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 `$P_R(x) = \{ T \in T | T \textrm{ has nonempty intersection with the closed ball of radius } R \textrm{ around } x \}$`

.

[Fre02]

FrettlĂ¶h, D

**Nichtperiodische Pflasterungen mit ganzzahligem Inflationsfaktor**

*Univ. Dortmund*
2002,