Here's a simple way to get 230 of them: For each of the 230 3D space groups select a generic point in space and generate its orbit by the group action. "Generic" in this context means that the point is not fixed by any element of the group (i.e., avoid points on mirror planes, rotation axes, etc.). To get a tiling of convex monotiles just take the Voronoi cells associated with the points in the group orbit. Veit On Oct 29, 2010, at 1:40 PM, rcs@xmission.com wrote:

Has anyone tackled the 3D version of tiling space with copies of a single convex polyhedral tile? The 2D problem turned out to be surprisingly complicated.

Does anyone know the smallest non-tiling polycube?

Rich

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