Fractal Dust and nSchottky Dancing University of Utah GSAC Colloquium 10.10.06

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Fractal Dust and nSchottky DancingUniversity
of Utah GSAC Colloquium 10.10.06
Josh Thompson
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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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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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Symmetry Abounds
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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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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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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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Kleinian Groups

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

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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) z1
• Hyperbolic Two fixed points, one attracting one
  repelling. Conjugate to multiplication
  (expansion) f(z) az, with a gt 1.
• Elliptic Two fixed points, both neutral.
  Conjugate to a rotation.

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Four Circles Tangent In A Chain
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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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Proof By Picture
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Extend the Circle Chain

• Given one Mobius transformation that takes C1 to
  C4, (and C2 to C3) there is a unique second
  Mobius transformation taking C1 to C2, (and C3 to
  C4) and the two transformations commute.

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Starting Arrangement of Four Circles and Images
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The Action of the Group
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The Orbit
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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 lta(z),b(z)gt generated
  by aand b is likely to be freely generated no
  relations in the group give the identity.

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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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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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Example of a Kleinian Group

• Two generators a(z) and b(z) pair four circles as
  follows
• a(outside of C1) inside of C2
• b(outside of C3) inside of C4
• 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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Swiss Cheese Schottky Tiling
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The Schottky Dance
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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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The Limit Set Is a Quasi-Circle
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Developing the Limit Set
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Kleinian Groups Artists

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

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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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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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• Meduim Resolution Double Cusp Group

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Acknowledgments

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