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1 Fractal Dust and Schottky Dancing Fractal Dust and nSchottky Dancing University of Utah GSAC Colloquium 10.10.06 Josh Thompson

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2 Geometric patterns have played many roles in history: ● Science ● Art ● Religious ● The symmetry we see is a result of underlying mathematical structure

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3 Symmetry ● Translation symmetry: invariance under a shift by some fixed length in a given direction. ● Rotational symmetry: invariance under a rotation about some point. ● Reflection symmetry: (mirror symmetry) invariance under flipping about a line ● Glide Reflection: translation composed with a reflection through the line of translation. Rigid Motions: transformations of the plane which preserve (Euclidean) distance.

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4 Symmetry Abounds

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8 How to Distinguish Transformations How to Distinguish Transformations ( look for what's left unchanged ) ● Translation – one point at infinity is fixed ● Rotation – one point (the center) in the interior fixed ● Reflection – a line of fixed points (lines perpendicular to the reflecting line are invariant) ● Glide Reflection – a line is invariant, no finite points fixed Note: The last two reverse orientation.

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9 Rigid Motions of the Plane ● Have form T(z) = az + b with a,b real, z complex ● Collection of transformations which preserve a pattern forms a group under composition. ● For example, the wallpaper shown before has a nice symmetry group:

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10 Mobius Transformations Mobius Transformations ( angle preserving maps ) They all have a certain algebraic form and the law of composition is equivalent to matrix multiplication. Mobius transformations can be thought of in many ways, one being the transformations that map {lines,circles} to {lines,circles}

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11 Kleinian Groups Mobius transformations are 'chaotic' or discrete A Kleinian group is a discrete group of Mobius transformations.

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12 Three types of Mobius Tranformations (Distinguished by the nature of the fixed points) Parabolic Only one fixed point. All circles through that fixed point and tangent to a specific direction are invariant. Conjugate to translation f(z) = z+1 Hyperbolic Two fixed points, one attracting one repelling. Conjugate to multiplication (expansion) f(z) = az, with |a| > 1. Elliptic Two fixed points, both neutral. Conjugate to a rotation.

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13 Four Circles Tangent In A Chain

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14 The four tangent points lie on a circle. Conjugate by a Mobius transformation so that one of the tangent points goes to infinity. The circles tangent there are mapped to parallel lines. The other three tangent points all lie on a straight line by Euclidean geometry, which goes through infinity the fourth tangent point.

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15 Proof By Picture

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16 Extend the Circle Chain Given one Mobius transformation that takes C 1 to C 4, (and C 2 to C 3 ) there is a unique second Mobius transformation taking C 1 to C 2, (and C 3 to C 4 ) and the two transformations commute.

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17 Starting Arrangement of Four Circles and Images

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18 The Action of the Group

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19 The Orbit

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20 Letting Two Mobius Transformations Play Allowing two Mobius transformations a(z), b(z) to interact can produce many Klienian groups. In general, the group G = generated by aand b is likely to be freely generated – no relations in the group give the identity.

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21 There Are Many Examples Since the determinants are taken to be 1, two transformations are specified by 6 complex parameters. (Three in each matrix.) After conjugation we only need 3 complex numbers to specify the two matices. A common choice of the three parameters is tr a, tr b, tr ab. Another choice for the third parameter is tr of the commutator.

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22 Geometry of the Group One way to visualize the geometry of the group is to plot a tiling, consists of taking a seed tile and plotting all the images under the elements of the group. This is the essence of a wallpaper pattern. Kleinian group tilings exhibit a new level of complexity over Euclidean wallpaper patterns. Euclidean tilings have one limit point. Kleinian tilings have infinitely many limit points, all arranged in a fractal.

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23 Example of a Kleinian Group Two generators a(z) and b(z) pair four circles as follows: a(outside of C 1 ) = inside of C 2 b(outside of C 3 ) = inside of C 4 This is known as a classical Schottky group. The tile we plot is the “Swiss cheese” common outside of all four circles.

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24 Swiss Cheese Schottky Tiling

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25 The Schottky Dance

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26 The Limit Set The limit set consists of all the points inside infintely nested sequences of circles. It is a Cantor set or fractal dust. The outside of all four circles is a fundamental (seed) tile for this tiling. The group identifies the edges of the tile to create a surface of genus two.

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27 The Limit Set Is a Quasi-Circle

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28 Developing the Limit Set

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29 Kleinian Groups Artists Jos Leys of Belgium has made an exhaustive study of Kleinian tilinigs and limit sets at this website: website And for the fanatics, there is even fractal jewelry to be had.fractal jewelry

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30 Double Cusp Group Next we look at one specific group that has a construction that demonstrates many aspects of the mathematics. Consider the following arrangement of circles.

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32 Deformation of Schottky Group The complement of the circle web consists of four white regions a,A,b,B. These now play the role of Schottky disks. This group is a deformation of a Schottky group – now a set curves on the surface are pinched.

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38 Meduim Resolution Double Cusp Group

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39 Acknowledgments (Most) Images by David Wright Resource Text: Indra's Pearls (Mumford, Series, Wright)

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