Topological Crystallography Commemorating the fourth centennial anniversary of the publication of Kepler‘s pamphlet (1611) New-Year's gift concerning six-

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Presentation transcript:

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

The purpose of this talk

Legacy of Ancient Greece in crystallography

Regular convex polyhedra

Archimedean Solids

Johaness Kepler ( )

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

Face-centered cubic lattice (fcc)

Discovery of crystal structures

Diamond and Lonsdaleite

Graphs associated with chemical compounds

Graphs

Fundamental finite graphs and building blocks

How to get the fundamental finite graphs

Canonical map

How to get the crystal from a building block

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

Example

Diamond and Honeycomb

Enumeration of vanishing subgroups

Kronecker’s dream in crystallography

Canonical placement

Canonical placement is most symmetric

Algorithm

Examples (Classical lattices)

A hypothetical crystal (1)

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

A hypothetical crystal (3)

A hypothetical crystal (4)

Open problems

Final Remarks

Graph-theoretic Abel-Jacobi maps

N.I.Lobachevskii