If you
bisect the kites and darts in a tiling, you get a tiling by two types of
triangles. The kite becomes two "golden triangles", and the dart
becomes two "golden gnomons".If you
bisect the kites and darts in a tiling, you get a tiling by two types of
triangles. The kite becomes two "golden triangles", and the dart
becomes two "golden gnomons".
If you start with these triangles, matching conditions have to be enforced in if you want to be sure that an aperiodic tiling results. If a tiling by kites and darts is composed, inflated, deflated or decomposed, then again you get a correspondingly altered triangle tiling. We use this to show the following:
Theorem: If the plane is tiled by kites and darts, the ratio of numbers of kites to darts is the golden ratio.
Let 
T designate the tile
produced if its sides are multiplied by a factor of .
It then turns out that two standard compositions of kites and darts yields the
result pictured above.
Now note that LA
= 2LA + SA and SA
= LA + SA. By this we mean that the
double-composed tiles can be decomposed into tiles of the given type and
number. If the process is continued another step, 2LA
= 5LA + 3SA and 2SA
= 3LA + 2SA. . A little induction will show
that nLA = F2n+1LA
+ F2nSA and nSA
= F2nLA + F2n-1SA , where Fn
is the nth Fibonacci number. The result follows from the fact that the
limit as n goes to infinity of Fn+1 /Fn is .
If a tiling by kites and darts were periodic, then there would
have to be a rational ratio of the tiles. But since
is irrational, a consequence of the above is:
Corollary: The kite and dart form an aperiodic set.
This fact can also be proved by other methods. What about this image?