Kenyon Tiling Construction (Substitution)

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This Demonstration shows how to create approximations to fractals by tile substitutions, using Kenyon's construction with self-similar tilings.

Contributed by: Dieter Steemann (March 2018)
Open content licensed under CC BY-NC-SA


Snapshots


Details

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.

Controls

: 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.

The corresponding generation of tiling boundaries is part of the Demonstration Kenyon Tiling Construction (Boundary) in the Related Links.

References

[1] R. Kenyon, The Construction of Self-Similar Tilings," Geometric & Functional Analysis, 6(3), 1996 pp. 471–488. doi:10.1007/BF02249260.

[2] Tilings Encyclopedia. "Richard Kenyon: Discovered Tilings." (Feb 28, 2018) tilings.math.uni-bielefeld.de/person/richard-kenyon.



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