A table of the colored crystallographic and icosahedral point groups, including their chirality and diamorphism

Abstract

All the colored crystallographic and icosahedral point groups that obey the van der Waerden and Burckhardt definition are tabulated here, using a symbolism based on a suggestion of Shubnikov and Koptsik. Each asymmetric domain of the object has a single color - or scalar quality. The symbol of each colored point group G(H' | H) contains G, the geometrical point group of the object; its sub-group H' that is the point group of each of the object's monochromatic domains; and H, the invariant subgroup of G that is the intersection of all the conjugate subgroups H'. If H' and H are the same, the symbol G(H' | H) is changed to G(H). If H is chiral, so is the colored point group. If G is not chiral, but H is, the chirality is only chromatic. These phenomena are listed in the table. If more color permutations are possible than occur in the operations of a colored point group, different arrangements of the same colors on the same geometrical object are fundamentally different: these are called diamorphs of each other; their number is listed. 'Color', as used here, symbolizes any scalar property that can vary from one asymmetric domain to another.