Ammann Bars for Octagonal Tilings

Many books on mathematics and art discuss a topic called demiregular tilings and claim that there are 14 such tilings. However, many of them give different lists of 14 tilings! In this paper we will compare the lists from some standard references that give a total of 18 such tilings. We will also show that unless we add further restrictions, there will in fact be infinitely many such tilings. The "fact" that there are 14 demiregular tilings has been repeated by many authors. The goal of this paper is to put an end to the concept of demiregular tilings.

Scripta Metallurgica, 1986

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.

Geometriae Dedicata, 1992

Journal of Geometry, 2007

Discrete & Computational Geometry, 2009

provided an important method to compute the boundaries of lattice reptiles as a 'recurrent set' on a free group of a finite alphabet. That is, those tilings are generated by lattice translations of a single tile, and there is an expanding linear map that carries tiles to unions of tiles. The boundary of the tile is identified with a sequence of words in the alphabet obtained from an expanding endomorphism (substitution) on the alphabet. In this paper, Dekking's construction is generalized to address tilings with more than one tile, and to have the elements of the tilings be generated by both translation and rotations. Examples that fall within the scope of our main result include self-replicating multi-tiles, self-replicating tiles for crystallographic tilings and aperiodic tilings.

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.

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.

2021

It is proved that homeomorphic images of certain two-dimensional aperiodic tilings, such as Ammann-A2 tilings, are recognizable, in both mathematical and practical senses. One implication of the results is that it is possible to search for distorted aperiodic structures in nature, where they may be hiding in plain sight.

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)