Fractal Dust and
nSchottky Dancing
University of Utah GSAC Colloquium 10.10.06
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Josh Thompson
Geometric patterns have
played many roles in history:
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Science
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Art
Religious
The symmetry we see
is a result of
underlying
mathematical structure
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Symmetry
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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
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Symmetry Abounds
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How to Distinguish Transformations
( look for what's left unchanged )
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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
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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) = 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 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 = <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 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)
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