Google Groups

Voronoi Polyhedron

John Conway Dec 22, 2003 10:14 AM
Posted in group: geometry.puzzles
On 19 Dec 2003, sujeet kumar wrote:

> What is the voronoi polyhedron for the hexagonal(3D) lattice?

     A hexagonal prism.  John Conway

   In general, there are only five possible topological shapes
the Voronoi cell of a 3-dimensional lattice can be, exemplified
by the truncated octahedron, the hexarhombic dodecahedron, the
rhombic dodecahedron, the hexagonal prism, and the cube.  You
can find a proof in my book "The Sensual Form"

    The Voronoi cell of the configuartion of carbon atoms in a diamond
(which is NOT a lattice) is a triakis truncated tetrahedron.

   John Conway