1. Topological Crystallography Commemorating the fourth centennial anniversary of the publication of Kepler‘s pamphlet (1611) New-Year's gift concerning six- cornered snow Toshikazu Sunada Meiji University August 26, 2011 at Sendai

2. The purpose of this talk

3. Legacy of Ancient Greece in crystallography

5. Regular convex polyhedra

7. Archimedean Solids

8. Johaness Kepler (1571- 1630)

9. “New Year’s gift concerning six-cornered snow” (1611)

12. Face-centered cubic lattice (fcc)

13. Discovery of crystal structures

15. Diamond and Lonsdaleite

16. Graphs associated with chemical compounds

17. Graphs

19. Fundamental finite graphs and building blocks

20. How to get the fundamental finite graphs

21. Canonical map

23. How to get the crystal from a building block

24. Examples of fundamental finite graphs Diamond and Lonsdaleite 3D Kagome lattice (comprised by corner-sharing tetrahedra)

28. Example

30. Diamond and Honeycomb

32. Enumeration of vanishing subgroups

34. Kronecker’s dream in crystallography

35. Canonical placement

37. Canonical placement is most symmetric

41. Algorithm

42. Examples (Classical lattices)

45. A hypothetical crystal (1)

46. A hypothetical crystal (2) (3D Kagome lattice)

47. A hypothetical crystal (3)

48. A hypothetical crystal (4)

50. Open problems

53. Final Remarks

55. Graph-theoretic Abel-Jacobi maps

57. N.I.Lobachevskii