Substitution Discrete Plane Tilings with 2n-Fold Rotational Symmetry for Odd n

Kari, Jarkko; Lutfalla, Victor H.

doi:10.1007/s00454-022-00390-z

Cited by 2 publications

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References 14 publications

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“…(Ammann et al, 1992;Beenker, 1982). In the case of type (10, 4), the prototile set consists of a thin 36-144 rhombus together with a thick 72-108 rhombus, which is the same set as appears in the quasiperiodic rhombic Penrose tiling of decagonal symmetry; to be precise, a generalized Penrose tiling (Ga ¨hler et al, 1994;Kari & Lutfalla, 2023), because a central patch of ten thin rhombuses, of vertex configuration ST (Zobetz & Preisinger, 1990), is not an allowed vertex configuration according to the aperiodicity-enforcing matching rules of the original Penrose tiling of thin and thick rhombuses. In the case of type (12, 5), the prototile set consists of a 30-150 rhombus together with an equilateral triangle and a square, which is the same set as appears in one of the quasiperiodic Stampfli tilings of dodecagonal symmetry (Schaad & Stampfli, 2021).…”

Section: Exceptional Cases: Regular and Aperiodic Tilingsmentioning

confidence: 99%

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

Hornfeck

2023

Acta Crystallogr A Found Adv

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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.

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