Ideas for other Fractal Designs

Here are some ideas for other fractal designs.

Other Substitution Tilings

45-degree pieces

45-degree design using 30-degree pieces

45-degree design using squares

Infinite set of 45-degree pieces



Ammann tiling with 45-degree pieces

Ammann tiling using 30-degree pieces

Ammann tiling using square pieces.

36-degree pieces


Penrose tiling uses 36-degree pieces


30-degree pieces

30-degree tiling using 20-degree pieces

30-degree tiling using triangles.

Infinite set of 30-degree tiles



Another tiling using 30-degree pieces

Other 30-degree tiling using 20 degree pieces

Other 30-degree tiling using triangles

20-degree pieces



Square-triangle tiling




Generalized Ammann tiling

Generalized Penrose tiling

Other transformations

There are other transformations besides the ones I listed. Many have the common property that the scaling size of the substitution tiling is the same for all the transformations. Many use the Y-shaped star.

Other transformations are possible that scale by a larger scaling factor. There are many possible other scaling factors. The following diagram shows some possible sizes of diamonds in the substitution tiling.

The first row shows the basic diamond. The second row shows the bigger diamond D1 used in many of the substitution tilings on this site. The third and fourth diamonds are slightly larger. A problem with using these as substitution tilings is that the sides are not symmetric, i.e. there are ways to put two sides together and not have the pieces match. The fifth and sixth diamonds solve that problem. In fact, the fifth diamond has size of D2 in many of the tilings on this site. The sixth diamond is the diamond D1 used in the 12 pointed star fractal. Besides these 6 diamonds there are, of course, larger diamonds.

Larger scaling factors may allow more than one possible tiling of a piece. Using multiple possible tilings in a pattern would create some new designs.

Other Ideas for Fractal Designs

You don't need a complete substitution tiling for a fractal. Since the fractal has holes, fractals don't need to be based on complete substitution tilings.

One design used a different piece (the X), but those pieces cannot be used for a substitution tiling. Other pieces could be invented compatible with the pattern block pieces that provide other fractals.

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Copyright 1998-2004 by Jim Millar