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Multiplicity‐weighted Euler's formula for symmetrically arranged space‐filling polyhedra
Author(s) -
Dauter Zbigniew,
Jaskolski Mariusz
Publication year - 2020
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/s2053273320007093
Subject(s) - polyhedron , multiplicity (mathematics) , euler's formula , mathematics , space (punctuation) , geometry , combinatorics , pure mathematics , mathematical analysis , computer science , operating system
The famous Euler's rule for three‐dimensional polyhedra, F − E + V = 2 ( F, E and V are the numbers of faces, edges and vertices, respectively), when extended to many tested cases of space‐filling polyhedra such as the asymmetric unit (ASU), takes the form Fn − En + Vn = 1, where Fn, En and Vn enumerate the corresponding elements, normalized by their multiplicity, i.e. by the number of times they are repeated by the space‐group symmetry. This modified formula holds for the ASUs of all 230 space groups and 17 two‐dimensional planar groups as specified in the International Tables for Crystallography , and for a number of tested Dirichlet domains, suggesting that it may have a general character. The modification of the formula stems from the fact that in a symmetrical space‐filling arrangement the polyhedra (such as the ASU) have incomplete bounding elements (faces, edges, vertices), since they are shared (in various degrees) with the space‐filling neighbors.