geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a tiling is a partition of the plane (or any other geometric setting) into closed sets (called ''tiles''), without gaps or overlaps (other than the boundaries of the tiles). A tiling is considered periodic if there exist translations in two independent directions which map the tiling onto itself. Such a tiling is composed of a single

fundamental unit A base unit (also referred to as a fundamental unit) is a Units of measurement, unit adopted for measurement of a ''base quantity''. A base quantity is one of a conventionally chosen subset of physical quantity, physical quantities, where no quanti ...

primitive cell In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessaril ...

which repeats endlessly and regularly in two independent directions. An example of such a tiling is shown in the adjacent diagram (see the image description for more information). A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called

aperiodic A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...

. The tilings obtained from an aperiodic set of tiles are often called aperiodic tilings, though strictly speaking it is the tiles themselves that are aperiodic. (The tiling itself is said to be "nonperiodic".) The first table explains the abbreviations used in the second table. The second table contains all known aperiodic sets of tiles and gives some additional basic information about each set. This list of tiles is still incomplete.

Explanations

List

References

External links

* Stephens P. W., Goldman A. I
''The Structure of Quasicrystals''
* Levine D., Steinhardt P. J
''Quasicrystals I Definition and structure''

''Tilings Encyclopedia''
{{DEFAULTSORT:Aperiodic sets of tiles Mathematics-related lists