Related papers: Some Generalizations of the Pinwheel Tiling

Two new series of substitution tilings are introduced in which the tiles appear in infinitely many orientations. It is shown that several properties of the well-known pinwheel tiling do also hold for these new examples, and, in fact, for…

We define a new family of non-periodic tilings with square tiles that is mutually locally derivable with some family of tilings with isosceles right triangles. Both families are defined by simple local rules, and the proof of their…

A new family of decagonal quasiperiodic tilings are constructed by the use of generalized point substitution processes, which is a new substitution formalism developed by the author [N. Fujita, Acta Cryst. A 65, 342 (2009)]. These tilings…

We introduce a fractal version of the pinwheel substitution tiling. There are thirteen basic prototiles, all of which have fractal boundaries. These tiles, along with their reflections and rotations, create a tiling space which is mutually…

The pinwheel triangle of Conway and Radin is a standard example for tilings with self-similarity and statistical circular symmetry. Many modifications were constructed, all based on partitions of triangles or rectangles. The fractal example…

We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two…

General substitution rules for non-periodic rhomb tilings are derived. From the requirement that all substitution tiles consist of a discrete number of prototiles, it follows that a substitution tile with angle s*pi/n must be built out of…

A new method for constructing aperiodic tilings is presented. The method is illustrated by constructing a particular tiling and its hull. The properties of this tiling and the hull are studied. In particular it is shown that these tilings…

We present a single, connected tile which can tile the plane but only non-periodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules…

Non-periodic tilings with Tile(1, 1) using the substitution method, as presented by Smith et al. in [2] and [3], can be converted into non-periodic tilings with three types of pentagons. When arbitrary replacements are excluded, the…

We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the…

This paper gives new solutions to the problem: 'Can we construct monohedral tilings of the disk such that a neighbourhood of the origin has trivial intersection with at least one tile?'

We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The…

We study nonperiodic tilings of the line obtained by a projection method with an interval projection structure. We obtain a geometric characterisation of all interval projection tilings that admit substitution rules and describe the set of…

Conway and Radin's "quaquaversal" tiling of R^3 is known to exhibit statistical rotational symmetry in the infinite volume limit. A finite patch, however, cannot be perfectly isotropic, and we compute the rates at which the anisotropy…

Given a tiling of a 2D grid with several types of tiles, we can count for every row and column how many tiles of each type it intersects. These numbers are called the_projections_. We are interested in the problem of reconstructing a tiling…

Rotationally symmetric tilings by a convex pentagonal tile belonging to both the Type 1 and Type 7 families are introduced. Among them are spiral tilings with two- and four-fold rotational symmetry. Those rotationally symmetric tilings are…

Inspired by the modelization of 2D materials systems, we characterize arrangements of identical nonflat squares in 3D. We prove that the fine geometry of such arrangements is completely characterized in terms of patterns of mutual…

It is shown that there are primitive substitution tilings with dense tile orientations invariant under n-fold rotation for n=2,3,4,5,6,8. The proof for dense tile orientations uses a general result about irrationality of angles in certain…

We show that a single prototile can fill space uniformly but not admit a periodic tiling. A two-dimensional, hexagonal prototile with markings that enforce local matching rules is proven to be aperiodic by two independent methods. The…