List of aperiodic sets of tiles
In geometry, 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).
[1] 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 or primitive cell which repeats endlessly and regularly in two independent directions.
[2] 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.
Click "show" for description.
A
periodic tiling
with a fundamental unit (triangle) and a primitive cell (hexagon) highlighted. A tiling of the entire plane can be generated by fitting copies of these triangular patches together. In order to do this, the basic triangle needs to be rotated 180 degrees in order to fit it edge-to-edge to a neighboring triangle. Thus a
triangular tiling
of fundamental units will be generated that is
mutually locally derivable
from the tiling by the colored tiles. The other figure drawn onto the tiling, the white hexagon, represents a primitive cell of the tiling. Copies of the corresponding patch of coloured tiles can be
translated
to form an infinite tiling of the plane. It is not necessary to rotate this patch in order to achieve this.
