Related papers: Ammann Bars for Octagonal Tilings

Aperiodic tilings support two classically studied but hitherto separately presented structures: matching rules, which enforce global order via local constraints, and height functions, which encode global geometry through integer-valued…

We present here an elementary construction of an aperiodic tile set. Although there already exist dozens of examples of aperiodic tile sets we believe this construction introduces an approach that is different enough to be interesting and…

A new kind of aperiodic tiling is introduced. It is shown to underlie a structure obtained as a superposition of waves with incommensurate periods. Its connections to other other tilings and quasicrystals are discussed.

A general construction principle of inflation rules for decagonal quasiperiodic tilings is proposed. The prototiles are confined to be polygons with unit edges. An inflation rule for a tiling is the combination of an expansion and a…

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of…

Mathematicians have been interested in non-periodic tilings of space for decades; however, it was the unexpected discovery of non-periodically ordered structures in intermetallic alloys which brought this subject into the limelight. These…

Cut and project sets are obtained by projecting an irrational slice through a lattice to a lower dimensional subspace. Under standard conditions, the resulting pattern has no translational periods even though it retains some regularity of…

Aperiodic tiling --- a form of complex global geometric structure arising through locally checkable, constant-time matching rules --- has long been closely tied to a wide range of physical, information-theoretic, and foundational…

Classical results on aperiodic tilings are rather complicated and not widely understood. Below, an alternative approach is discussed in hope to provide additional intuition not apparent in classical works.

We establish a structure theorem for the family of Ammann A2 tilings of the plane. Using that theorem we show that every Ammann A2 tiling is self-similar in the sense of [B. Solomyak, Nonperiodicity implies unique composition for…

Sets of three types of convex pentagons that are aperiodic with no matching conditions on the edges are created from a chiral aperiodic monotile Tile(1, 1). This method divides the interior of Tile(1,1) into five convex polygons with five…

Bobbin lace is a fibre art form in which threads are braided together to form a fabric, often with a very detailed and complex design. In traditional practice, each region of the fabric is filled with a periodic texture. We establish the…

We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both…

We propose a new simple construction of an aperiodic tile set based on self-referential (fixed point) argument. People often say about some discovery that it appeared "ahead of time", meaning that it could be fully understood only in the…

We present a method for generating hexagonal aperiodic tilings that are topologically equivalent to the triangular and dice lattices. This approach incorporates aperiodic sequences into the spacing between three sets of grids for the…

We study the cohomology and hence $K$-theory of the aperiodic tilings formed by the so called 'cut and project' method, i.e., patterns in $d$ dimensional Euclidean space which arise as sections of higher dimensional, periodic structures.…

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…

This paper presents a tileset of 3 squares with local constraints on their borders and corners that enforce non-periodic tiling. We start with a description of the tileset and we demonstrate that it can tile the entire plane…

We construct the first aperiodic tiles for two amenable 3-dimensional Lie groups: Sol and the Heisenberg group. Our construction relies on the use of higher-dimensional uniformly finite homology. In particular, we settle completely the…

An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics…