TY - JOUR

T1 - A periodic-like table of space groups

AU - Kahr, Bart

N1 - Funding Information:
The following funding is acknowledged: National Science Foundation, Directorate for Mathematical and Physical Sciences (grant No. DMR-2003968 to Bart Kahr).
Publisher Copyright:
© 2023 International Union of Crystallography. All rights reserved.

PY - 2023/2/7

Y1 - 2023/2/7

N2 - There are about 100 chemical elements, and 200 space groups, rounding to the nearest hundreds. The elements, by virtue of the iconic periodic table, which hangs in schoolrooms worldwide, are part of our common culture. Few on the outside of solid-state science and discrete mathematics know of the remarkable enumeration of the space groups and how they organize solids, the chemical elements included, that can be conformed to translationally periodic lattices in three dimensions. Perhaps the comparative inaccessibility of space groups to non-specialists can be attributed to the absence of a single chart that captures and organizes the space groups in a glance, as does the periodic table for the chemical elements? Here is designed a periodic-like table of the space groups. The symmorphic space groups are organized as the coordinates of two nonlinear, non-orthogonal axes that capture the point group symmetry and the order of the group, the multiplicity of general positions. The related non-symmorphic groups are treated as ‘isotopes’ of the symmorphic groups and are enumerated with a subscript. The intent is to inspire better designs and to bring the crystallographic space groups to the attention of a wider audience.

AB - There are about 100 chemical elements, and 200 space groups, rounding to the nearest hundreds. The elements, by virtue of the iconic periodic table, which hangs in schoolrooms worldwide, are part of our common culture. Few on the outside of solid-state science and discrete mathematics know of the remarkable enumeration of the space groups and how they organize solids, the chemical elements included, that can be conformed to translationally periodic lattices in three dimensions. Perhaps the comparative inaccessibility of space groups to non-specialists can be attributed to the absence of a single chart that captures and organizes the space groups in a glance, as does the periodic table for the chemical elements? Here is designed a periodic-like table of the space groups. The symmorphic space groups are organized as the coordinates of two nonlinear, non-orthogonal axes that capture the point group symmetry and the order of the group, the multiplicity of general positions. The related non-symmorphic groups are treated as ‘isotopes’ of the symmorphic groups and are enumerated with a subscript. The intent is to inspire better designs and to bring the crystallographic space groups to the attention of a wider audience.

KW - Barlow

KW - Fedorov

KW - periodic table

KW - point groups

KW - Schoenflies

KW - space groups

KW - symmorphic

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U2 - 10.1107/S2056989023000786

DO - 10.1107/S2056989023000786

M3 - Article

AN - SCOPUS:85159637027

SN - 2056-9890

VL - 79

SP - 124

EP - 128

JO - Acta Crystallographica Section E: Crystallographic Communications

JF - Acta Crystallographica Section E: Crystallographic Communications

IS - Pt 3

ER -