Plane square tilings

Quotient maps of 2-uniform tilings of the plane on the torus

debashis bhowmik

2021

A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${\rm Aut}(X)$ acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of two types, then the map is said to be a $2$-semiequivelar map. A 2-uniform tiling is an edge-to-edge tiling of regular polygons having $2$ distinct transitivity classes of vertices. Clearly, a $2$-uniform map is $2$-semiequivelar. The converse of this is not true in general. We know that there are infinitely many tilings of $2$-semiequivelar type. In this article, we study $2$-semiequivelar maps on the torus that are the quotient of the plane's $2$-unform latices. We have found bounds of the number of orbits of vertices of the quotient $2$-semiequivelar maps on the torus under the automorphism groups (symmetric groups).

Combinatorial construction of tilings by barycentric simplex orbits ( D symbols) and their realizations in Euclidean and other homogeneous spaces

Emil Molnár

Acta Crystallographica Section A Foundations of Crystallography, 2005

A new method, developed in previous works by the author (partly with coauthors), is presented which decides algorithmically, in principle by computer, whether a combinatorial space tiling (T , À) is realizable in the d-dimensional Euclidean space E d (think of d = 2, 3, 4) or in other homogeneous spaces, e.g. in Thurston's 3-geometries: E 3 ; S 3 ; H 3 ; S 2 ÂR; H 2 ÂR; g SL 2 R SL 2 R; Nil; Sol: Then our group À will be an isometry group of a projective metric 3-sphere PS 3 ðR; h ; iÞ, acting discontinuously on its above tiling T. The method is illustrated by a plane example and by the well known rhombohedron tiling ðT ; ÀÞ, where À = R " 3 3m is the Euclidean space group No. 166 in International Tables for Crystallography.

An algebraic invariant for substitution tiling systems, Geometriae Dedicata 73

1998

Abstract. We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems. Mathematics Subject Classifications (1991): 52C22, 52C20. Key words: tiling, dynamics, invariant.

SL$_2 (\mathbb Z)$-tilings of the torus, Coxeter–Conway friezes and Farey triangulations

L’Enseignement Mathématique, 2015

e notion of SL 2-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic SL 2-tilings that contain a rectangular domain of positive integers. Every such SL 2-tiling corresponds to a pair of frieze patterns and a unimodular 2 2-matrix with positive integer coe cients. We relate this notion to triangulated n-gons in the Farey graph.

$ SL_2 (\ mathbb {Z}) $-tiling of the torus, Coxeter-Conway friezes and Farey triangulations

The notion of SL 2 -tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic SL 2 -tilings that contain a rectangular domain of positive integers. Every such SL 2 -tiling corresponds to a pair of frieze patterns and a unimodular 2 × 2-matrix with positive integer coefficients. We relate this notion to triangulated n-gons in the Farey graph.

Symmetry Groups of a Class of Spherical Folding Tilings

Ana Breda

2009

We classify the group of symmetries of all dihedral folding tilings by spherical triangles and spherical parallelograms, obtained in [2], [3] and . The transitivity classes of isogonality and isohedrality are also determined, see Table .1.

Institute for Mathematical Physics an Algebraic Invariant for Substitution Tiling Systems an Algebraic Invariant for Substitution Tiling Systems

2007

We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We deene an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems.

A Characterization of Rotation Number on One-Dimensional Tiling Spaces

arXiv: Dynamical Systems, 2017

Identity-homotopic self-homeomorphisms of a space of non-periodic 1-dimensional tiling are generalizations of orientation-preserving self-homeomorphisms of circles. We define the analogue of rotation numbers for such maps. In constrast to the classical situation, additional assumptions are required to make rotation numbers globally well-defined and independent of initial conditions. We prove that these conditions are sufficient, and provide counterexamples when these conditions are not met.

SL2 (Z)-Tilings of the Torus, Coxeter-Conway Friezes and Farey Triangulations

The notion of SL 2-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic SL 2-tilings that contain a rectangular domain of positive integers. Every such SL 2-tiling corresponds to a pair of frieze patterns and a unimodular 2 × 2-matrix with positive integer coefficients. We relate this notion to triangulated n-gons in the Farey graph.

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References (12)

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Related papers

On tilings of the plane

Andreas Dress

Geometriae Dedicata, 1987

The paper discusses homeomorphic types of (periodic) filings of the plane in terms of their associated Delaney symbol. Such a symbol consists of a (finite) set ~ on which three involutions ao,al and tr2 act from the right such that troa ~ = a2ao and there are two maps mol ,m12 :~ ~ ~ satisfying certain compatibility conditions. It is shown how the barycentric subdivision of a tiling can be used to define its Delaney symbol and that the symbol characterizes the tiling up to (equivariant) homeomorphisms. Furthermore, it is shown how properties of the tiling can be recognized from corresponding properties of the symbol and how this technique can be used to enumerate various types of tilings with specific properties. If necessary, this enumeration can be done by appropriate computer programs. Among other results, we have been able to vindicate the results by Griinbaum et al., announced in [8]. Finally, some recursive enumeration formulas, based on the Delaney symbol technique,, are stated.

A Guide to the Symmetry Structure of Quasiperiodic Tiling Classes

Joachim Hermisson

1997

Based upon the torus parametrization which was introduced recently, we present a recipe allowing for a complete analysis of the symmetry structure of quasiperiodic local isomorphism classes in one and two dimensions. A number of results is provided explicitly, including some widely used planar tiling classes with 8-, 10- and 12-fold rotational symmetry.

The classification of face-transitive periodic three-dimensional tilings

Emil Molnár

Acta Crystallographica Section A Foundations of Crystallography, 1993

Co-authrs are: Andreas W. M. Dress, Daniel H. Huson, Emil Molnár It has long been known that there exists an infinite number of types of tile-transitive periodic three-dimensional tilings. Here, it is shown that, by contrast, the number of types of face-transitive periodic three-dimensional tilings is finite. The method of Delaney symbols and the properties of the 219 isomorphism classes of crystallographic space groups are used to find exactly 88 equivariant types that fall into seven topological families. There have been many connections, continuations of this paper, see e.g. the report of Daniel H. Huson in the list of Academia edu. (This remark is by Emil Molnár, Budapest, 14. 04, 2025)

Quotient maps of 2,3-uniform tilings of the plane on the torus

debashis bhowmik

2021

A 2-uniform tiling is an edge-to-edge tiling by regular polygons having 2 distinct transitivity classes of vertices. There are 20 distinct 2-uniform tilings (these are of 14 different types) on the plane, and since the plane is the universal cover of the torus, it is natural to explore maps on the torus that correspond to the 2-uniform tilings. In this article, we discuss that if a map is the quotient of a plane's 2-uniform lattice then what would be the bounds of the number of vertex orbits. A 3-uniform tiling is an edge-to-edge tiling by regular polygons having 3 distinct transitivity classes of vertices. There are 61 distinct 3-uniform tilings on the plane. In this article, we discuss that if a map is the quotient of a plane's 3-uniform lattice then what would be the bounds of the number of vertex orbits.

Rotation Numbers and Rotation Classes on One-Dimensional Tiling Spaces

Annales Henri Poincaré

We extend rotation theory of circle maps to tiling spaces. Specifically, we consider a 1-dimensional tiling space Ω with finite local complexity and study self-maps F that are homotopic to the identity and whose displacements are strongly pattern equivariant (sPE). In place of the familiar rotation number we define a cohomology class [µ] ∈Ȟ 1 (Ω, R). We prove existence and uniqueness results for this class, develop a notion of irrationality, and prove an analogue of Poncaré's Theorem: If [µ] is irrational, then F is semi-conjugate to uniform translation on a space Ω µ of tilings that is homeomorphic to Ω. In such cases, F is semi-conjugate to uniform translation on Ω itself if and only if [µ] lies in a certain subspace ofȞ 1 (Ω, R).

A class of spherical dihedral f-tilings

Patrícia Ribeiro

2008

The classification of all dihedral triangular f-tilings of the Riemannian sphere S 2 whose prototiles are an equilateral triangle and an isosceles triangle and the identification of their symmetry groups are given. We also determine their classes of isogonality and isohedrality.

A geometrical approach of quasiperiodic tilings

Christophe Oguey

Communications in Mathematical Physics, 1988

Tilings provide generalized frames of coordinates and as such they are used in different areas of physics. The aim of the present paper is to present a unified and systematic description of a class of tilings which have appeared in contexts as disconnected as crystallography and dynamical systems. The tilings of this class show periodic or quasiperiodic ordering and the tiles are related to the unit cube through affine transformations. We present a section procedure generating canonical quasiperiodic tilings and we prove that true tilings are indeed obtained. Moreover, the procedure provides a direct and simple characterization of quasiperiodicity which is suitable for tilings but which does not refer to Fourier transform.

SL2(Z)-tiling of the torus, Coxeter-Conway friezes and Farey triangulations

2014

The notion of SL2-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic SL2-tilings that contain a rectangular domain of positive integers. Every such SL2-tiling corresponds to a pair of frieze patterns and a unimodular 2 × 2-matrix with positive integer coefficients. We relate this notion to triangulated n-gons in the Farey graph.

An Algebraic Invariant for Substitution Tiling Systems

Geometriae Dedicata, 1998

We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems.

Some Generalizations of the Pinwheel Tiling

Discrete & Computational Geometry, 1998

We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases, the tiles appear in a finite number of sizes and an infinite number of orientations. These tilings generally do not meet full-edge to full-edge, but can be forced through local matching rules. In a countable number of cases, the tiles appear in a finite number of orientations but an infinite number of sizes, all within a set range, while in an uncountable number of cases both the number of sizes and the number of orientations is infinite.