A census of edge-transitive planar tilings

Acta Crystallographica Section A Foundations and Advances, 2019

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 {\cal T} 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 {\cal T} 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.

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.

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)

2017

In my earlier two articles on Tessellations – Covering the Plane with Repeated Patterns Parts I and II – which appeared in At Right Angles (March 2014 and July 2014), I tried to provide a glimpse into the topic of recognizing regular polygons that can tessellate. In an accompanying article, Enumeration of Semi-regular Tessellations (AtRiA, March 2014), the authors arithmetically enumerated the combinations of regular polygons whose interior angles could fit together to make a complete angle (i.e., 360°); 17 such combinations were listed. However, not all of these tessellate. The present article may be viewed as an extension of the earlier ones; I have tried to identify and shortlist the combinations that extend to create semi-regular tiling patterns

2008

A locally finite face-to-face tiling of euclidean d-space by convex polytopes is called combinatorially multihedral if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local characterization of combinatorially multihedral tilings in terms of centered coronas. This generalizes the Local Theorem for Monotypic Tilings, established in an earlier paper, which characterizes the case of combinatorial tile-transitivity.

Geometriae Dedicata, 1992

2018

A planar monohedral tiling is a decomposition of R2 into congruent tiles. We say that such a tiling has the flag property if for each triple of tiles that intersect pairwise, the three tiles intersect in a common point. We show that for convex tiles, there exist only three classes of tilings that are not flag, and they all consist of triangular tiles; in particular, each convex tiling using polygons with n ≥ 4 vertices is flag. We also show that an analogous statement for the case of non-convex tiles is not true by presenting a family of counterexamples.

Journal of Geometry, 2007

2002

We consider tilings of quadriculated regions by dominoes and of triangulated regions by lozenges. We present an overview of results concerning tileability, enumeration and the structure of the space of tilings.

1998

We consider tilings of quadriculated regions by dominoes and of triangulated regions by lozenges. We present an overview of results concerning tileability, enumeration and the structure of the space of tilings.