Scaling Factor

Each substitution tiling scales the object by a certain factor. This page demonstrates the scaling factor for a few of the designs.

Pattern Blocks

Many of the fractal designs on this site have the same scaling factor. The scaling factor is determined by the length of the side of the first substitution tiling, i.e. the length of the side of D₁, T₁, or Y₁.

In these larger objects the length of the side is the length of the long diagonal of the tan diamond, which is, or roughly 1.9.

All dimensions scale by this amount. For the tan diamond, the length of the short diagonal is, the reciprocal of the length of the long diagonal. In the larger tan diamond the short diagonal is scaled to length 1.

If these objects are grown again, the object is scaled by the same amount.

You can see that these objects D₂, T₂ and Y₂ have side length. And in fact.

Octagon Blocks

In the octagonal system, the scaling factor the length of the long diagonal in the diamond.

The diamond has long diagonal of length, the scaling factor. The short diagonal has length.

Back to Mathematics behind the Designs.