Two tilings are called mld (mutually locally derivable), if one is obtained from the other in a unique way by local rules, and vice versa. For example, a tiling by Penrose Rhomb is obtained from a Robinson Triangle tiling easily: just delete the shortest and longest edges, keeping only the medium ones yields the Penrose Rhomb tiling; and vice versa: in a Penrose Rhomb tiling, add in each fat rhomb the long diagonal, and in each thin rhomb add the short diagonal. This gives again the Robinson Triangle tiling.
The proper definition is as follows [BSJ91]
: A tiling $T$ is locally derivable from a tiling $S$ (with radius $r$), if for all $x$ holds: If $S$ intersected with $B_r(x) = (S \textrm{ intersected with } B_r(y)) + (x-y)$, then $T$ intersected with ${x} = (T \textrm{ intersected with } {y}) + (x-y)$. Here, $B_r(x)$ denotes the open ball around $x$ of radius $r$. ‘$T$ intersected with $M$’ is to be understood as the set of all tiles in $T$ which have nonempty intersection with M.
$S$ and $T$ are mld, if $S$ is locally derivable from $T$ and vice versa.
It is easy to see that mld defines an equivalence relation. Sometimes it is more convenient to deal with mld-classes than with individual tilings.
[BSJ91]
Baake, M and Schlottmann, M and Jarvis, P D
Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability
J. Phys. A
1991,
19,
pp. 4637-4654,
MR1132337