Subdivision HYPLANE and K=-1 surfaces with symmetry

1. Subdivision HYPLANE andK=-1 surfaces with symmetry Kazushi AHARA, and Keita SAKUGAWA (Meiji University)

2. What’s hyplane? Hyplane is a polyhedron such that(a) Vertex curvature is negative and constant, and (b) vertices are ‘configured uniformly.’ Hyplane is a polyhedral analogue of K=-1(negative constant curvature) surface in R3. Sep./6/2007

3. Vertex curvature K(v) Let v be an internal vertex of a polyhedron. ThenK(v) = 2π - ∑vA (figure) A: face with v Sep./6/2007

4. Gauss theorem on a polyhedron TheoremIf a triangle ABC is on a polyhedron and it bounds a triangle region on polyhedron, then ∠A+ ∠B+ ∠C = π+ ∑K(v) (figure) v∈ΔABC Sep./6/2007

5. Hyplane on triangle tessellation Let (a,b,c) be a triad of positive integers such that (a) 1/a + 1/b + 1/c < 1/2 (b) If a is odd then b=c. If b is odd then c=a. If c is odd then a=b. Then we can consider a hyperbolic tessellation of triangles with angles (2π/a, 2π/b, 2π/c ). Sep./6/2007

6. (a, b, c)=(4, 6, 14) Sep./6/2007

7. Hyplane on triangle tessellation Let a triangle ABC be such that ∠A=bcπ/(ab+bc+ca) ∠B=caπ/(ab+bc+ca) ∠C=abπ/(ab+bc+ca) And make a polyhedron P such that (1) all faces are congruent to ΔABC, (2) Any two faces side by side are symmetric, and (3) there are a faces meeting together at each vertex corresponding to A, (and similarly for B and C.) Sep./6/2007

8. (a,b,c)=(6,6,7) case ΔABC is a triangle with angles 6π/20, 7π/20, 7π/20, (54 degree, 63 degree, 63 degree) Software ‘hyplane’ is on this model. Sep./6/2007

9. Hyplane and K=-1 surface If the faces of the polyhedron are congruent to each other , we may consider that the second condition (b) vertices are ‘configured uniformly’is satisfied. (Yes, I think so.) So we may consider a hyplane as an polyhedral analogue of a surface of the negative constant curvature. Sep./6/2007

10. Acyclic hyplane When we have a hyperbolic tessellation by triangles of the same size and figure, we can construct a hyplane. Here is an acyclic example: tessellation of rhombi with angles (6π/11,4π/11). Sep./6/2007

11. Sep./6/2007

12. Examples of K=-1 surfaces (1) Surface of revolution 1 (pseudo sphere, revolution of tratrix, tractoid) Sep./6/2007

13. Examples of K=-1 surfaces (2) Surface of revolution 2 (Hyperboloid type) Sep./6/2007

14. Examples of K=-1 surfaces (3) Surface of revolution 3 (conic type) Sep./6/2007

15. Examples of K=-1 surfaces (4) Kuen surface Sep./6/2007

16. Examples of K=-1 surfaces (5) Dini surface Sep./6/2007

17. Subdivision of hyplane Hyplane (polyhedron) corresponds to a triangle tessellation on the hyperbolic plane. If the sum of internal angles of the triangle in the tessellation get smaller, the area get larger, the vertex curvature of the hyplane get larger, and it gets difficult to figure up the hyplane model. (Mission) To obtain a K=-1 surface from hyplane, make ‘subdivision’ of hyplane. Sep./6/2007

18. Difficulty for subdivision We need to preserve the conditions on subdivision: (a) vertex curvature is negative and constant. (b) vertices are ‘configured uniformly.’ So we need to ‘rescale’ the size of all faces. From Gauss-Bonnet theorem, the total of vertex curvature must be constant, so the vertex curvature of each vertex must be near 0 (, since the number of vertices get large.) Sep./6/2007

19. Rescaling We first consider subdivision into 4 triangles (figure) but the middle point on the edge must not the midpoint of the edge. To satisfy (a), we need to rescale all small faces in the subdivision. Sep./6/2007

20. Assumption We consider the following condition. That is, in the figure below, the same alphabets mean congruent, and all small triangles are isosceles. (figure) Sep./6/2007

21. Equations Let Ai and Bi be as in the figure below. A4 and (π-A1)/2 is determined directly from the new vertex curvature. About otherangles, we can determinethem easily. Sep./6/2007

22. But ….. The area of triangles are not the same. There are no solution for satisfying (b) vertices are ‘configured uniformly’ in any subdivision (in any meaning.) So we use the above solution for subdivision. Sep./6/2007

23. Surfaces of revolution and hyplane Hyperboloid type Sep./6/2007

24. Surfaces of revolution and hyplane Hyperboloid type Sep./6/2007

25. conic type and pseudo-sphere There exists a hyplane model for conic type , but pseudo-sphere. Sep./6/2007

26. Surface with C3 symmetry There are NOT known K=-1 surface with C3(=cyclic group of order 3) symmetry (other than surfaces of revolution.) But there are some examples of hyplane with C3 symmetry. We call them ‘omusubi’ hyplane. Sep./6/2007

27. Omusubi type (1) Sep./6/2007

28. Omusubi type (2) Sep./6/2007

29. Surface with A4 symmetry There are NOT known K=-1 surface with A4(=Alternating Group of lengthe 4) symmetry. But there are some examples of hyplane with A4 symmetry. We call them ‘chimaki’ hyplane. Sep./6/2007

30. Chimaki type Sep./6/2007

31. Symmetry axis and surface If K=-1 surface S has a symmetry (of order more than 2) of rotation and let X be the axis of the symmetry. Then X never intersect with S. Because if they intersect, then the intersection point must be umbilical and hence has positive curvature. Sep./6/2007

32. Punctured examples (1) Sep./6/2007

33. Punctured examples (1) Sep./6/2007

34. Punctured examples (1) Sep./6/2007

35. Singular type If there exists a smooth K=-1 surface with such (C3 or A4) symmetry, each singular point on the symmetry axis never be cusp shape. Because on such K=-1 surface, there exists a tessellation of triangles and in the tessellation viewpoint, the point is not singular. Sep./6/2007

36. Problem! Find a good coordinate on these ‘surfaces.’ To get a K=-1 surface, we need a specified coordinate (, where the second fundamental form is reduced) and a solution of sine-Gordon equation ωuv=sin ω with a certain boundary condition. Sep./6/2007