Chiral spiral cyclic twins. II. A two-parameter family of cyclic twins composed of discrete circle involute spirals.

Abstract

A mathematical toy model of chiral spiral cyclic twins is presented, describing a family of deterministically generated aperiodic point sets. Its individual members depend solely on a chosen pair of integer parameters, a modulus m and a multiplier μ. By means of their specific parameterization they comprise local features of both periodic and aperiodic crystals. In particular, chiral spiral cyclic twins are composed of discrete variants of continuous curves known as circle involutes, each discrete spiral being generated from an integer inclination sequence. The geometry of circle involutes does not only provide for a constant orthogonal separation distance between adjacent spiral branches but also yields an approximate delineation of the intrinsically periodic twin domains as well as a single aperiodic core domain interconnecting them. Apart from its mathematical description and analysis, e.g. concerning its circle packing densities, the toy model is studied in association with the crystallography and crystal chemistry of α-uranium and CrB-type crystal structures.