Equidecomposability of polyhedra with reference to crystallographic groups
type: Item
instance of: scholarly article
title: Equidecomposability of polyhedra with reference to crystallographic groups
zbMATH Open document ID: 0647.20051
DOI: 10.1007/BF02187920
author: Carsten Müller
published in: Discrete \& Computational Geometry
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/131058
review text: Let G be a crystallographic group of the d-dimensional real Euclidean space E d (d\(\geq 2)\); so G will be a discrete transformation group containing a subgroup of translations which is generated by d linearly independent vectors. Two sets of points A, B in E d are called G-equal if there exists a transformation g of G such that \(B=g(A)\); A, B are called G-equidecomposable if they can be dissected into a finite number of subsets being pairwise G-equal. A polyhedron P is called G-paving if there exists a subgroup H (elements h) of G such that (1) the union of all h(P) is the space E d, and (2) \(h_ 1(P)\cap h_ 2(P)\) contains no interior points if \(h_ 1\neq h_ 2\). The main result is presented in Theorem 3: Two G-paving polyhedra in E d (G being a crystallographic group) are G-equidecomposable if (and only if) they are of equal volume. As a corollary one gets (Theorem 2): Let \(W_ 2\) be the crystallographic group of \(E_ 2\) which is generated by two noncollinear vectors and one point reflection; then any two lattice polygons of \(W_ 2\) having equal area are \(W_ 2\)-equidecomposable.
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 52Bxx
Mathematics Subject Classification ID: 51M25
zbMATH DE Number: 4055908
zbMATH Keywords: equidecomposability
zbMATH Keywords: crystallographic group
zbMATH Keywords: discrete transformation group
zbMATH Keywords: polyhedron
zbMATH Keywords: lattice polygons
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4055908