Theorem:
Except for 7 tiles at the center, every tile in a cartwheel tiling is contained
in a patch of tiles that has symmetry group D5 .Theorem:
Except for 7 tiles at the center, every tile in a cartwheel tiling is contained
in a patch of tiles that has symmetry group D5 .
Proof: The is easily seen to be true for C4 . If this tiling is inflated, a patch of tiles with symmetry D5 is inflated into a patch with symmetry D5 . The result follows by induction.
Theorem: In any tiling with kites and darts, every tile lies in a cartwheel of order n for every positive integer n.
Proof: Every point in the tiling is in a tile that is part of an ace. Now compose the tiling 2n times. The given point is part of an ace in the composed tiling. If we now decompose, the ace becomes a cartwheel, the original tiling is recovered, and the point is part of a cartwheel which is discovered by this decomposition.