Relatives of the Sierpinski Gasket

One of the many ways to generate the Sierpinski gasket is to begin with a “seed” such as an L-shape; make three copies of the seed and arrange them in an L-shape, and repeat the process an infinite (or at least pleasingly large) number of times.

Four iterations toward the Sierpinski gasket:

For other interesting variations, each of the three square “blocks” in the L-shape can be rotated or flipped, possibly with a different transformation for each square. The eight possible transformations of the seed are:

(In the Mathematica code I call the transformations Identity, MapReverseTranspose, ReverseMapReverse, ReverseTranspose, TransposeReverseMapReverse, Reverse, Transpose, MapReverse, not that you asked.)

To generate a picture, choose a transformation for the NW, SW, and SE quarters of the L-shape, and iterate the transformations. Two randomly selected pictures:

Taking every combination of transformations, removing duplicates, we find these 456 Sierpinski relatives:

See Peitgen, Jürgens, and Saupe, Chaos and Fractals: New Frontiers of Science, Springer-Verlag, 1992; pp. 244–251.

Designed and rendered with Mathematica.

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