• 25 dec 2009
Penrose-like Tilings

# Original triangles

The originals are isosceles triangles with all standing sides equal. The equal angles are 36° and 72° respectively.
If the standing sides are equal to 1, then the basis is equal to 1.61803399 = tau=. -:>:%:5  and 0.618034 = tau - 1 .

low triangle                                               high triangle

# Next generation

This process is similar to the deflation of Penrose. First all triangles are multiplied by tau and then divided as follows.
the low triangle is divided in 2 low and 1 high triangle                the high triangle becomes 1 high and 1 low triangle.

For each generation this multiplication and division is repeated.

# Aperiodicity

If we start with a low triangle, and we administer the number of high and low triangles, we get

 generation # high # low graph 0 0 1 1 1 2 2 3 5 3 8 13

It is trivial to see that in each next generation, the next 2 fibonnacci numbers are generated.
The ratio between these two numbers tend to tau, so the ultimate tiling of the plane is aperiodic since this ratio is not rational.

# Result of tiling the plane

RE Boss/J-blog/PenroselikeTiling (last edited 2010-01-24 18:06:54 by RE Boss)