## Repetitive

A tiling $T$ is called repetitive, if for every $r>0$ there is $R>0$, such that a copy of every $r$-patch in $T$ is contained in every $R$-patch in $T$. In plain words, this means that each local part of the tiling occurs ‘everywhere’ in the tiling. In even plainer words: If you stand on a repetitive tiling, then your local surrounding do not tell you the in the slightest way where you are, even if you have a map of the whole tiling:

If $R$ depends on $r$ linearly, say, $R=ar+b$ for some $a,b>0$, then $T$ is called linearly repetitive. A weaker version is the following, which we prefer to call weakly repetitive: For every patch $P$ in $T$, there is $R>0$ such that every $R$-patch in $T$ contains a copy of $P$. Unfortunately, the latter is sometimes referred to as ‘repetitive’ in the literature, sometimes as ‘weakly repetitive’. Every repetitive tiling that is locally finite is of flc. Every primitive substitution tiling is weakly repetitive. For each substitution tiling $T$ holds: $T$ flc $\Leftrightarrow$ $T$ repetitive $\Leftrightarrow$ $T$ linearly repetitive.