Kenyon Tiling Construction (Boundary)

Use this Demonstration to create approximations of fractal boundaries using Kenyon's construction of self-similar tilings.

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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 boundary constructions for polynomials using various parametrized combinations of , , and . Kenyon's construction still achieves suitable tilings with relaxed restrictions on , , , so far as the polynomial exhibits complex roots and the constructed boundary avoids self-intersection. His method also works for negative ; noninteger numbers or zero values may be allowed for , and . An exemplary noninteger variation of is introduced by using the "" slider. Kenyon's construction produces backtracking paths. These are not part of the final tiling boundary and thus must be canceled. In this code, the cancellation can be switched off.
More parameter combinations could be used, but the results would look more or less like distorted versions of the given examples.
The basic routine for the boundary calculation is based on a program by Bagula [2].
Controls
iteration depth: Use this slider to set the iteration. At the prototiles are shown; higher levels approximate the boundary fractal.
: Use this popup menu to modify the polynomial exponents.
: Use this popup menu to select the set of the polynomial coefficients.
: Use this slider to modify the coefficient by .
tile: Use this popup menu to select one of the three tile types; at higher levels, the tiles may become similar.
cancel: Use this checkbox to toggle the display of the backtracking paths.
fill: Use this checkbox to toggle the display of the boundary polygon as filled or empty.
Approximations of well-known fractals are achieved by setting the parameter to:
, , → Kenyon 121 fractal [3]; this is the example given in [1].
, , → Rauzy fractal.
, {, → ammonite (curly) fractal.
The corresponding substitution tilings filling these boundaries are part of the Demonstration Kenyon Tiling Construction (Substitution) 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] R. Bagula. "Perron Number Tiling Systems" from the Wolfram Library Archive—A Wolfram Web Resource. library.wolfram.com/infocenter/MathSource/5642.
[3] Tilings Encyclopedia. "Richard Kenyon: Discovered Tilings." (Feb 28, 2018) tilings.math.uni-bielefeld.de/person/richard-kenyon.
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