fractal tilings - college of the redwoods

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


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

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.


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.


• 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

].


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.


• 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

]


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)}.


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.


Example 2If we have M =

[1 2−1 1

]and r1 =

[00

], r2 =

[10

], and r3 =

[20

].

r1 r2 r3

(2, 1)

(1,−1)



The tiling produced will be three tiles stacked horizontally.

Figure: Horizontal Tiling.


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.


• 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.


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.


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)



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.



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).


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: Horizontal Tiling.


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)



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.


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.


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.


Fractal are fun! (and pretty)

fractal tilings - college of the redwoods

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