Uniqueness and Symmetry in Problems of Optimally Dense Packings

Details Uniqueness and Symmetry in Problems of Optimally Dense Packings Uniqueness and Symmetry in Problems of Optimally Dense Packings

2003-02-05

arxiv.org/abs/math/0302056v1

Mathematics

Metric Geometry

Plain TeX, 27 pages of text followed by 11 figures

Scientific paper

We analyze the general problem of determining optimally dense packings, in a Euclidean or hyperbolic space, of congruent copies of some fixed finite set of bodies. We are strongly guided by examples of aperiodic tilings in Euclidean space and a detailed analysis of a new family of examples in the hyperbolic plane. Our goal is to understand qualitative features of such optimum density problems, in particular the appropriate meaning of the uniqueness of solutions, and the role of symmetry in classfying optimally dense packings.

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