geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, an Ammann A1 tiling is a

tiling Tiling may refer to: *The physical act of laying tiles *Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester The ...

from the 6 piece

prototile In mathematics, a prototile is one of the shapes of a tile in a tessellation. Definition A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint interiors. Some of the tiles m ...

set shown on the right. They were found in 1977 by

Robert Ammann Robert Ammann (October 1, 1946 – May, 1994) was an List of amateur mathematicians, amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings. Ammann attended Brandei ...

. Ammann was inspired by the Robinson tilings, which were found by

Robinson Robinson may refer to: People and names * Robinson (name) Fictional characters * Robinson Crusoe, the main character, and title of a novel by Daniel Defoe, published in 1719 Geography * Robinson projection, a map projection used since the 19 ...

in 1971. The A1 tiles are one of five sets of tiles discovered by Ammann and described in ''

Tilings and patterns ''Tilings and patterns'' is a book by mathematicians Branko Grünbaum and Geoffrey Colin Shephard published in 1987 by W.H. Freeman. The book was 10 years in development, and upon publication it was widely reviewed and highly acclaimed. Structu ...

''. The A1 tile set is

aperiodic A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the tr ...

, i.e. they tile the whole

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

, but only without ever creating a

periodic tiling A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensiona ...

Generation through matching

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 In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the ''left child'' and the ''right child''. That is, it is a ''k''-ary tree with . A recursive definition using set theory i ...

that is continued indefinitely. The markings on the tiles in the pictures emphasize this

hierarchical structure A hierarchy (from Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an importan ...

, however, they have only illustrative character and do not represent additional

matching rules 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 (or prototiles) is aperiodic if copies of these tiles can form only non- period ...

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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

of the

Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformati ...

, e.g.

translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...

and

rotations Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersec ...

. 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 A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect, it results from specular reflection off from ...

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

\Sigma^

for the

alphabet An alphabet is a standard set of letter (alphabet), letters written to represent particular sounds in a spoken language. Specifically, letters largely correspond to phonemes as the smallest sound segments that can distinguish one word from a ...

\Sigma=\

, where ''g'' indicates the green choice while ''r'' indicates the red choice. These are in bijection to a

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Throu ...

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 In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger tha ...

many different A1 tilings, all of them necessarily nonperiodic. Since there are only

countably In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...

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. AmmannA1FinalTiling3D

Further pictures

References

{{Tessellation

Generation through matching

Further pictures

See also

References