Convex Bodies which Tile Space

P. McMullen
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P. McMullen: University College

A chapter in The Geometric Vein, 1981, pp 123-128 from Springer

Abstract: Abstract We say that the convex body (compact convex set with nonempty interior) K tiles d-dimensional Euclidean space E d (by translation) if there is some family T of translation vectors, such that (i) K covers E d , and (ii) if t i ∈ T (i = 1,2) with t 1 ≠ t 2, then K + t 1 and K + t 2 have disjoint interiors; that is, K is simultaneously a covering and packing of E d . We call K a tiling of E d (by translation), and call K and its translates in K tiles. A particularly important case is when T is a lattice (discrete additive subgroup of E d ), when we call K a lattice tiling.

Keywords: Convex Body; Translation Vector; Finite Union; Star Body; Dimensional Face (search for similar items in EconPapers)
Date: 1981
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DOI: 10.1007/978-1-4612-5648-9_6

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