In geometry, a chair tiling (or L tiling) is a nonperiodic

substitution tiling In geometry, a tile substitution is a method for constructing highly ordered Tessellation, tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational ...

created from L-tromino

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 ...

s. These prototiles are examples of

rep-tile In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by ...

s and so an iterative process of decomposing the L tiles into smaller copies and then rescaling them to their original size can be used to cover patches of the plane. Chair tilings do not possess

translational symmetry In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation. Analogously, an operato ...

, i.e., they are examples of ''nonperiodic tilings'', but the chair tiles are not aperiodic tiles since they are not forced to tile nonperiodically by themselves. The ''trilobite'' and ''cross'' tiles are aperiodic tiles that enforce the chair tiling substitution structure and these tiles have been modified to a simple aperiodic set of tiles using matching rules enforcing the same structure. Barge et al. have computed the

Čech cohomology In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open set, open cover (topology), covers of a topological space. It is named for the mathematician Eduard Čech. Moti ...

of the chair tiling and it has been shown that chair tilings can also be obtained via a cut-and-project scheme.

References

External links

* Tilings Encyclopedia
Chair
Aperiodic tilings {{geometry-stub