Theorem: Every sequence of 0's and 1's that does not contain the string 11 is the index sequence for some point in a tiling by kites and darts.
The proof is essentially constructive. Suppose, for example, that you want a tiling that starts 0001000. To get the zeros, enlarge the triangle. To get the one, do nothing.
Theorem: The index sequences of two points in the same kite and dart tiling differ only for a finite number of terms in the beginning.
Theorem. If two points have identical index sequences, then the points lie in congruent tilings (or if in the same tiling, then there is a symmetry of the tiling that takes the tile of one point to the tile of the other).
Theorem: There are uncountably many distinct tilings by kites and darts.
Further, if the index sequences for two points agree for some finite number of terms, then the tiles live in congruent patches. The larger number of identical terms, the larger the patch.