• 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

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