1

Fractal Dust Fractal Dust andand nSchottky DancingSchottky DancingUniversity of Utah GSAC Colloquium 10.10.06

Josh Thompson

2

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

3

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.

4

Symmetry AboundsSymmetry Abounds

5

6

7

8

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.

9

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:

10

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}

11

Kleinian GroupsKleinian Groups

Mobius transformations are 'chaotic' or discrete

A Kleinian group is a discrete group of Mobius transformations.

12

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.

13

Four Circles Tangent In A ChainFour Circles Tangent In A Chain

14

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.

15

Proof By PictureProof By Picture

16

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.

17

Starting Arrangement of Four Starting Arrangement of Four Circles and ImagesCircles and Images

18

The Action of the GroupThe Action of the Group

19

The OrbitThe Orbit

20

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.

21

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.

22

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.

23

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.

24

Swiss Cheese Schottky TilingSwiss Cheese Schottky Tiling

25

The Schottky DanceThe Schottky Dance

26

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.

27

The Limit Set Is a Quasi-CircleThe Limit Set Is a Quasi-Circle

28

Developing the Limit SetDeveloping the Limit Set

29

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.

30

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.

31

32

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.

33

34

35

36

37

38

Meduim Resolution Double Cusp Group

39

AcknowledgmentsAcknowledgments

(Most) Images by David Wright

Resource Text: Indra's Pearls(Mumford, Series, Wright)