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The kisrhombille tiling is the dual tessellation of one of the semi‐regular tessellations. It consists of right‐angled triangle tiles with 12 different orientations. An adequate coordinate system for the tiles of the grid has been defined that allows a formal description of the grid. In this paper, two tiles are considered to be neighbors if they share at least one point in their boundary. Paths are sequences of tiles such that any two consecutive tiles are neighbors. The digital distance is defined as the minimum number of steps in a path between the tiles, and the distance formula is proven through constructing minimum paths. In fact, the distance between triangles is almost twice the hexagonal distance of their embedding hexagons. [ABSTRACT FROM AUTHOR]

A brief introductory review is provided of the theory of tilings of 3‐periodic nets and related periodic surfaces. Tilings have a transitivity [p q r s] indicating the vertex, edge, face and tile transitivity. Proper, natural and minimal‐transitivity tilings of nets are described. Essential rings are used for finding the minimal‐transitivity tiling for a given net. Tiling theory is used to find all edge‐ and face‐transitive tilings (q = r = 1) and to find seven, one, one and 12 examples of tilings with transitivity [1 1 1 1], [1 1 1 2], [2 1 1 1] and [2 1 1 2], respectively. These are all minimal‐transitivity tilings. This work identifies the 3‐periodic surfaces defined by the nets of the tiling and its dual and indicates how 3‐periodic nets arise from tilings of those surfaces. [ABSTRACT FROM AUTHOR]

Substitution tilings with pure discrete spectrum are characterized as regular model sets whose cut‐and‐project scheme has an internal space that is a product of a Euclidean space and a profinite group. Assumptions made here are that the expansion map of the substitution is diagonalizable and its eigenvalues are all algebraically conjugate with the same multiplicity. A difference from the result of Lee et al. [Acta Cryst. (2020), A76, 600–610] is that unimodularity is no longer assumed in this paper. [ABSTRACT FROM AUTHOR]

The 3D facets of the Delone cells of the root lattice D6 which tile the 6D Euclidean space in an alternating order are projected into 3D space. They are classified into six Mosseri–Sadoc tetrahedral tiles of edge lengths 1 and golden ratio τ = (1 + 51/2)/2 with faces normal to the fivefold and threefold axes. The icosahedron, dodecahedron and icosidodecahedron whose vertices are obtained from the fundamental weights of the icosahedral group are dissected in terms of six tetrahedra. A set of four tiles are composed from six fundamental tiles, the faces of which are normal to the fivefold axes of the icosahedral group. It is shown that the 3D Euclidean space can be tiled face‐to‐face with maximal face coverage by the composite tiles with an inflation factor τ generated by an inflation matrix. It is noted that dodecahedra with edge lengths of 1 and τ naturally occur already in the second and third order of the inflations. The 3D patches displaying fivefold, threefold and twofold symmetries are obtained in the inflated dodecahedral structures with edge lengths τn with n ≥ 3. The planar tiling of the faces of the composite tiles follows the edge‐to‐edge matching of the Robinson triangles. [ABSTRACT FROM AUTHOR]

Primitive substitution tilings on whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut‐and‐project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut‐and‐project scheme. [ABSTRACT FROM AUTHOR]

Tilings based on the cut‐and‐project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite‐volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut‐and‐project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffraction for this example is recapitulated. Further, recent developments are discussed for cut‐and‐project structures with an inflation symmetry, which are based on an internal counterpart of the renormalization cocycle. Finally, a brief review is given of the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures. [ABSTRACT FROM AUTHOR]

The vertex graph of the Ammann–Beenker tiling is a well‐known quasiperiodic graph with an eightfold rotational symmetry. The coordination sequence and coordination shells of this graph are studied. It is proved that there exists a limit growth form for the vertex graph of the Ammann–Beenker tiling. This growth form is an explicitly calculated regular octagon. Moreover, an asymptotic formula for the coordination numbers of the vertex graph of the Ammann–Beenker tiling is also proved. [ABSTRACT FROM AUTHOR]

This work considers the scaling properties characterizing the hyperuniformity (or anti‐hyperuniformity) of long‐wavelength fluctuations in a broad class of one‐dimensional substitution tilings. A simple argument is presented which predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and numerical computations confirm that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra and limit‐periodic tilings. Quasiperiodic or singular continuous cases can be constructed with α arbitrarily close to any given value between −1 and 3. Limit‐periodic tilings can be constructed with α between −1 and 1 or with Fourier intensities that approach zero faster than any power law. This work examines the long‐wavelength scaling properties of self‐similar substitution tilings, placing them in their hyperuniformity classes. Quasiperiodic, non‐PV (Pisot–Vijayaraghavan number) and limit‐periodic examples are analyzed. Novel behavior is demonstrated for certain limit‐periodic cases. [ABSTRACT FROM AUTHOR]

An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. The first application is to the two kinds of vertices (tetravalent and trivalent) in the Cairo (or dual‐32.4.3.4) tiling. The coordination sequence for a tetravalent vertex turns out, surprisingly, to be 1, 4, 8, 12, 16, ..., the same as for a vertex in the familiar square (or 44) tiling. The authors thought that such a simple fact should have a simple proof, and this article is the result. The method is also used to obtain coordination sequences for the 32.4.3.4, 3.4.6.4, 4.82, 3.122 and 34.6 uniform tilings, and the snub‐632 tiling. In several cases the results provide proofs for previously conjectured formulas. This article presents a simple method for finding formulas for coordination sequences, based on coloring the underlying graph according to certain rules. It is illustrated by applying it to several uniform tilings and their duals. [ABSTRACT FROM AUTHOR]

In this article, a framework is presented that allows the systematic derivation of planar edge‐to‐edge k‐isocoronal tilings from tile‐s‐transitive tilings, s ≤ k. A tiling is k‐isocoronal if its vertex coronae form k orbits or k transitivity classes under the action of its symmetry group. The vertex corona of a vertex x of is used to refer to the tiles that are incident to x. The k‐isocoronal tilings include the vertex‐k‐transitive tilings (k‐isogonal) and k‐uniform tilings. In a vertex‐k‐transitive tiling, the vertices form k transitivity classes under its symmetry group. If this tiling consists of regular polygons then it is k‐uniform. This article also presents the classification of isocoronal tilings in the Euclidean plane. This article presents a method to determine planar edge‐to‐edge k‐isocoronal tilings – tilings whose vertex coronae form k orbits or k transitivity classes under the action of the symmetry group. [ABSTRACT FROM AUTHOR]

This work introduces the idea of symmetry order, which describes the rotational symmetry types of tilings in the hull of a given substitution. Definitions are given of the substitutions σ6 and σ7 which give rise to aperiodic primitive substitution tilings with dense tile orientations and which are invariant under six‐ and sevenfold rotations, respectively; the derivation of the symmetry orders of their hulls is also presented. [ABSTRACT FROM AUTHOR]

2-Periodic self-dual tilings are important in fields ranging from crystal chemistry to mathematical physics. They have been systematically enumerated using combinatorial tiling theory, and 1, 5 and 62 uninodal, binodal and trinodal self-dual tilings have been found. This paper illustrates all uninodal and binodal self-dual tilings and selected trinodal self-dual tilings. Most of these structures are described for the first time. [ABSTRACT FROM AUTHOR]

Tilings with a singular point are obtained by applying conformal maps on regular tilings of the Euclidean plane and their symmetries are determined. The resulting tilings are then symmetrically colored by applying the same conformal maps on colorings of regular tilings arising from sublattice colorings of the centers of the tiles. In addition, conditions are determined in order that the coloring of a tiling with singularity that is obtained in this manner is perfect. [ABSTRACT FROM AUTHOR]

The generalized Penrose tiling is, in fact, an infinite set of decagonal tilings. It is constructed with the same rhombs (thick and thin) as the conventional Penrose tiling, but its long-range order depends on the so-called shift parameter ( s∈〈0; 1)). The structure factor is derived for the arbitrarily decorated generalized Penrose tiling within the average unit cell approach. The final formula works in physical space only and is directly dependent on the s parameter. It allows one to straightforwardly change the long-range order of the refined structure just by changing the s parameter and keeping the tile decoration unchanged. This gives a great advantage over the higher-dimensional method, where every change of the tiling (change in the s parameter) requires the structure model to be built from scratch, i.e. the fine division of the atomic surfaces has to be redone. [ABSTRACT FROM AUTHOR]

This work investigates symmetry and color symmetry properties of Kepler, Heesch and Laves tilings embedded on a flat torus and their geometric realizations as tilings on a round torus in Euclidean 3-space. The symmetry group of the tiling on the round torus is determined by analyzing relevant symmetries of the planar tiling that are transformed to axial symmetries of the three-dimensional tiling. The focus on studying tilings on a round torus is motivated by applications in the geometric modeling of nanotori and the determination of their symmetry groups. [ABSTRACT FROM AUTHOR]

This work investigates the symmetry properties of single-wall carbon nanotubes and their structural analogs, which are nanotubes consisting of different kinds of atoms. The symmetry group of a nanotube is studied by looking at symmetries and color fixing symmetries associated with a coloring of the tiling by hexagons in the Euclidean plane which, when rolled, gives rise to a geometric model of the nanotube. The approach is also applied to nanotubes with non-hexagonal symmetry arising from other isogonal tilings of the plane. [ABSTRACT FROM AUTHOR]