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Fractal Dust Fractal Dust andand nSchottky DancingSchottky DancingUniversity of Utah GSAC Colloquium 10.10.06
Josh Thompson
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Geometric patterns have Geometric patterns have played many roles in history:played many roles in history:
● Science● Art
● Religious● The symmetry we see
is a result of underlying
mathematical structure
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SymmetrySymmetry
● 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 AboundsSymmetry Abounds
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How to Distinguish TransformationsHow 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 PlaneRigid 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 TransformationsMobius 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 GroupsKleinian Groups
Mobius transformations are 'chaotic' or discrete
A Kleinian group is a discrete group of Mobius transformations.
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Three types of Three types of Mobius TranformationsMobius 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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Four Circles Tangent In A ChainFour Circles Tangent In A Chain
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The four tangent The four tangent points lie on a circle.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 PictureProof By Picture
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Extend the Circle ChainExtend 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 Starting Arrangement of Four Circles and ImagesCircles and Images
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The Action of the GroupThe Action of the Group
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The OrbitThe Orbit
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Letting Two Mobius Letting Two Mobius Transformations PlayTransformations Play
Allowing two Mobius transformations a(z), b(z) to interact can produce many Klienian groups.
In general, the group G = <a(z),b(z)> generated by aand b is likely to be freely generated – no relations in the group give the identity.
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There Are There Are ManyMany Examples 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 GroupGeometry of the GroupOne 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 GroupExample 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 TilingSwiss Cheese Schottky Tiling
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The Schottky DanceThe Schottky Dance
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The Limit SetThe 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-CircleThe Limit Set Is a Quasi-Circle
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Developing the Limit SetDeveloping the Limit Set
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Kleinian Groups ArtistsKleinian 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 GroupDouble 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 GroupDeformation 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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AcknowledgmentsAcknowledgments
(Most) Images by David Wright
Resource Text: Indra's Pearls(Mumford, Series, Wright)
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