Random Triangulations and Emergent Conformal Structure Discrete - - PowerPoint PPT Presentation

Random Triangulations and Emergent Conformal Structure

Discrete Differential Geometry, Berlin, July 2007 Ken Stephenson, University of Tennessee

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Outline

Background on Circle Packing: Giving geometry to combinatorics Classical Conformal Geometry — and Companion Notions Discrete Conformal Geometry Emergent Conformal Geometry Applications

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Outline

Background on Circle Packing: Giving geometry to combinatorics Classical Conformal Geometry — and Companion Notions Discrete Conformal Geometry Emergent Conformal Geometry Applications

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• 1. Circle Packing Basics

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Circle Packing Structures

Def: A circle packing P is a configuration of circles with a specified pattern of

• tangencies. (Initiated by Koebe, Andreev, and (principally) Bill Thurston.)

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Circle Packing Structures

Def: A circle packing P is a configuration of circles with a specified pattern of

• tangencies. (Initiated by Koebe, Andreev, and (principally) Bill Thurston.)

The pattern of P is given by a (simplicial) complex K which triangulates an

• riented topological surface.

The configuration P has a circle Cv for each vertex v ∈ K. When u, v is an edge of K, then Cu and Cv are tangent. When u, v, w is an oriented face of K, then Cu, Cv, Cw is an oriented triple of mutually tangent circles. The radii are given in a label R. (Computing R is where the work goes; compatibility depends on angle sums — centers are secondary.) Typical operation: given K − → compute R − → lay out P

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Packing Plasticity

The theory has been extended with boundary conditions and branching (not pertinent here) to give amazing plasticity.

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Packing Plasticity

The theory has been extended with boundary conditions and branching (not pertinent here) to give amazing plasticity.

"Maximal" packing P_K Disc Sphere Specified boundary radii Specified Boundary angles Common Combinatorics K

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Thurston’s Excellent Idea

Ω

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Thurston’s Excellent Idea

Ω

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Thurston’s Excellent Idea

Ω

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Thurston’s Excellent Idea

f Ω

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Thurston’s Excellent Idea

f Ω

Thurston’s Conjecture: If increasingly fine hexagonal circle packings Pn are used in Ω and the maps fn are appropriately normalized, then fn converges uniformly on compact subsets of D to the classical conformal mapping F : D −

→ Ω.

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K

1

K K

2n

Ω P

P f

2n 4n

f f

P P

2n 4n

P P

4n 1

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Enabling Theory

Koebe-Andreev-Thurston Theorem: For any triangulation K of a

sphere, there exists an associated univalent circle packing e P of the Riemann sphere, unique up to Möbius transformations

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Enabling Theory

Koebe-Andreev-Thurston Theorem: For any triangulation K of a

sphere, there exists an associated univalent circle packing e P of the Riemann sphere, unique up to Möbius transformations Thurston’s Conjecture on convergence to conformal mapping: proved by Burt Rodin and Dennis Sullivan using quasiconformal mapping theory.

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Enabling Theory

Koebe-Andreev-Thurston Theorem: For any triangulation K of a

sphere, there exists an associated univalent circle packing e P of the Riemann sphere, unique up to Möbius transformations Thurston’s Conjecture on convergence to conformal mapping: proved by Burt Rodin and Dennis Sullivan using quasiconformal mapping theory. Convergence extended by various authors to more general combinatorics, still using quasiconformal theory

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Enabling Theory

Koebe-Andreev-Thurston Theorem: For any triangulation K of a

sphere, there exists an associated univalent circle packing e P of the Riemann sphere, unique up to Möbius transformations Thurston’s Conjecture on convergence to conformal mapping: proved by Burt Rodin and Dennis Sullivan using quasiconformal mapping theory. Convergence extended by various authors to more general combinatorics, still using quasiconformal theory Culminating in a theorem of Zheng-Xu He and Oded Schramm that removes the quasiconformal theory, implying:

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Enabling Theory

Koebe-Andreev-Thurston Theorem: For any triangulation K of a

sphere, there exists an associated univalent circle packing e P of the Riemann sphere, unique up to Möbius transformations Thurston’s Conjecture on convergence to conformal mapping: proved by Burt Rodin and Dennis Sullivan using quasiconformal mapping theory. Convergence extended by various authors to more general combinatorics, still using quasiconformal theory Culminating in a theorem of Zheng-Xu He and Oded Schramm that removes the quasiconformal theory, implying: The Koebe-Andreev-Thurston Theorem is equivalent to the Riemann Mapping Theorem for plane domains.

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And ...

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And ...

Circle packings, refinements, and convergence results are extended to Riemann surfaces

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And ...

Circle packings, refinements, and convergence results are extended to Riemann surfaces With notion of branch points, circle packings provide wide ranging “discrete analytic functions”: discrete rational maps, inner functions, entire functions, covering maps, etc.

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And ...

Circle packings, refinements, and convergence results are extended to Riemann surfaces With notion of branch points, circle packings provide wide ranging “discrete analytic functions”: discrete rational maps, inner functions, entire functions, covering maps, etc. Indeed, a fairly comprehensive theory of discrete analytic functions emerges:

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And ...

Circle packings, refinements, and convergence results are extended to Riemann surfaces With notion of branch points, circle packings provide wide ranging “discrete analytic functions”: discrete rational maps, inner functions, entire functions, covering maps, etc. Indeed, a fairly comprehensive theory of discrete analytic functions emerges: Circle Packing: “quantum” complex analysis, classical in the limit.

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And ...

Circle packings, refinements, and convergence results are extended to Riemann surfaces With notion of branch points, circle packings provide wide ranging “discrete analytic functions”: discrete rational maps, inner functions, entire functions, covering maps, etc. Indeed, a fairly comprehensive theory of discrete analytic functions emerges: Circle Packing: “quantum” complex analysis, classical in the limit. Important to our story: the existence of practical (and provable) algorithms for computing circle packings and software CirclePack for manipulating them.

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• 2. Classical Conformal Structure and

Companion Notions

Conformal maps Brownian motion Harmonic measure Extremal length

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f harmonic measure

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g f harmonic measure L H extremal length = L/H

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• 3. Discrete Conformal Structure —

Discrete Companions

Discrete conformal maps Random walks Discrete harmonic measure Discrete extremal length

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f

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g f

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Discrete Conformal Mappings

Definition: A discrete conformal mapping is a map f : Q −

→ P between univalent circle packings associated with the same complex K.

f K P Q

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Discrete Conformal Mappings

Definition: A discrete conformal mapping is a map f : Q −

→ P between univalent circle packings associated with the same complex K.

f K P Q

Proposal: A discrete conformal structure for an oriented topological surface

S is a simplicial complex K which triangulates S.

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• 4. Emergent Conformal Structure

A random idea Experimental support Intuition What is a “random” triangulation

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Packing Triangulations

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Packing Triangulations −

→ Random Triangulations

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Random Triangulations — and Companions

Random discrete maps Random walks Discrete harmonic measure Discrete extremal length

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g f

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Emergent Conformal Structure

Setting: Let Ω be a bounded simply connected plane domain, z1, z2 ∈ Ω, and let

F : Ω − → D be the unique conformal mapping with F(z1) = 0, F(z2) > 0.

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Emergent Conformal Structure

Setting: Let Ω be a bounded simply connected plane domain, z1, z2 ∈ Ω, and let

F : Ω − → D be the unique conformal mapping with F(z1) = 0, F(z2) > 0.

Random Maps: For n >> 1, define a “random” map fn : Ω −

→ D as follows: Select a random triangulation Kn of Ω having n vertices Compute the maximal circle packing Pn for Kn (in D) Define fn : Kn − → carrier(Pn) (An appropriate φ ∈ Auto(D) applied to Pn ensures fn(z1) = 0, fn(z2) > 0)

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Emergent Conformal Structure

Setting: Let Ω be a bounded simply connected plane domain, z1, z2 ∈ Ω, and let

F : Ω − → D be the unique conformal mapping with F(z1) = 0, F(z2) > 0.

Random Maps: For n >> 1, define a “random” map fn : Ω −

→ D as follows: Select a random triangulation Kn of Ω having n vertices Compute the maximal circle packing Pn for Kn (in D) Define fn : Kn − → carrier(Pn) (An appropriate φ ∈ Auto(D) applied to Pn ensures fn(z1) = 0, fn(z2) > 0)

Conjecture: When Ω, F, and fn are as above, then fn

P

− → F as n → ∞; that is, the random maps converge “in probability” to the conformal map F.

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Emergent Conformal Structure

Setting: Let Ω be a bounded simply connected plane domain, z1, z2 ∈ Ω, and let

F : Ω − → D be the unique conformal mapping with F(z1) = 0, F(z2) > 0.

Random Maps: For n >> 1, define a “random” map fn : Ω −

→ D as follows: Select a random triangulation Kn of Ω having n vertices Compute the maximal circle packing Pn for Kn (in D) Define fn : Kn − → carrier(Pn) (An appropriate φ ∈ Auto(D) applied to Pn ensures fn(z1) = 0, fn(z2) > 0)

Conjecture: When Ω, F, and fn are as above, then fn

P

− → F as n → ∞; that is, the random maps converge “in probability” to the conformal map F.

Speculation: This should extend readily to general Riemann surfaces for an

appropriate notion of “random triangulation”.

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f f f Ω

2n n 4n

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Distribution of Dilatations

f

Color coding by qc-dilatation k; faces with dilatation k > 2 are blue.

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Random Square Construction

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Random Square Construction

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Random Square Construction

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Random Square Construction

L H log (aspect)=log(H/L)

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Square Map Detail

Create random K

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Square Map Detail

Create random K − → circle pack it

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Square Map Detail

Create random K − → circle pack it − → the carrier is equivalent to K

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Square Map Detail

Create random K − → circle pack it − → the carrier is equivalent to K Disregard the circles, leaving the “carrier”.

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Square Map Detail

Create random K − → circle pack it − → the carrier is equivalent to K

f

Disregard the circles, leaving the “carrier”. The map f is a piecewise affine map between the random triangulation and the “carrier” of the circle packing.

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Experiments with the Square

5000 trials with 3200 random vertices per trial yield this histogram:

−0.64 −0.62 −0.6 −0.58 −0.56 −0.54 −0.52 −0.5 −0.48 −0.46 50 100 150 200 250 300

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Experiments with the Square

5000 trials with 3200 random vertices per trial yield this histogram:

−0.64 −0.62 −0.6 −0.58 −0.56 −0.54 −0.52 −0.5 −0.48 −0.46 50 100 150 200 250 300

Visually and with QQ-plot the distribution appears to be gaussian.

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Varying the Complexity

−0.1 −0.08 −0.06 −0.04 −0.02 0.02 0.04 0.06 0.08 0.1 10 20 30 40 50 60 70 80

Here are plots, 5000 trials each for N = 200, 400, 800, 1600, 3200, 6400, 12800.

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Varying the Complexity

−0.1 −0.08 −0.06 −0.04 −0.02 0.02 0.04 0.06 0.08 0.1 10 20 30 40 50 60 70 80

1 2 3 4 5 6 7 8 −12 −11 −10 −9 −8 −7 −6

Here are plots, 5000 trials each for N = 200, 400, 800, 1600, 3200, 6400, 12800. A log-log plot of variance shows: “double N and you halve the variance.”

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A Rectangles of Aspect 2

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A Rectangles of Aspect 2

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Trials for Aspect 2

5000 random trials each for N = 200, 800, 3200, 12800. (Truth log(2) ≈ 0.6931)

0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 10 20 30 40 50 60 70 80

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Torus Triangulations

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 5 10 15 20 25

5000 trials with various N for torus of modulus (1+4i)/2: variance 3200 800 200 N mean (true=2.0616) .00039 .00162 .00642 2.0638 2.0605 2.0617

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Back to Ω

g f

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Extremal Length Trials

Measure extremal length of the paths between the red and blue arcs in ∂Ω

−0.75 −0.7 −0.65 −0.6 −0.55 −0.5 −0.45 −0.4 −0.35 −0.3 5 10 15 20 25 30 35

5000 trials each, N = 200, 800, 3200, 12800.

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Harmonic Measure Trials

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Harmonic Measure Trials

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7

200 vertices

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Harmonic Measure Trials

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7

200 vertices

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7

800 vertices 3200 vertices 12800 vertices

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Harmonic Measure Trials

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7

200 vertices

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7

800 vertices 3200 vertices 12800 vertices

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7

51200 vertices

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Intuition

1 2 3 4

200 points each

~

800 points

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• 5. Applications

What is a Random Triangulation?

Subdivision Tilings Brain Mapping Random surfaces

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Subdivision Tilings

Studied by Jim Cannon, Bill Floyd, and Walter Parry in the context of Thurston’s Geometrization Conjecture and Kleinian groups.

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Subdivision Tilings

Studied by Jim Cannon, Bill Floyd, and Walter Parry in the context of Thurston’s Geometrization Conjecture and Kleinian groups. Their Twisted Pentagonal example goes like this:

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Subdivision Tilings

Studied by Jim Cannon, Bill Floyd, and Walter Parry in the context of Thurston’s Geometrization Conjecture and Kleinian groups. Their Twisted Pentagonal example goes like this:

Subdivide

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Circle Packed at Stage 7

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Unexpected Self-Similarity

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Unexpected Self-Similarity

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Brain Flattening

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DCM Proj DCM

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Random Surfaces in Physics

A “stepped surface”

• Cf. Rick Kenyon and Andrei Okounkov

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Acknowledgements

Rick Kenyon for images (Kenyon/Okounkov) Jim Cannon, Bill Floyd, Walter Parry for tiling data Gerald Orick (Tennessee) for his new packing/layout algorithm Brain collaborators: Monica Hurdal, Phil Bowers, De Witt Sumners, David Rottenberg, Chuck Collins NSF for their support of this research

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