INCIDENCE GEOMETRIES Part II

1
INCIDENCE GEOMETRIESPart II

• Further Examples and Properties

2
Reye Configuration

• Reye Configuration of points, lines and planes in
  the 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 -
3
Theodor Reye

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

4
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.(?)
• If the radii are the same, the external center is
  at infinity.

5
Residual geometry

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

G
Gx
x
6
Reye Configuration -Revisited

• Reye 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.

7
Exercises

• N1. Let there be three circles in a plane giving
  rise to 3 internal and 3 external centers of
  similitude. Prove that the three external center
  of similitude are colinear.

8
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.
• A maximal flag (flag of rank I is called a
  chamber. A flag of rank I-1 is called a wall.
• To each geometry G we can associate the chamber
  graph
• Vertices chambers
• Two chamber are adjacent if and only if they
  share a common wall.
• (See Egon Shulte, ..., Titts systems)

9
The 4-Dimensional Cube Q4.
0010
0001
0000
0100
1000
10
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

11
4-dimensional Cube.
0110
0010
0111
1110
0011
1010
1111
1011
0001
1101
1001
0000
0100
1100
1000
12
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.

13
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.

14
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.

15
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 -
16
Exercises

• N1 Determine all residual geometries of Reyeve
  configuration
• N2 Determine all residual geometries of Q4.
• N3 Determine all residual geometries of
  Platonic solids.
• N4 Determine the Levi graph of the geometry for
  the grup Z2 Z2 Z2, with three cyclic
  subgroups, generated by 100, 010, 001,
  respectively.
• (Add Exercises for truncations!!!)

17
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.

18
Ranked Posets

• Note that a ranked poset (P,) or 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.
• If we are given a poset (P, ) with a 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.

19
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.

20
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 rank function is changed by 1 at
  each step of the path.

21
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(x2), x lt y there exist exactly two elements
  y1, y2 such that x lt y1 lt z, x lt y2 lt z.
• Each nonempty interval I(x,y) is connected.
• Note that abstract poytopes are a special case of
  posets but they form also a generalization of the
  convex polytopes.

22
Exercises

• 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,

23
The Poset
1

• In the Hasse diagram we have the following local
  picture

t2
h
t1
t3
t4
t5
t6
e2
e1
e3
e4
e5
e6
f2
f1
f3
f4
f5
f6
v2
v0
v1
v3
v4
v5
v6
0