Equality in tiles

Equality in tiles is a property frequently used, in particular when transformations are applied into tiles. Transformations such as translates and copies. Before to continue our discussion, we needs to mention that all patches have a set of triangles, in which there are all necessary copies of triangles. We call $C(M)$ the set that contains all triangles (even if there exists more than one copy of some triangle).

Definition 4   Let $M_1$ and $M_2$ two tiles in $lca(2,1)_{110}$ . We say that $M_1$ is graphically-equal to $M_2$ , typed $M_1\eqcirc M_2$ , when:

1. Sets that contain tiles of $M_1$ and $M_2$ , are equals. $C(M_1)=C(M_2)$
2. For each triangle $t_n\in M_1$ there exists another triangle $t_n\in M_2$ that preserves the same relative location to the others elements of its tile.