## Nonperiodic

A tiling `$T$`

is called nonperiodic, if from `$T + x = T$`

it follows that `$x=0$`

. In other words, if no translation (other than the trivial one) maps the tiling to itself. In the theory of nonperiodic tilings usually the repetitive ones are the objects to be investigated.
Usual constructions for repetitive nonperiodic tilings are substitutions, cut and project methods, and matching rules. Another well-studied class of nonperiodic (non-repetitive) tilings are random tilings, which can also be viewed as being generated by matching rules.