Kenyon  gives an explicit construction for self-similar tilings based on complex Perron numbers
are natural numbers. His method creates a closed polygonal path that can be filled by parallelograms (prototiles) with edges related to the vectors
. The iteration of this method converges to a fractal boundary.
This Demonstration shows the tiling of such boundaries by parallelograms for fixed
and five chosen combinations of
. The iterated substitution is extended so that prototiles at every iteration level can be merged into super-tiles that also tile the boundary of higher iteration levels.
: Use this popup menu to select the set of polynomial coefficients.
depth: Use this slider to set the iteration. At
only prototiles are shown; higher levels approximate the boundary fractal by substitution of prototiles.
super: Use this slider to select the iteration level where the prototiles are merged to super-tiles. At
the basic tiling by parallelograms is shown.
tile: Use this popup menu to select one of the three tile types; at higher levels, the tiles may become similar.
Approximations of well-known fractals are achieved by setting the parameters to:
→ Kenyon 121 fractal ; this is the example given in .
→ Rauzy fractal.
→ ammonite (curly) fractal.
 R. Kenyon, The Construction of Self-Similar Tilings," Geometric & Functional Analysis
(3), 1996 pp. 471–488. doi:10.1007/BF02249260