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Michael Mampusti
Software to Produce Fractal Tilings of the Plane
University of Wollongong
- Dr. Samuel Webster and Dr. Michael Whittaker
Software to Produce Fractal Tilings of the Plane University of - - PowerPoint PPT Presentation
Software to Produce Fractal Tilings of the Plane University of - - PowerPoint PPT Presentation
Michael Mampusti Software to Produce Fractal Tilings of the Plane University of Wollongong Dr. Samuel Webster and Dr. Michael Whittaker Why make Fractal Tilings? they look so awesome! aperiodic tilings are hard to construct. We can build
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Why make Fractal Tilings?
◮ they look so awesome! ◮ aperiodic tilings are hard to construct. We can build an
infinity of substitution tilings from aperiodic tilings.
◮ ...
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Why make Fractal Tilings?
◮ they look so awesome! ◮ aperiodic tilings are hard to construct. We can build an
infinity of substitution tilings from aperiodic tilings.
◮ ... ◮ FRACTALZ!!!
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Tilings of the Plane
◮ a cover of R2 with shapes, called tiles.
Example (Tiled bathroom floor)
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More formally, a tiling of the plane is a set of compact subsets of R2, T := {ti : i ∈ N} such that
- i∈N
ti = R2, and int(ti) ∩ int(tj) = ∅ for all i = j. Each ti is called a tile.
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We only care about tilings which:
◮ are made up of finitely many tile types (up to rigid motions),
called prototiles; and
◮ are aperiodic (there exists no translation of the tiling which
matches up with the original tiling).
Example (Tiled bathroom floor)
This is not an aperiodic tiling since translation by length of the tile gives you back the same tiling.
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Substitution Tilings
◮ Defined using substitution rules on the set of prototiles:
◮ expand, tile and contract.
Example (Tiled bathroom floor)
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A nice example of a substitution tiling is the Penrose tiling. It consists of 4 prototiles. The substitution rules are as follows:
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This is an aperiodic tiling of the plane.
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Programming in Mathematica
◮ write prototiles (p1, . . . , pn) as a vector. ◮ developed a special matrix which we applied to the vector.
Special Matrix (The Tile Breaker)
◮ ijth entry contains all the rotations and translations of pj in
substitution of pi.
◮ application of The Tile Breaker breaks up the prototiles into
parts according to the substitution rules.
◮ discovering The Tile Breaker and implementing it in
Mathematica was non-trivial.
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Fractals
Contraction Mappings
◮ a map f : R2 → R2 which brings points closer together
Iterated Function Systems (IFSs)
◮ let f1, . . . , fn be contractions on R2. ◮ R2 with these contractions is an iterated function system.
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Consider H(R2), the space of compact subsets of R2.
Theorem ([1], Theorem 7.1)
Given an IFS (R2 with f1, . . . , fn contractions), the function f : H(R2) → H(R2) defined by f (B) =
n
- i=1
fi(B), is a contraction on H(R2). There exists a unique fixed point A ∈ H(R2) of f called the attractor. It satisfies A = lim
n→∞ f n(B), for any B ∈ H(R2).
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Examples (Koch Curve and Sierpinski Gasket)
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Fractal Tilings
◮ tilings of the plane. ◮ tiles have fractal edges; fractiles. ◮ we build them from existing substitution tilings.
Example (Half-Hex Tiling)
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◮ inscribe dual graph in each prototile.
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◮ take a few substitutions of original tiling with graphs inscribed
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◮ choose edges in substitution which resembles original graph
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◮ iterate edges infinitely. →
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◮ inscribe final fractal edges into original prototiles.
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◮ tile plane with fractal edges inscribed.
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Programming in Mathematica
◮ take The Tile Breaker from substitution tiling and build a
special edge matrix, called The Fractalix.
◮ apply it to the vector of edges similar to the prototiles. ◮ choosing edges deletes entries in this special edge matrix. ◮ each entry in the matrix is not only a rotation and translation,
it is also a contraction. In this way, we have an (generalized) IFS with the fractal edges as the attractors!
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- 5
5 10 5 5 10 , 2 1 1 2 3 3 2 1 1 2 3 , 0.5 0.5 1.0 1.5 0.5 0.5 1.0 1.5 , 0.2 0.4 0.6 0.8 1.0 1.2 0.2 0.2 0.4 0.6 0.8 1.0 1.2 , 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 , 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 , 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
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Here is a fractal tiling built from the Penrose substitution.
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Octagonal Tiling
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Here is a fractal tiling built from the Octagonal substitution.
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Equithirds Tiling
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Here is a fractal tiling built from the Equithirds substitution.
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Further directions I plan to explore:
- 1. use Mathematica to extract the fractiles and substitution;
- 2. put code on the Wolfram website for public use;
- 3. update the Tiling Encylopedia to include some of the fractal
tilings we made;
- 4. look at other types of tilings; and
- 5. most importantly, write up work for publication.
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- M. F. Barnsley, Fractals Everywhere, Second edition,
Academic Press Professional, Boston, 1993.
- N. P. Frank and M. F. Whittaker, A Fractal Version of the
Pinwheel Tiling, Math. Intellig. 33:2 (2011), 7-17.
- L. Sadun, Topology of tiling spaces, AMS University Lecture