fractal tilings - college of the redwoods
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Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Fractal Tilings
Katie Moe and Andrea Brown
December 13, 2006
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Table of ContentsIntroduction
Examples of Fractal TilingsExample 1Example 2
Creating the TilingsShort Summary of Important IdeasExample 3
Tiles with Radial SymmetryExample 4Example 5
Similarity MapsExample 6-Case IExample 7-Case II
Variations
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Introduction
In this presentation we will be generating tilings with individualtiles called fractiles whose boundaries are fractal curves.Fractal curves are objects or quantities that displayself-similarity, in a somewhat technical sense, on all scales.This means that it looks the same at any scale. We will use aniterative process, involving repeated compositions of two ormore functions and those, in turn, will generate the fractal tiling.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Examples of Fractal Tilings
• Start with a matrix M =
[a −bb a
]where a and b are
chosen so that a2 + b2 > 1.
• We must understand that[x1x2
]and
[ab
]are points in the
complex plane and M[x1x2
]=
[ax1 − bx2ax1 + bx2
]represents the
complex multiplication of x1 + ix2 by a + ib.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
• Next, we must find a collection of vectors that will translatethe copies of the fractile so that they are positionedcorrectly in the tiling.
• We will define the set ξ = {rj} and the vectors in this sethave integer coordinates that lie in or on S but not on thetwo outer edges that don’t have the origin as a vertex. ξhas exactly m vectors.
• The unit square that is determined by the vectors[10
]and[
01
]is mapped onto the square S with area m = a2 + b2
and is spanned by the vectors v1 =
[ab
]and v2 =
[−ba
].
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Example 1
• Let M =
[1 −1−1 1
]then m = 2.
• We can determine that the two translation vectors are
r1 =
[00
]and r2 =
[10
]
r1 r2
(1, 1)
(1,−1)
Figure: Finding Equivalent Residue Vectors.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
• Now we have ξ = {r1, r2}.• For z = (x1, x2), where z is our initial point of translation,
we can define our mappings as fj(z) := rj + M−1(z) forj = 1, 2. That is,
f1 :=
[x1x2
]7→
[00
]+
[.5 −.5.5 .5
] [x1x2
]f2 :=
[x1x2
]7→
[10
]+
[.5 −.5.5 .5
] [x1x2
]
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
The collections of functions {fj} is called an iterated functionsystem. To initiate this process an initial point zo is randomlyselected in the plane and is used to evaluate f1(zo) and f2(zo).For n ≥ 1, we make sure to choose recursively and randomlyso that znε{f1(zn−1), f2(zn−1)}.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Points will be lying near the tiling after a few iterations, butthousands of iterations will be needed to generate the desiredtiling. The result of the iterated function system for this examplecan be seen in the following Figure.
Figure: Residue Vectors.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Example 2If we have M =
[1 2−1 1
]and r1 =
[00
], r2 =
[10
], and r3 =
[20
].
r1 r2 r3
(2, 1)
(1,−1)
Figure: Residue Vectors.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
The tiling produced will be three tiles stacked horizontally.
Figure: Horizontal Tiling.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Creating the Tilings
• To generate a tiling we need a matrix to be an invertibleinteger matrix that is an expansive map, i.e. all eigenvalueshave modulus larger than 1.
• The matrix we will choose will be M =
[a bc d
].
• The translation vectors are chosen with the followingprocess. For a matrix M as above,|det(M)| = |ad − bc| = m is the area of parallelogram P
spanned by the two vectors v1 =
[ac
]and v2 =
[bd
].
• These vectors are called principal residue vectors. Thevectors in {rj} form a complete residue system for M.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
• Generally, as long as y1 = r1 =
[00
]and yj ≈ rj for
j = 2, ...m, then the collection of vectors {yj} will also forma complete residue system for matrix M.
• The location of the residue vectors determines thelocations of the fractiles but the shape of the tilings maychange drastically with the different choices of residuesystems.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Short Summary of Important Ideas
• M represents an expansive map• {y1, ...ym} is a complete residue system for M• fj(z) := rj + M−1(z).
• The attractor set A = ∪j=1m Aj is the tiling of m tiles Aj .
These tiles are now called m-rep tiles.
These ideas will now be used to create a tiling of m-rep tiles.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Example 3Let M =
[2 −11 2
]; then m = 5. Here the principal residue
vectors are r1 =
[00
], r2 =
[01
], r3 =
[11
], r4 =
[02
], and r5 =
[12
]
r1
r2 r3
r4y3 y4
y5
(2, 1)
(−1, 2)
Figure: Residue Vectors.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Example 3For a more symmetric tiling, we choose the following equivalentresidue vectors for our residue system out of the collection {yj}.Our next tiling is created by using y1 = r1, y2 = r2, y3 =[−10
]≈ r3, y4 =
[10
]≈ r4, and y5 =
[0−1
]≈ r5. The vectors
{y1, y2, y3, y4, y5} are symmetric about r1.
Figure: Residue Vectors.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Tiles with Radial Symmetry
When m = 2, 3, 4, 5, and 7, we are able to create a tiling thathas radial symmetry.In order to have radial symmetry we need a change of basematrix (B).
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Example 4Let M =
[2 −22 0
]and B =
[1 −1/20
√3/2
]. New residue vectors
By1 =
[00
]By2 =
[10
]By3 =
[−1−1
]By4 =
[01
]are formed by the equation
fj(z) = Byj + h−1(z)
where h = BMB−1.
r1r2
r3r4
y2
y3
(2, 2)
(−2, 0)
Figure: Residue Vectors.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Figure: Horizontal Tiling.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Example 5
M =
[1 −22 3
]B =
[1 1/20 −
√3/2
]By1 =
»00
–By2 =
»01
–By3 =
»−11
–By4 =
»−10
–By5 =
»0−1
–By6 =
»1−1
–By7 =
»10
–
r1
r2
r3
r4
r5
r6
r7
y3
y4
y5 y6
y7
(1, 2)
(−2, 3)
Figure: Residue Vectors.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Figure: Residue Vectors.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Similarity Maps
There are two cases when you are developing similarity maps:• M has two real eigenvalues with independent eigenvectors• M has a pair of complex conjugate eigenvalues
The formatfj(z) = Byj + h−1(z)
where h = BMB−1 and B−1 is the eigenvectors is used.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Example 6
M =
[2 21 −2
]B =
[1 1/20 −
√2/2
]By1 =
[00
]By2 =
[10
]By3 =
[20
]By4 =
[2−1
]By5 =
[1−1
]By6 =
[3−1
]
Figure: Similarity Tiling.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Example 7
M =
[1 −11 2
]B−1 =
[1 1
(√
6− 2)/2 −(√
6 + 2)/2
]By1 =
[00
]By2 =
[01
]By3 =
[10
]
Figure: Similarity Tiling.
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations
Fractal are fun! (and pretty)