Self-replicant fractal growth patterns

Fractal growth of a pentagonal annulus. Underlying image from Bowers and Stephenson, "A 'regular' pentagonal tiling of the plane."

Same image as above with some of the un-needed boundaries erased.

Peripheral and non-simple fractal growth.

By self-replicant fractal growth I mean growth that incorporates at each molt or growth stage an increment that makes the shape a larger version of its earlier self. It is easier to think about fractal growth time-reversed as a repeated subdivision of a large tile into smaller tiles. The algorithm is simple: subdivide the large tile in a way that leaves a sub-tile that is geometrically similar to the large one; reapply the subdivision to the similar tile; repeat step two. (To completely specify the subdivision procedure it may be necessary to mark a point on the boundary of the large tile, and the corresponding point on the boundary of the similar tile.)

These patterns are either central or peripheral according to whether the intersection of the boundaries of the large tile and the similar tile is empty or not. The peripheral patterns can be simple or non-simple according to whether or not the boundaries intersection is connected. The pentagonal example above is peripheral and non-simple.

Peripheral simple fractal growth of a triangle.

Peripheral simple fractal growth of a triangle.

Peripheral non-simple fractal growth of a triangle.

Central fractal growth of a triangle.