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Arithmetic proof of the multiplicity‐weighted Euler characteristic for symmetrically arranged space‐filling polyhedra
Author(s) -
Naskręcki Bartosz,
Dauter Zbigniew,
Jaskolski Mariusz
Publication year - 2021
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.742
H-Index - 83
ISSN - 2053-2733
DOI - 10.1107/s2053273320016186
Subject(s) - mathematics , euler's formula , polyhedron , multiplicity (mathematics) , combinatorics , unit (ring theory) , lattice (music) , euler characteristic , dirichlet distribution , pure mathematics , mathematical analysis , physics , mathematics education , acoustics , boundary value problem
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, χ m = 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.