## Patch

Dealing with tilings, it is useful to consider finite parts of a tiling. These are called patches (or clusters). The definition is simply: a patch is a finite subset of a tiling. Sometimes one requires, in addition, that the support of a patch should be connected, or homeomorphic to a ball. The latter leads to problems in higher dimensions. There are tilings where no patch with more than one tile is homeomorphic to a ball (although the prototiles are), see Fig. 3 in [Fre02] . Frequently, $R$-patches are used. The $R$-patch around $x$ in a tiling $T$ is the set $P_R(x) = \{ T \in T | T \textrm{ has nonempty intersection with the closed ball of radius } R \textrm{ around } x \}$.

### References

[Fre02]
Frettlöh, D
Nichtperiodische Pflasterungen mit ganzzahligem Inflationsfaktor
Univ. Dortmund 2002,