Chiral Spiral Cyclic Twins.

12^th December 2025, 15:00
Wolfgang Hornfeck
Institute of Physics of the Czech Academy of Sciences

Abstract

Chiral Spiral Cyclic Twins [1] are a mathematical toy model describing the ideal atomic arrangement of actual and potential cyclic twins by means of a deterministically generated two-dimensional aperiodic point set. The model is based on a complex number representation of discrete spiral nodes. The spatial distribution of spiral nodes is governed by an integer inclination sequence, determining a spiral's coordinate evolution with increasing distance from its origin in a turtle geometry fashion. The integer inclination sequence itself is determined by a baseline step-function of linearly varying step lengths. All together, each chiral spiral cyclic twin is completely fixed by four integer parameters.
Apart from the mathematical description of the model and its systematic derivation, my talk will illustrate the model on some examples from crystallography, namely the actual cyclic twins of NiZr and related binary intermetallic compounds, as well as the potential cyclic twins of cyclododecasulfur molecules. I will further show the model's relation to the quasiperiodic rhombic tilings of Ammann-Beenker, Penrose, and Stampfli, which themselves are toy models for axial quasicrystals of octagonal, decagonal, and dodecagonal symmetry. In this context I will also highlight its relation to the novel concept of Z module twinning. Finally, I will discuss potential applications outside of crystallography.
The main idea of the talk is to show how the material science inspired experimental study of alloys can guide the open mind to the theoretical study of more abstract realms inhabitated by aesthetically and conceptually beautiful mathematical entities, which in turn can be used to highlight hidden patterns in the coordinate description of crystal structures, going beyond the common classification based on space group symmetry.
[1] Hornfeck, W. Chiral spiral cyclic twins. III. Twins galore. Struct Chem (2025). https://doi.org/10.1007/s11224-025-02637-w (open access)