In geometry, an Ammann A1 tiling is a tiling from the 6 piece prototile set shown on the right. They were found in 1977 by Robert Ammann. [1] Ammann was inspired by the Robinsion tilings, which were found by Robinson in 1971. The A1 tiles are one of five sets of tiles discovered by Ammann and described in Tilings and patterns . [2]
The A1 tile set is aperiodic, [2] i.e. they tile the whole Euclidean plane, but only without ever creating a periodic tiling.
The prototiles are squares with indentations and protrusions on the sides and corners that force the tiling to form a pattern of a perfect binary tree that is continued indefinitely. The markings on the tiles in the pictures emphasize this hierarchical structure, however, they have only illustrative character and do not represent additional matching rules as this is already taken care of by the indentations and protrusions.
However, the tiling produced in this way is not unique, not even up to isometries of the Euclidean group, e.g. translations and rotations. When going to the next generation, one has choices. In the picture to the left, the initial patch in the left upper corner highlighted in blue can be prolonged by either a green or a red tile, which are mirror images of each other and instances of the prototile labeled b. Then there are two more choices in the same spirit but with prototile e. The remainder of the next generation is then fixed. If one would deviate from the pattern for this next generation, one would run into configurations that will not match up globally at least at some later stage.
The choices are encoded by infinite words from for the alphabet , where g indicates the green choice while r indicates the red choice. [3] These are in bijection to a Cantor set and thus their cardinality is the continuum. Not all choices lead to a tiling of the plane. E.g. if one only sticks to the green choice one would only fill a lower right corner of the plane. If there are sufficiently generic infinitely many alteration between g and r one will however cover the whole plane. This still leaves uncountably many different A1 tilings, all of them necessarily nonperiodic. Since there are only countably many possible Euclidean isometries that respect the squares underlying the tiles to relate these different tilings, there are uncountable many A1 tilings even up to isometries.
Additionally an A1 tiling may have faults (also called corridors) going off to infinity in arms. This additionally increases the numbers of possible A1 tilings, but the cardinality remains that of the continuum. Note that the corridors allow for some part with binary tree hierarchy to be rotated compared to the other such parts.
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types is aperiodic if copies of these tiles can form only non-periodic tilings.
Robert Ammann was an amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings.
In geometry, the Heesch number of a shape is the maximum number of layers of copies of the same shape that can surround it with no overlaps and no gaps. Heesch's problem is the problem of determining the set of numbers that can be Heesch numbers. Both are named for geometer Heinrich Heesch, who found a tile with Heesch number 1 and proposed the more general problem.
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.
In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol {3,6}:e.
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.
In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.
In geometry, a shape is said to be anisohedral if it admits a tiling, but no such tiling is isohedral (tile-transitive); that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling. A tiling by an anisohedral tile is referred to as an anisohedral tiling.
In geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite polyhedron or a tiling of the plane.
In geometry, an apeirogonal antiprism or infinite antiprism is the arithmetic limit of the family of antiprisms; it can be considered an infinite polyhedron or a tiling of the plane.
In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker. They are one of the five sets of tilings discovered by Ammann and described in Tilings and patterns.
A Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and a tiling is aperiodic if it does not contain arbitrarily large periodic regions or patches. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s.
A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, explaining its name. It is commonly used as a pattern for floor tiles. When used for this, it is also known as a hopscotch pattern or pinwheel pattern, but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern.
A set of prototiles is aperiodic if copies of the prototiles can be assembled to create tilings, such that all possible tessellation patterns are non-periodic. The aperiodicity referred to is a property of the particular set of prototiles; the various resulting tilings themselves are just non-periodic.
The Voderberg tiling is a mathematical spiral tiling, invented in 1936 by mathematician Heinz Voderberg (1911-1945). Karl August Reinhardt asked the question of whether there is a tile such that two copies can completely enclose a third copy. Voderberg, his student, answered in the affirmative with Form eines Neunecks eine Lösung zu einem Problem von Reinhardt ["On a nonagon as a solution to a problem of Reinhardt"].
In geometry of the Euclidean plane, the 3-4-6-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular triangles, squares, hexagons and dodecagons, arranged in two vertex configuration: 3.4.6.4 and 4.6.12.
