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

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 mu. 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 alpha-uranium and CrB-type crystal structures.

Automated summary

This paper presents a mathematical toy model of chiral spiral cyclic twins, which describes a family of deterministically generated aperiodic point sets. The model is controlled by a pair of integer parameters and includes both local features of periodic and aperiodic crystals. The spiral twins are generated from integer inclination sequences, forming discrete variants of continuous circle involutes. The geometry of circle involutes ensures a constant orthogonal separation distance between adjacent spiral branches, and provides an approximate delineation of the intrinsically periodic twin domains as well as a single aperiodic core domain interconnecting them.