Equidecomposability of polyhedra with reference to crystallographic groups
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.
scientific article; zbMATH DE number 4055908
scientific article