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Multiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedra.

ABSTRACT: 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.

SUBMITTER: Dauter Z

PROVIDER: S-EPMC7459769 | biostudies-literature | 2020 Sep

REPOSITORIES: biostudies-literature

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Multiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedra.

Dauter Zbigniew Z Jaskolski Mariusz M

Acta crystallographica. Section A, Foundations and advances 20200709 Pt 5

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 g ...[more]

PMID: 32869755

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