Uniform partitions of 3-space, their relatives and embedding

1999/06/06 by Michel Deza, M. Deza, Deza, M. +3
Engineering · Mathematics · #Advanced Mathematical Identities #Advanced Mathematical Theories #graph theory and CDMA systems #math.CO #math.MG

paper · pdf · doi:10.48550/arxiv.math/9906034

11 pages

arxiv created 1999/06/06 · arxiv updated 2009/11/30

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.

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