A periodic-like table of space groups

Abstract

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 non-linear, 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.

Foreword to the special virtual issue on Modern approaches and tools for teaching crystallography

“…Bart Kahr (Kahr, 2023) presents us with a very interesting idea: the design of a periodic-like table of space groups. The table is constructed based on the symmorphic space groups arranged over two non-linear non-orthogonal axes.…”