Arithmetic proof of the multiplicity‐weighted Euler characteristic for symmetrically arranged space‐filling polyhedra. Issue 2 (5th February 2021)

• Record Type: Journal Article
• Authors: Naskręcki, Bartosz; Dauter, Zbigniew; Jaskolski, Mariusz
• Is Part Of: Acta crystallographica. Volume 77:Issue 2(2021)
• Page Start: 126
• Page End: 129
• Publication Date: 2021-02-05
• Subjects: Euler's formula -- multiplicity‐weighted Euler characteristic -- space‐filling polyhedra -- polytopes -- asymmetric unit -- Dirichlet domains; Crystallography -- Periodicals; Condensed matter -- Periodicals; 548
• DOI: 10.1107/S2053273320016186
• Languages: English
• ISSNs: 2053-2733

Abstract

Abstract : A mathematical proof based on arithmetic argument is presented for the modified Euler characteristic (where the first summation runs from 0‐dimensional vertices to the N ‐dimensional cell or `interior'), applicable to symmetrically arranged space‐filling polytopes in N ‐dimensional space, where the contribution of each j th i ‐dimensional element of the polytope is weighted by a factor inversely proportional to its multiplicity m ( ij ). 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 = V m − E m + F m − I m = 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.