Kenyon [1] gives an explicit construction for self-similar tilings based on complex Perron numbers

that satisfy

where

,

and

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

,

and

. 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 [2]; this is the example given in [1].

,

→ Rauzy fractal.

{

,

→ ammonite (curly) fractal.

[1] R. Kenyon, The Construction of Self-Similar Tilings,"

*Geometric & Functional Analysis*,

**6**(3), 1996 pp. 471–488.

doi:10.1007/BF02249260.