1. INCIDENCE GEOMETRIES Part 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. P=12L=16  P=12-46 L=163-3  64-

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  =( , ~, 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 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  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)

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

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 = 2 n e = n 2 n-1

11. 4-dimensional Cube. 0000 1000 0010 0100 0001 1100 1110 0110 0111 0011 1001 1101 11111011 1010

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 Q 4. Vertices (Q 0 ) of Q 4 : 16 Edges (Q 1 )of Q 4 : 32 Squares (Q 2 ) of Q 4 : 24 Cubes (Q 3 ) of Q 4 : 8 Total: 80 The Levi graph of Q 4 has 80 vertices and is colored with 4 colors.

15. Residual geometries of Q 4. VESQ3.Q3.  (V) -464  (E) 2-33  (S) 44-2  (Q 3 ) 8126-

16. Exercises N1: Determine all residual geometries of Reyeve configuration N2: Determine all residual geometries of Q 4. N3: Determine all residual geometries of Platonic solids. N4: Determine the Levi graph of the geometry for the grup Z 2 £ Z 2 £ Z 2, 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 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.

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 = z 0, z 1,..., z m = 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(x+2), x < y there exist exactly two elements y 1, y 2 such that x < y 1 < z, x < y 2 < 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: v 0, v 1, v 2,..., v 6. 12 edges: e 1, e 2,..., e 6, f 1, f 2,..., f 6 7 faces: h,t 1,t 2,t 3,.., t 6 1 e 1 = v 1 v 2, e 2 = v 2 v 3, e 3 = v 3 v 4, e 4 = v 4 v 5, e 5 = v 5 v 6, e 6 = v 6 v 1, f 1 = v 1 v 0, f 2 = v 2 v 0,f 3 = v 3 v 0, f 4 = v 4 v 0, f 5 =v 5 v 0, f 6 = v 6 v 0. h = v 1 v 2 v 3 v 4 v 5 v 6, t 1 = v 1 v 2 v 0, t 2 = v 2 v 3 v 0, t 3 = v 3 v 4 v 0, t 4 = v 4 v 5 v 0, t 5 = v 5 v 6 v 0, t 6 = v 6 v 1 v 0,

23. The Poset In the Hasse diagram we have the following local picture: v2v2 v0v0 v1v1 v3v3 v4v4 v5v5 v6v6 e2e2 e1e1 e3e3 e4e4 e5e5 e6e6 f2f2 f1f1 f3f3 f4f4 f5f5 f6f6 t2t2 ht1t1 t3t3 t4t4 t5t5 t6t6 1 0