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.

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) = Å{G(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 • Thegraph 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 1011 1111 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.

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 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(G) = 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 = 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(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.

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, f5=v5v0, 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: h t1 t2 t3 t4 t5 t6 e1 e2 e3 e4 e5 e6 f1 f2 f3 f4 f5 f6 v0 v1 v2 v3 v4 v5 v6 0