Monohedral Tilings Research Articles

Periodic diffraction from an aperiodic monohedral tiling - the Spectre tiling. Addendum.

This article describes the diffraction pattern (2-periodic Fourier transform) from the vertices of a large patch of the recently discovered `Spectre' tiling - a strictly chiral aperiodic monotile. It was reported recently that the diffraction pattern of the related weakly chiral aperiodic `Hat' monotile was 2-periodic with chiral plane-group symmetry p6 [Kaplan et al. (2024). Acta Cryst. A80, 72-78]. The diffraction periodicity arises because the Hat tiling is a systematic aperiodic deletion of vertices from the 2-periodic hexagonal mta tiling. Despite the similarity of the Hat and Spectre tilings, the Spectre tiling is not aligned with a 2-periodic lattice, and its diffraction pattern is non-periodic with chiral point symmetry 6 about the origin.

Periodic diffraction from an aperiodic monohedral tiling.

The diffraction pattern from the recently reported aperiodic `einstein', or `hat', monohedral tiling [Smith et al. (2023). arXiv:2303.10798v1] has been analyzed. The structure is the hexagonal mta net, a kite tiling, with aperiodic vertex deletions. A large model's diffraction pattern displays a robust sixfold periodicity in plane group p6. A repeating, roughly triangular motif of `diffused intensity' arises between the strongest Bragg peaks. The motif contains high-density regions of discrete `satellite' peaks, rather than continuous `diffuse scattering', breaking mirror symmetry, consistent with the chiral hat tiling.

On monohedral tilings of a regular polygon

A tiling of a topological disc by topological discs is called monohedral if all tiles are congruent. Maltby (J Comb Theory Ser A 66:40–52, 1994) characterized the monohedral tilings of a square by three topological discs. Kurusa et al. (Mediterr J Math 17:156, 2020) characterized the monohedral tilings of a circular disc by three topological discs. The aim of this note is to connect these two results by characterizing the monohedral tilings of any regular n-gon with at most three tiles for any n≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 5$$\\end{document}.

Monohedral tilings of a convex disc with a smooth boundary

In this paper we give a complete description about normal monohedral tilings of a convex disc with smooth boundary where we have at most three topological discs as tiles. This result is a far-reaching generalization of the results of Kurusa, Lángi and Vígh [8]. Some further partial results are proved for non-normal tilings.

Spirals in Periodic Tilings

Spiral tilings, as appealing as they are for their aesthetics, have not been studied well mathematically. One of the difficulties in this area of tiling theory is providing a mathematical definition of spiral tilings. A recently published attempt at providing a formal definition distinguishes a so-called L-spiral tiling (L-tiling) and an S-spiral tiling (S-tiling), with the two types being characterized by special properties of tile set partitions. Based on these existing definitions, we investigate the spiral structure in periodic tilings. Unlike spiral tilings, periodic tilings lend themselves easily to a definition and have been well studied. We first prove that it is not possible for periodic tilings to be S-tilings. We then study a subset of periodic tilings that can be L-tilings. In particular, we demonstrate that there exist examples for each type of isohedral tilings (a subset of periodic monohedral tilings) that are L-spirable.

Tiling a Circular Disc with Congruent Pieces

In this note, we prove that any monohedral tiling of the closed circular unit disc with k le 3 topological discs as tiles has a k-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain the center, and the first step towards answering a question of Stein appearing in the problem book of Croft, Falconer, and Guy in 1994.

How to Define a Spiral Tiling?

SummaryA precise mathematical definition is given for spiral plane tilings. It is not restricted to monohedral tilings and is tested on a series of examples from the literature. Unwanted cases from regular tilings can be excluded. In case of one single arm a modified definition can be applied. Also the special case of locally infinite tilings with one singular point can be treated with any number of spiral arms. The question whether such a definition in mathematical terms could be given was posed by Grünbaum and Shephard in the late 1970s.

Rotationally symmetric tilings with convex pentagons and hexagons

In contrast to many known results concerning periodic tilings of the Euclidean plane with pentagons, here tilings with rotational symmetry are investigated. A certain class of convex pentagons is introduced. It can be shown that for any given symmetry type $\mathbf{C}_{n}$ or $\mathbf{D}_{n}$ there exists a monohedral tiling generated by a pentagon from this class. For $n>1$ each of these tilings is also a spiral tiling with $n$ arms. As a byproduct it follows that the same holds for convex hexagons.

Anisohedral spherical triangles and classification of spherical tilings by congruent kites, darts and rhombi

We classify all spherical monohedral (kite/dart/rhombus)-faced tilings, as follows: The set of spherical monohedral rhombus-faced tilings consists of (1) the central projection of the rhombic dodecahedron, (2) the central projection of the rhombic triacontahedron, (3) a series of non-isohedral tilings, and (4) a series of tilings which are topologically trapezohedra (here a trapezohedron is the dual of an antiprism.). The set of spherical tilings by congruent kites consists of (1) the central projection $T$ of the tetragonal icosikaitetrahedron, (2) the central projection of the tetragonal hexacontahedron, (3) a non-isohedral tiling obtained from $T$ by gliding a hemisphere of $T$ with $pi/4$ radian, and (4) a continuously deformable series of tilings which are topologically trapezohedra. The set of spherical tilings by congruent darts is a continuously deformable series of tilings which are topologically trapezohedra. In the above explanation, unless otherwise stated, the tilings we have enumerated are isohedral and admit no continuous deformation. We prove that if a spherical (kite/dart/rhombus) admits an edge-to-edge spherical monohedral tiling, then it also does a spherical isohedral tiling. We also prove that the set of anisohedral, spherical triangles (i.e., spherical triangles admitting spherical monohedral triangular tilings but not any spherical isohedral triangular tilings) consists of a certain, infinite series of isosceles triangles $I$ , and an infinite series of right scalene triangles which are the bisections of $I$ .

JPD-Coloring of the Monohedral Tiling for the Plane

We introduce a definition of coloring by using joint probability distribution “JPD-coloring” for the plane which is equipped by tilingI. We investigate the JPD-coloring of ther-monohedral tiling for the plane by mutually congruent regular convex polygons which are equilateral triangles atr= 3 or squares atr= 4 or regular hexagons atr= 6. Moreover we present some computations for determining the corresponding probability values which are used to color in the three studied cases by MAPLE-Package.

Right Triangular Spherical Dihedral f–Tilings with Two Pairs of Congruent Sides

By a dihedral folding tiling (f-tiling, for short) of the sphere S whose prototiles are spherical right triangles, T1 and T2, we mean a polygonal subdivision τ of S such that each cell (tile) of τ is congruent to T1 or T2 and the vertices of τ satisfy the angle-folding relation, i.e., each vertex of τ is of even valency and the sums of alternating angles around each vertex are π. In fact, the crease pattern associated to the subjacent graph of a spherical f-tiling satisfy the Kawasaki’s condition at any vertex v. In this paper we shall discuss dihedral f-tilings by spherical right triangles, considering two distinct cases of adjacency. We assume that from all the sides of the prototiles involved there are two pairs of congruent sides. The 3-dimensional representations of the obtained f-tilings are presented as well as the combinatorial structure. f-tilings are intrinsically related to the theory of isometric foldings of Riemannian manifolds, introduced by S. A. Robertson [6] in 1977. The classification of f-tilings was initiated by Ana Breda [1], with a complete classification of all spherical monohedral f-tilings. Later on, in 2002, Y. Ueno and Y. Agaoka [10] have established the complete classification of all triangular monohedral tilings (without any restrictions on angles). The study of dihedral f-tilings by spherical right triangles is a very extensive and exhaustive work and some particular cases were recently obtained in papers [4, 5]. A list of all dihedral f-tilings of the sphere by triangles and parallelograms including the combinatorial structure of each tiling can be found in [2]. Robert Dawson has also been interested in special classes of spherical tilings, see [7–9] for instance. We shall denote by Ω (T1, T2) the set, up to an isomorphism, of all dihedral f-tilings of S whose prototiles are T1 and T2. From now on T1 is a spherical right triangle of internal angles π2 , α and β, with edge lengths a (opposite to β), b (opposite to α) and c (opposite to π2 ), and T2 is a spherical right triangle of internal angles π2 , γ and δ, with edge lengths d (opposite to δ), e (opposite to γ) and f (opposite to π2 ) (see Figure 1). We will assume throughout the text that T1 and T2 are distinct triangles, i.e., (α, β) = (γ, δ) and (α, β) = (δ, γ). The case α = β or γ = δ was analyzed in [4], and so we will assume further that α = β and γ = δ.

Sigma Coloring of the Monohedral Tiling

Sigma Coloring of the Monohedral Tiling

Tilable nature of virus capsids and the role of topological constraints in natural capsid design

Virus capsids are highly specific assemblies that are formed from a large number of often chemically identical capsid subunits. In the present paper we ask to what extent these structures can be viewed as mathematically tilable objects using a single two-dimensional tile. We find that spherical viruses from a large number of families-eight out of the twelve studied-qualitatively possess properties that allow their representation as two-dimensional monohedral tilings of a bound surface, where each tile represents a subunit. This we did by characterizing the extent to which individual spherical capsids display subunit-subunit (1) holes, (2) overlaps, and (3) gross structural variability. All capsids with T numbers greater than 1 from the Protein Data Bank, with homogeneous protein composition, were used in the study. These monohedral tilings, called canonical capsids due to their platonic (mathematical) form, offer a mathematical segue into the structural and dynamical understanding of not one, but a large number of virus capsids. From our data, it appears as though one may only break the long-standing rules of quasiequivalence by the introduction of subunit-subunit structural variability, holes, and gross overlaps into the shell. To explore the utility of canonical capsids in understanding structural aspects of such assemblies, we used graph theory and discrete geometry to enumerate the types of shapes that the tiles (and hence the subunits) must possess. We show that topology restricts the shape of the face to a limited number of five-sided prototiles, one of which is the "bisected trapezoid" that is a platonic representation of the most ubiquitous capsid subunit shape seen in nature (the trapezoidal jelly-roll motif). This motif is found in a majority of seemingly unrelated virus families that share little to no host, size, or amino acid sequence similarity. This suggests that topological constraints may exhibit dominant roles in the natural design of biological assemblies, while having little effect on amino acid sequence similarity.

The Local Theorem for Monotypic Tilings

A locally finite face-to-face tiling $\cal T$ of euclidean $d$-space ${\Bbb E}^d$ is monotypic if each tile of $\cal T$ is a convex polytope combinatorially equivalent to a given polytope, the combinatorial prototile of $\cal T$. The paper describes a local characterization of combinatorial tile-transitivity of monotypic tilings in ${\Bbb E}^d$; the result is the Local Theorem for Monotypic Tilings. The characterization is expressed in terms of combinatorial symmetry properties of large enough neighborhood complexes of tiles. The theorem sits between the Local Theorem for Tilings, which describes a local characterization of isohedrality (tile-transitivity) of monohedral tilings (with a single isometric prototile) in ${\Bbb E}^d$, and the Extension Theorem, which gives a criterion for a finite monohedral complex of polytopes to be extendable to a global isohedral tiling of space.

Crumpled Cube and Solid Horned Sphere Space Fillers

We construct two classes of wildly embedded space fillers of R3. First, every crumpled cube is shown to have an embedding in R3 that admits a monohedral tiling of R3. Second, a solid Alexander horned sphere with a topologically trivial interior is shown to admit a monohedral tiling of a cube and hence R3. By joining a solid horned sphere with compact polyhedral 3-submanifolds of R3 with one boundary component, we construct space fillers homeomorphic to the polyhedral submanifolds but of different embedding types. Using the suitably embedded crumpled cubes instead of a solid horned sphere, space fillers of even more different topological types can be produced.

Holey Coronas A Solution of the Grünbaum-Shephard Conjecture on Convex Isohedral Tilings

In an article published recently in the section of Unsolved Problems of this MONTHLY, Branko Griunbaum [1] proposes the following conjecture: If S9 is an isohedral tiling of the Euclidean d-dimensional space by copies of a convex tile, then the corona of each tile is topological ball. The same conjecture is discussed as an open question in the article of Geoffrey C. Shephard [3], in which the author proves its stronger version for d = 2. We shall therefore refer to it as the Griinbaum-Shephard Conjecture. A tiling S9' is a collection of closed topological balls (disks, for d = 2) with mutually disjoint interiors (the tiles of S) such that the union of all tiles is the whole space. If all tiles are congruent, then S9 is monohedral, and their common shape is said to be the prototile of Y A tiling is isohedral provided its symmetry group acts transitively on the tiles; it is convex provided each of its tiles is convex. A lattice (of vectors) in R d is the collection of linear combinations with integer coefficients of a basis for R d. A typical example of a lattice is the collection of all vectors with integer coordinates, called the integer lattice (every lattice is affinely equivalent to the integer lattice). A monohedral tiling consisting of translates of the prototile by the vectors of a lattice is called a lattice tiling. Obviously, every lattice tiling is isohedral, while the converse is not true even in the plane. The corona of a tile T of S is the union of all tiles that meet T, including T itself. For example, in the usual (face-to-face) integer-lattice tiling of d-dimensional space by unit cubes, the corona of each tile is a cube of edge-length 3. Obviously, in an isohedral tiling, the coronas of any two tiles are congruent. See Griinbaum and Shephard [2] for a nearly encyclopedic survey of the theory of plane tilings; also see Schulte [4] and Schattschneider and Senechal [5], which deal with tilings in higher dimensional spaces. Grunbaum [1] observes that a confirmation of his conjecture for d = 2 (even without the assumption of convexity) can be established by inspection, since all types of isohedral tilings of the plane are known [2, Chapter 6]. He also conjectures that in the plane even more is true, namely that in every monohedral tiling of the plane the corona of every tile is simple (meaning simply-connected, hence homeomorphic to a disk). Shephard [3] found a non-convex counterexample, but also proved that in convex monohedral plane tilings every corona is simple [3, p. 358].

Tilings with the neighborhood property

The neighborhood N(T) of a tile T is the set of all tiles which meet T in at least one point. If for each tile T there is a different tile T1 such that N(T) = N(T1) then we say the tiling has the neighborhood property (NEBP). Grünbaum and Shepard conjecture that it is impossible to have a monohedral tiling of the plane such that every tile T has two different tiles T1, T2 with N(T) = N(T1) = N(T2). If all tiles are convex we show this conjecture is true by characterizing the convex plane tilings with NEBP. More precisely we prove that a convex plane tiling with NEBP has only triangular tiles and each tile has a 3‐valent vertex. Removing 3‐valent vertices and the incident edges from such a tiling yields an edge‐to‐edge planar triangulation. Conversely, given any edge‐to‐edge planar triangulation followed by insertion of a vertex and three edges that triangulate each triangle yields a convex plane tiling with NEBP. We exhibit an infinite family of nonconvex monohedral plane tilings with NEBP. We briefly discuss tilings of R3 with NEBP and exhibit a monohedral tetrahedral tiling of R3 with NEBP.

Tiling complexity of small n-ominoes ( N < 10)

Using a computer, the author studies small polyominoes that admit a monohedral tiling, both with reflection and without. In either case, he defines the complexity of a tile to be the smallest number of congruent copies that can be assembled into a patch that tiles by translation. He presents examples of the different tiling types he finds, and presents a table of the number of each kind.

The eighty-one types of isohedral tilings in the plane

1. Atilingis a collection= {Ti|i= 1, 2, …} of closed topological discs which covers the Euclidean planeE2, and of which the individualtiles Tihave disjoint interiors. We shall assume throughout that the intersection of any two tiles is a connected set. If each tile iscongruent(directly or reflectively isometric) to a given setT, then the tilingis calledmonohedralandTis called theprototileof. Clearly every monohedral tiling is locally finite.