Fractal Tilings Katie Moe and Andrea Brown December 13, 2006 - PowerPoint PPT Presentation

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 Contents Introduction Examples of Fractal Tilings Example 1 Example 2 Creating the Tilings Short Summary of Important Ideas Example 3 Tiles with Radial Symmetry Example 4 Example 5 Similarity Maps Example 6-Case I Example 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 individual tiles called fractiles whose boundaries are fractal curves. Fractal curves are objects or quantities that display self-similarity, in a somewhat technical sense, on all scales. This means that it looks the same at any scale. We will use an iterative process, involving repeated compositions of two or more 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 � a � − b • Start with a matrix M = where a and b are b a chosen so that a 2 + b 2 > 1. � x 1 � � a � • We must understand that and are points in the x 2 b � x 1 � � ax 1 − bx 2 � complex plane and M = represents the ax 1 + bx 2 x 2 complex multiplication of x 1 + ix 2 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 translate the copies of the fractile so that they are positioned correctly in the tiling. • We will define the set ξ = { r j } and the vectors in this set have integer coordinates that lie in or on S but not on the two outer edges that don’t have the origin as a vertex. ξ has exactly m vectors. � 1 � • The unit square that is determined by the vectors and 0 � 0 � is mapped onto the square S with area m = a 2 + b 2 1 � a � � − b � and is spanned by the vectors v 1 = and v 2 = . b a

Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations Example 1 � 1 � − 1 • Let M = then m = 2. − 1 1 • We can determine that the two translation vectors are � 0 � � 1 � r 1 = and r 2 = 0 0 (1 , 1) r 1 r 2 (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 = { r 1 , r 2 } . • For z = ( x 1 , x 2 ) , where z is our initial point of translation, we can define our mappings as f j ( z ) := r j + M − 1 ( z ) for j = 1 , 2 . That is, � x 1 � � 0 � � . 5 − . 5 � � x 1 � f 1 := �→ + x 2 0 . 5 . 5 x 2 � x 1 � � 1 � � . 5 � � x 1 � − . 5 f 2 := �→ + x 2 0 . 5 . 5 x 2

Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations The collections of functions { f j } is called an iterated function system. To initiate this process an initial point z o is randomly selected in the plane and is used to evaluate f 1 ( z o ) and f 2 ( z o ) . For n ≥ 1, we make sure to choose recursively and randomly so that z n ǫ { f 1 ( z n − 1 ) , f 2 ( z n − 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, but thousands of iterations will be needed to generate the desired tiling. The result of the iterated function system for this example can 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 2 � 1 � � 0 � � 1 � � 2 � 2 If we have M = and r 1 = , r 2 = , and r 3 = . − 1 1 0 0 0 (2 , 1) r 1 r 2 r 3 (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 invertible integer matrix that is an expansive map, i.e. all eigenvalues have modulus larger than 1. � a � b • The matrix we will choose will be M = . c d • The translation vectors are chosen with the following process. For a matrix M as above, | det ( M ) | = | ad − bc | = m is the area of parallelogram P � a � � b � spanned by the two vectors v 1 = and v 2 = . c d • These vectors are called principal residue vectors. The vectors in { r j } form a complete residue system for M .

Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations � 0 � • Generally, as long as y 1 = r 1 = and y j ≈ r j for 0 j = 2 , ... m , then the collection of vectors { y j } will also form a complete residue system for matrix M . • The location of the residue vectors determines the locations of the fractiles but the shape of the tilings may change drastically with the different choices of residue systems.

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 • { y 1 , ... y m } is a complete residue system for M • f j ( z ) := r j + M − 1 ( z ) . • The attractor set A = ∪ j = 1 m A j is the tiling of m tiles A j . 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 3 � 2 − 1 � Let M = ; then m = 5. Here the principal residue 1 2 � 0 � � 0 � � 1 � � 0 � � 1 � vectors are r 1 = , r 2 = , r 3 = , r 4 = , and r 5 = 0 1 1 2 2 ( − 1 , 2) (2 , 1) r 2 r 3 y 3 r 1 y 4 r 4 y 5 Figure: Residue Vectors.

Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations Example 3 For a more symmetric tiling, we choose the following equivalent residue vectors for our residue system out of the collection { y j } . Our next tiling is created by using y 1 = r 1 , y 2 = r 2 , y 3 = � 0 � − 1 � � 1 � � ≈ r 3 , y 4 = ≈ r 4 , and y 5 = ≈ r 5 . The vectors 0 0 − 1 { y 1 , y 2 , y 3 , y 4 , y 5 } are symmetric about r 1 . 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 that has radial symmetry. In order to have radial symmetry we need a change of base matrix ( B ).

Introduction Examples of Fractal Tilings Creating the Tilings Tiles with Radial Symmetry Similarity Maps Variations Example 4 � 2 � � 1 − 1 / 2 � − 2 √ Let M = and B = . New residue vectors 2 0 0 3 / 2 � 0 � � 1 � � − 1 � � 0 � B y 1 = B y 2 = B y 3 = B y 4 = 0 0 − 1 1 are formed by the equation f j ( z ) = B y j + h − 1 ( z ) where h = BMB − 1 . (2 , 2) r 4 r 3 ( − 2 , 0) r 2 r 1 y 2 y 3

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 � 1 − 2 � � 1 1 / 2 � √ M = B = 2 3 0 − 3 / 2 » – » – » – » – » – » – » – 0 0 − 1 − 1 0 1 1 B y 1 = B y 2 = B y 3 = B y 4 = B y 5 = B y 6 = B y 7 = 0 1 1 0 − 1 − 1 0 r 5 ( − 2 , 3) r 7 r 3 (1 , 2) r 6 r 4 y 3 r 2 y 4 r 1 y 7 y 5 y 6

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 format f j ( z ) = B y j + 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 � 2 � � 1 1 / 2 � 2 √ M = B = − 2 1 0 − 2 / 2 � 2 � 1 � 3 � 0 � � 1 � � 2 � � � � B y 1 = B y 2 = B y 3 = B y 4 = B y 5 = B y 6 = 0 0 0 − 1 − 1 − 1 Figure: Similarity Tiling.