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Erschienen in:
2013 | OriginalPaper | Buchkapitel
Atoms for Parallelohedra
verfasst von : Jin Akiyama, Midori Kobayashi, Hiroshi Nakagawa, Gisaku Nakamura, Ikuro Sato
Erschienen in: Geometry — Intuitive, Discrete, and Convex
Verlag: Springer Berlin Heidelberg
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A parallelohedron is a convex polyhedron which tiles 3-dimensional space by translations only. A polyhedron
σ
is said to be an atom for the set Π of parallelohedra if for each parallelohedron
P
in Π, there exists an affine-stretching transformation
A
: ℝ
3
→ ℝ
3
such that
A
(
P
) is the union of a finite number of copies of
σ
. In this paper, we will present two different atoms for the parallelohedra, and determine the number of these atoms used to make up each parallelohedron. We will also show an arrangement of the parallelohedra in lattice-like order and introduce the notion of indecomposability.