
149
Point Configuration

• A point configuration in R2 is a collection of
  points affinely spanning R2.
• In other words not all points are collinear.

150
Line Arrangement

• A line arrangement is a partitioning of the plane
  R2 into connected regions (cells, edges, and
  vertices) induced by a finite set of lines.

151
Area of a Triangle

• Area of the green trapezoid
• A12 (1/2)(y2 y1) (x2 x1)
• In the same way
• A23 (1/2)(y2 y3) (x3 x2)
• A13 (1/2)(y3 y1) (x3 x1)
• Area of the triangle
• T A12 A23 A13.

P2(x2,y2)
y2
y1
P1(x1,y1)
y3
P3(x3,y3)
O
x2
x1
x3
152
Area of a Triangle
P2(x2,y2)
y2
y1
P1(x1,y1)
y3
P3(x3,y3)
O
x2
x1
x3
153
Triple of Collinear Points
P2(x2,y2)
y2
y1
P1(x1,y1)
y3
P3(x3,y3)
O
x2
x1
x3
The points P1(x1,y1), P2(x2,y2), P3(x3,y3), are
collinear if and only if T 0.
154
Point Configurations Line Arrangements

• Each point configuration S gives rise to a line
  arrangement A(S). The lines are determined by all
  pairs of points.
• Another line arrangement A3(S) is determined by
  triples of collinear points.

155
Polarity with Respect to a Circle
p

• Let us consider the extended plane and a circle K
  in it. There is a mapping from points to lines
  (and vice versa). p p a P.
• p polar
• P pole

P
p
P
p
P
156
Polarity with respect to the unit circle

• Given P(a,b) the equation of the polar is
• p y (-a/b)x (1/b)
• p by ax 1
• In general Circle K(p,qr)
• p y(b-q) x(a-p) p(a-p) q(b-q) r2.
• Given
• p y kx n
• P(a,b)
• a -k/n
• b 1/n
• In general
• a p-kr2/(kp n q)
• b q r2/(kp n q)

157
Natural Parameters p,q,r

• For a given point configuration S the center of
  the circle(p,q) is determined as the barycenter
  of S while the radius is given as the average
  distance from the center.

158
Polarity in General

• A general polarity is defined with respect to a
  conic section (ellipse, hyperbola, or parabola).

159
Polar Duality of Vectors and Central Planes in R3.

• A polar duality is a mapping associating a vector
  v 2 R3 with an oriented central plane having v as
  its normal vector and vice versa.

160
A Standard Affine Polar-Duality

• A standard affine polar duality is a mapping
  between non-vertical lines and points of R2
  associating the non-vertical line y ax b with
  the point (a,-b) and vice versa.

161
Polar Duality of Points and Lines in the Affine
Space.

• General rule Take a polar-duality of vectors and
  central planes and consider the intersetion with
  some affine plane in R3 .

162
Homogeneous Coordinates

• Take the affine plane z 1. A point with
  Euclidean coordinates (x,y) can be assigned the
  homogeneous coordinates (x,y,1). Ideal points get
  homogeneous coordinates (x,y,0).

(z0x0,z0y0,z0)
(x0,y0,1)
(x0,y0)
163
Equation of a plane through the origin

• Recall general plane
• ax by cz d.
• Equation of a plane through the origin
• ax by cz 0-
• Another meaning
• (x,y,z) homogeneous coordinates of a projective
  point
• a,b,c homogeneous coordinates of a projective
  line.

164
Point on a Line

• Let (a,b,c) be homogeneous coordinates of a point
  P and let A,B,C be homogeneous coordinates of a
  line p.
• Then P lies on p if and only if aA bB cC 0.

• Let P(a,b,c) and P(a,b,c). The equation of a
  line through P Æ P. is defined by the cross
  product A,B,C (a,b,c) (a,b,c).
• Similarly we get the intersection of two lines.

165
Example

• Polarity of a point configuration consisting of
  the points of a 10 10 grid.
• Parameters of the circle are determined
  automatically.

166
Star Polygons (n/k).

• By (n/k) we denote star polygons.
• Note that each of them defines an incidence
  structure. in which the points are the vertices
  and intersections while the lines are the edges
  of a polygon.

3/1
4/1
5/1
5/2
6/1
6/2
7/1
7/2
7/3
167
Fano Plane

• We obtain the Fano plane from F23. There are
  obviously 7 (non-zero) points Any pair of points
  defines a unique line that contians exactly one
  additional point.

0 0 1 0 1 1 1
0 1 0 1 0 1 1
1 0 0 1 1 0 1
168
Exercises 10-1

• A polarity maps a point configuration to a line
  arrangement and vice versa.
• N1Take an equilateral triangle ABC with sides
  a,b,c. Find a polarity, such that a a A, b a B
  and c a C.
• N2 Determine the polar figure of point
  configuration determined by the vertices of a
  regular n-gon with respect to its inscribed
  circle.
• N3 Determine the polar of an ideal point and the
  pole of the ideal line.

169
Exercises 10-2

• N4 Determine the number of points and lines of
  the incidence structure defined by the star
  polygon 5/2.
• N5() Determine the number of points and lines
  of the incidence structure determined by the star
  polygon n/k.

170
Homework 10

• H1 Prove that the number of points on each line
  of any finite projective plane P is constant, say
  q1. Then q is called the order of P.
• H2 Which axioms of a projective planes are valid
  for a near-pencil N(n)?

171
11. Pappus and Desargues Theorem
172
Pappus Theorem
C

• Let A, B, C be three collinear points and let A',
  B' , C' be another triple of collinear points.
  Let A'' be the intersection of (BC') and (B'C),
  B'' the intersection (A,C') and (A'C), C'' the
  intersetion of (AB') and (A'B). Then the points
  A'', B'' and C'' are collinear.

B
A
C''
B''
A''
A'
B'
C'
173
Desargues Theorem
B''
B'

• Let ABC and A'B'C' be two triangles. Let A'' be
  the intersection of BC and B'C', let B'' be the
  intersection of AC and A'C' and C'' be the
  intersection of AB and A'B'. The lines AA',BB'
  and CC' intersect in a common point O if and only
  if A'', B'' and C'' are collinear.

A'
C''
A
O
B
C
C'
A''
174
Ternary ring coordinatization.
b

• Ternary operation, desrcibed in geometric terms.
• Properties
• (a) x0b 0xbb
• (b)x10 1 x 0 x
• (c) Given x,y,a, there is a unique b such that y
  xab
• (d) Given x,x,y,y with x ¹ x there is a unique
  ordered pair (a,b) such that y xab and
  yxab.
• (e) Given a,a,b,b with a ¹ a, there is a
  unique x such that xabxab.

0,abc
b
0,c
1,b
0,b
0,0
1,0
a,0
175
Pappian and Desarguesian Projective Planes

• Thm. A projective plane is desarguesian if and
  only if the ternary ring is a field or a
  sqew-field.
• Thm. A projective plane is pappian if an only if
  the ternary ring is a field.

176
Non-Desarguesian Projective Plane

• F.R.Multon (1902)
• Points points in the real projective plane.
• Lines
• y