Abstract
The puzzling observation that the famous Euler's formula for three-dimensional polyhedra V - E + F = 2 or Euler characteristic χ = V - E + F - I = 1 (where V, E, F are the numbers of the bounding vertices, edges and faces, respectively, and I = 1 counts the single solid itself) when applied to space-filling solids, such as crystallographic asymmetric units or Dirichlet domains, are modified in such a way that they sum up to a value one unit smaller (i.e. to 1 or 0, respectively) is herewith given general validity. The proof provided in this paper for the modified Euler characteristic, χm = Vm - Em + Fm - Im = 0, is divided into two parts. First, it is demonstrated for translational lattices by using a simple argument based on parity groups of integer-indexed elements of the lattice. Next, Whitehead's theorem, about the invariance of the Euler characteristic, is used to extend the argument from the unit cell to its asymmetric unit components.
Highlights
The famous Euler characteristic that gives a simple relation between the numbers of geometrical elements of an isolated polytope in N-dimensional space can be expressed as follows: 1⁄4 PN ðÀ1Þini 1⁄4 1 i1⁄40 where ni is the number of i-dimensional cells building up the polytope
The classical N-dimensional Euler characteristic of a polytope was derived by Schlafli (1901)
In a previous paper (Dauter & Jaskolski, 2020) in this journal we demonstrated, by analyzing standardized asymmetric units (ASUs) in all planar and space groups in the International Tables for Crystallography, Vol A (Aroyo, 2016), as well as selected Dirichlet domains, that the polytopes in a lattice built from symmetrically arranged, space-filling
Summary
The famous Euler characteristic that gives a simple relation between the numbers of geometrical elements of an isolated polytope in N-dimensional space can be expressed as follows: 1⁄4 PN ðÀ1Þini 1⁄4 1 i1⁄40 where ni is the number of i-dimensional cells (elements) building up the polytope. It is important to stress that, while Coxeter’s proof is sufficient for translational m 1⁄4 Vm À Em þ Fm À Im 1⁄4 0 lattices, or space filled with unit cells (hyper-parallelepipeds), it cannot be automatically extended to situations where the where Vm , Em , Fm , Im represent total fractions of elements of unit cell is subdivided into smaller polytopes, such as the ASU
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