INCIDENCE GEOMETRIESPart II

Further Examples and Properties

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.

P=12 L=16

P=12 - 4 6

L=16 3 - 3

6 4 -

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.”

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.

Residual geometry

• Each incidence geometry• =(, ~, c, I)• (,~) a simple graph• c, proper vertex coloring,• I collection of colors.• c: V ! I

• Each element x 2 V determines a residual geometry x. defined by an induced graph defined on the neighborhood of x in .

x

x

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.

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.

Flags and Residuals

• In an incidence geometry a clique on m vertices (complete subgraph) is called a flag of rank m.

• Residuum can be definied for each flag F ½ V(). (F) = Å{(x) = x |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 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)

The 4-Dimensional Cube Q4.

0000

1000

0010

0100

0001

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

4-dimensional Cube.

0000

1000

0010

0100

0001

1100

1110

0110

01110011

1001 1101

11111011

1010

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.

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.

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.

Residual geometries of Q4.

V E S Q3.

(V) - 4 6 4

(E) 2 - 3 3

(S) 4 4 - 2

(Q3) 8 12 6 -

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!!!)

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.

Ranked Posets

• Note that a ranked poset (P,·) or rank n has the property that there exists a rank function r:P ! {-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() = P.• x ~ y if and only if x < y. • c(x) := r(x). Vertex color is just the rank.

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.

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.

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 < y there

exist exactly two elements y1, y2 such that x < y1 < z, x < y2 < 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.

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, f5=v5v0, f6 = v6v0.

• h = v1v2v3v4v5v6,

• t1 = v1v2v0, t2 = v2v3v0, t3 = v3v4v0, t4 = v4v5v0, t5

= v5v6v0, t6 = v6v1v0,

The Poset

• In the Hasse diagram we have the following local picture:

v2v0 v1 v3 v4 v5 v6

e2e1 e3 e4 e5 e6 f2f1 f3 f4 f5 f6

t2h t1 t3 t4 t5 t6

1

0