Cartwheels, Part 4

Theorem:  Every patch of tiles in a tiling by kites and darts is congruent to infinitely many patches in any tiling by kites and darts.

Proof: Let Tⁿ  designate the nth composition of tiling T.  For a given tiling T, pick n large enough so that the minimal distance between two vertices of Tⁿ  exceeds the diameter of the given patch P. Then P will contain at most one vertex V of Tⁿ  . If P does not contain a vertex, extend P until it does.  Now the tiles that meet at V contain P.  Consider any other tiling S.  Let V₁  be any vertex of Sⁿ  such that the vertex type of V₁  is the same as that of V.  Then decomposing V₁  n times will produce a patch identical to the patch at V when it is decomposed n times.  This will give a patch in S that is congruent to the original patch in T.  Since  there are infinitely many vertices of each type, this proves the theorem.

Conway  has claimed that the distance from one patch to an identical patch in a tiling is always less than 0.5+G, where G is the golden ratio.

A consequence of this theorem is that a look at finite pieces of two tilings is not enough to determine whether they are identical.  Contrast this with tiling by regular hexagons.

Empires   Aperiodic Tiling