Uniform partitions of 3-space, their relatives and embedding

M. Deza; M. I. Shtogrin

Uniform partitions of 3-space, their relatives and embedding

Metric Geometry 2007-05-23 v1 Combinatorics

Authors: M. Deza , M. I. Shtogrin

Abstract

We review 28 uniform partitions of 3-space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice ${\bf Z}_n$ . We also consider some relatives of those 28 partitions, including Achimedean 4-polytopes of Conway-Guy, non-compact uniform partitions, Kelvin partitions and those with unique vertex figure (i.e. Delaunay star). Among last ones we indicate two continuums of aperiodic tilings by semi-regular 3-prisms with cubes or with regular tetrahedra and regular octahedra. On the way many new partitions are added to incomplete cases considered here.

Keywords

uniform partitions of 3-space isometric embedding cubic lattice archimedean 4-polytopes kelvin partitions delaunay star aperiodic tilings semi-regular prisms

Cite

@article{arxiv.math/9906034,
  title  = {Uniform partitions of 3-space, their relatives and embedding},
  author = {M. Deza and M. I. Shtogrin},
  journal= {arXiv preprint arXiv:math/9906034},
  year   = {2007}
}

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11 pages

Uniform partitions of 3-space, their relatives and embedding

Abstract

Keywords

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