{Reference Type}: Journal Article
{Title}: Arithmetic proof of the multiplicity-weighted Euler characteristic for symmetrically arranged space-filling polyhedra.
{Author}: Naskręcki B;Dauter Z;Jaskolski M;
{Journal}: Acta Crystallogr A Found Adv
{Volume}: 77
{Issue}: 0
{Year}: Mar 2021 1
{Factor}: 2.331
{DOI}: 10.1107/S2053273320016186
{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.