Circles in the sand Lionel Levine (Cornell University) Wesley - - PowerPoint PPT Presentation

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Circles in the sand Lionel Levine (Cornell University) Wesley - - PowerPoint PPT Presentation

May 30, 2023 20 likes •996 views

Circles in the sand Lionel Levine (Cornell University) Wesley Pegden (Carnegie Mellon) Charles Smart (Cornell University) Harvard, April 15, 2015 The Abelian Sandpile (BTW 1987, Dhar 1990) Start with a pile of n chips at the origin in Z d .

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SLIDE 1

Circles in the sand

Lionel Levine (Cornell University) Wesley Pegden (Carnegie Mellon) Charles Smart (Cornell University) Harvard, April 15, 2015

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SLIDE 2

The Abelian Sandpile (BTW 1987, Dhar 1990)

◮ Start with a pile of n chips at the origin in Zd. ◮ Each site x = (x1, . . . , xd) ∈ Zd has 2d neighbors

x ± ei, i = 1, . . . , d.

◮ Any site with at least 2d chips is unstable, and topples by

sending one chip to each neighbor.

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SLIDE 3

The Abelian Sandpile (BTW 1987, Dhar 1990)

◮ Start with a pile of n chips at the origin in Zd. ◮ Each site x = (x1, . . . , xd) ∈ Zd has 2d neighbors

x ± ei, i = 1, . . . , d.

◮ Any site with at least 2d chips is unstable, and topples by

sending one chip to each neighbor.

◮ This may create further unstable sites, which also topple. ◮ Continue until there are no more unstable sites.

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SLIDE 4

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 12

1 1 1 1

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SLIDE 5

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 8

2 2 2 2

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SLIDE 6

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 4

3 3 3 3

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SLIDE 7

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 4

4 4 4

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SLIDE 8

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 4

4 4 4 4

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Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 4

4 4 4 4

  • 1

4 1 1 4 4 1

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SLIDE 10

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 4

4 4 4

  • 1

4 1 1 4 4 1 4

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SLIDE 11

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 4

4 4 4

  • 1

4 1 2 4 2 1 1

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SLIDE 12

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 4

4 4 4

  • 1

4 1 2 4 2 1 1 4

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SLIDE 13

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 4

4 4 4

  • 1

4 1 1 3 1 2 2 1

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SLIDE 14

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 4

4 4 4

  • 1

4 1 1 3 1 2 2 1 4

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SLIDE 15

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 4

4 4 4

  • 1

2 2 1 4 1 2 2 1

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SLIDE 16

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 4

4 4 4

  • 1

2 2 1 4 1 2 2 1 4

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SLIDE 17

Toppling to Stabilize A Sandpile

◮ Example: n=16 chips in Z2. ◮ Sites with 4 or more chips are unstable.

16

  • 4

4 4 4

  • 1

2 1 2 1 1 1 1 2 1 2 1

Stable.

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SLIDE 18

Abelian Property

◮ The final stable configuration does not depend on the order of

topplings.

◮ Neither does the number of times a given vertex topples.

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Sandpile of 1, 000, 000 chips in Z2

◮ Ostojic 2002, Fey-Redig 2008, Dhar-Sadhu-Chandra 2009,

L.-Peres 2009, Fey-L.-Peres 2010, Pegden-Smart 2011

◮ Open problem: Determine the limit shape! (It exists.)

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Limit shape 1: The sandpile computes an area-minimizing tropical curve through n given points Caracciolo-Paoletti-Sportiello 2010, Kalinin-Shkolnikov 2015

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SLIDE 21

Limit shape 2: Identity element of the sandpile group of an n × n square grid Le Borgne-Rossin 2002. Sportiello 2015+

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SLIDE 22

Sandpiles of the form h + nδ0 h = 2 h = 1 h = 0

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SLIDE 23

What about h = 3?

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3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

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3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 4 4 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

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3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 5 5 3 3 3 4 4 4 3 3 3 5 5 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3

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3 3 3 4 3 3 3 3 3 5 5 3 3 3 5 1 4 1 5 3 4 4 4 4 3 5 1 4 1 5 3 3 3 5 5 3 3 3 3 3 4 3 3 3

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3 3 5 5 3 3 3 5 1 4 1 5 3 5 1 5 5 1 5 4 4 4 5 1 5 5 1 5 3 5 1 4 1 5 3 3 3 5 5 3 3 . . . Never stops toppling!

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3 5 1 4 1 5 3 5 1 5 5 1 5 1 5 1 4 1 5 1 4 4 4 4 1 5 1 4 1 5 1 5 1 5 5 1 5 3 5 1 4 1 5 3 . . . Never stops toppling!

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5 1 5 5 1 5 1 5 1 4 1 5 1 5 1 5 5 1 5 4 4 4 5 1 5 5 1 5 1 5 1 4 1 5 1 5 1 5 5 1 5 . . . Never stops toppling!

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1 5 1 4 1 5 1 5 1 5 5 1 5 1 5 1 4 1 5 1 4 4 4 4 1 5 1 4 1 5 1 5 1 5 5 1 5 1 5 1 4 1 5 1 . . . Never stops toppling!

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A dichotomy Any sandpile τ : Zd → N is either

◮ stabilizing: every site topples finitely often ◮ or exploding: every site topples infinitely often

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An open problem

◮ Given a probability distribution µ on N, decide whether the

i.i.d. sandpile τ ∼

x∈Z2 µ is stabilizing or exploding. ◮ For example, find the smallest λ such that i.i.d. Poisson(λ) is

exploding.

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SLIDE 34

How to prove an explosion

◮ Claim: If every site in Zd topples at least once, then

every site topples infinitely often.

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How to prove an explosion

◮ Claim: If every site in Zd topples at least once, then

every site topples infinitely often.

◮ Otherwise, let x be the first site to finish toppling.

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SLIDE 36

How to prove an explosion

◮ Claim: If every site in Zd topples at least once, then

every site topples infinitely often.

◮ Otherwise, let x be the first site to finish toppling. ◮ Each neighbor of x topples at least one more time, so x

receives at least 2d additional chips.

◮ So x must topple again. ⇒⇐

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SLIDE 37

The Odometer Function

◮ u(x) = number of times x topples.

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The Odometer Function

◮ u(x) = number of times x topples. ◮ Discrete Laplacian:

∆u(x) =

  • y∼x

u(y) − 2d u(x)

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The Odometer Function

◮ u(x) = number of times x topples. ◮ Discrete Laplacian:

∆u(x) =

  • y∼x

u(y) − 2d u(x) = chips received − chips emitted

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The Odometer Function

◮ u(x) = number of times x topples. ◮ Discrete Laplacian:

∆u(x) =

  • y∼x

u(y) − 2d u(x) = chips received − chips emitted = τ∞(x) − τ(x) where τ is the initial unstable chip configuration and τ∞ is the final stable configuration.

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SLIDE 41

Stabilizing Functions

◮ Given a chip configuration τ on Zd and a function

u1 : Zd → Z, call u1 stabilizing for τ if τ + ∆u1 ≤ 2d − 1.

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SLIDE 42

Stabilizing Functions

◮ Given a chip configuration τ on Zd and a function

u1 : Zd → Z, call u1 stabilizing for τ if τ + ∆u1 ≤ 2d − 1.

◮ If u1 and u2 are stabilizing for τ, then

τ + ∆min(u1, u2) ≤ τ + max(∆u1, ∆u2) ≤ 2d − 1 so min(u1, u2) is also stabilizing for τ.

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Least Action Principle

◮ Let τ be a sandpile on Zd with odometer function u. ◮ Least Action Principle:

If v : Zd → Z≥0 is stabilizing for τ, then u ≤ v.

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Least Action Principle

◮ Let τ be a sandpile on Zd with odometer function u. ◮ Least Action Principle:

If v : Zd → Z≥0 is stabilizing for τ, then u ≤ v.

◮ So the odometer is minimal among all nonnegative stabilizing

functions: u(x) = min{v(x) | v ≥ 0 is stabilizing for τ}.

◮ Interpretation: “Sandpiles are lazy.”

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The Green function of Zd

◮ G : Zd → R and ∆G = −δ0. ◮ In dimensions d ≥ 3,

G(x) = E0#{k|Xk = x} is the expected number of visits to x by simple random walk started at 0.

◮ As |x| → ∞,

G(x) ∼ g(x) =

  • cd|x|2−d

d ≥ 3 c2 log |x| d = 2.

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SLIDE 46

An integer obstacle problem

◮ The odometer function for n chips at the origin is given by

u = nG + w where G is the Green function of Zd, and w is the pointwise smallest function on Zd satisfying w ≥ −nG ∆w ≤ 2d − 1 w + nG is Z-valued

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SLIDE 47

An integer obstacle problem

◮ The odometer function for n chips at the origin is given by

u = nG + w where G is the Green function of Zd, and w is the pointwise smallest function on Zd satisfying w ≥ −nG ∆w ≤ 2d − 1 w + nG is Z-valued

◮ What happens if we replace Z by R?

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SLIDE 48

Abelian sandpile Divisible sandpile (Integrality constraint) (No integrality constraint)

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Scaling limit of the abelian sandpile in Zd

◮ Consider sn = nδ0 + ∆un, the sandpile formed from n chips at

the origin.

◮ Let r = n1/d and

¯ sn(x) = sn(rx) (rescaled sandpile) ¯ wn(x) = r−2un(rx) − nG(rx) (rescaled odometer)

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SLIDE 50

Theorem (Pegden-Smart, 2011)

◮ There are functions w, s : Rd → R such that as n → ∞,

¯ wn → w locally uniformly in C(Rd) ¯ sn → s weakly-∗ in L∞(Rd). Moreover s is a weak solution to ∆w = s.

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Two Sandpiles of Different Sizes n = 100, 000 n = 200, 000 (scaled down by √ 2)

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Locally constant “steps” of s correspond to periodic patterns:

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Limit of the least action princpile w = min{v ∈ C(Rd) | v ≥ −g and D2(v + g) ∈ Γ}.

◮ g encodes the initial condition (rotationally symmetric!) ◮ Γ is a set of symmetric d × d matrices, to be described.

It encodes the sandpile “dynamics.”

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Limit of the least action princpile w = min{v ∈ C(Rd) | v ≥ −g and D2(v + g) ∈ Γ}.

◮ g encodes the initial condition (rotationally symmetric!) ◮ Γ is a set of symmetric d × d matrices, to be described.

It encodes the sandpile “dynamics.”

◮ D2u ∈ Γ is interpreted in the sense of viscosity:

D2φ(x) ∈ Γ whenever φ is a C ∞ function touching u from below at x (that is, φ(x) = u(x) and φ − u has a local maximum at x).

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SLIDE 55

The set Γ of stabilizable matrices

◮ Γ = Γ(Zd) is the set of d × d real symmetric matrices A for

which there exists a slope b ∈ Rd and a function v : Zd → Z such that ∆v(x) ≤ 2d − 1 and v(x) ≥ 1

2x · Ax + b · x

for all x ∈ Zd.

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The set Γ of stabilizable matrices

◮ Γ = Γ(Zd) is the set of d × d real symmetric matrices A for

which there exists a slope b ∈ Rd and a function v : Zd → Z such that ∆v(x) ≤ 2d − 1 and v(x) ≥ 1

2x · Ax + b · x

for all x ∈ Zd.

◮ How to test for membership in Γ?

◮ Start with v(x) =

1

2x · Ax + b · x

  • .

◮ For each x ∈ Zd such that ∆v(x) ≥ 2d, increase v(x) by 1.

Repeat.

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Testing for membership in Γ

◮ A ∈ Γ if and only if there exists b such that the sandpile

sA,b = ∆⌈qA,b⌉ stabilizes, where qA,b(x) = 1

2x · Ax + b · x. ◮ if A and b have rational entries, then sA,b is periodic. ◮ Topple until stable, or until every site has toppled at least

  • nce.
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SLIDE 58

The structure of Γ(Z2)

Parameterize 2 × 2 real symmetric matrices by M(a, b, c) = 1 2 c + a b b c − a

  • .
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SLIDE 59

The structure of Γ(Z2)

Parameterize 2 × 2 real symmetric matrices by M(a, b, c) = 1 2 c + a b b c − a

  • .

Note that if A ≤ B (that is, B − A is positive semidefinite) and B ∈ Γ then A ∈ Γ. In particular, Γ = {M(a, b, c) | c ≤ γ(a, b)} for some function γ : R2 → R.

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Graph of γ(a, b)

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SLIDE 61
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SLIDE 62
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SLIDE 63

Cross section

  • 2
  • 1

2

  • 1

1

  • 2

1

  • 2
  • Γ
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SLIDE 64

Cross section

The Laplacian of v(x) = 1

2x1(x1 + 1) + 1 2x2(x2 + 1)

is 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

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SLIDE 65

Cross section

The Laplacian of v(x) = 1

2x1(x1 + 1) + 1 2x2(x2 + 1) + ⌈εx2 2⌉

is 2 3 2 3 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 3 2 2 3

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SLIDE 66

Cross section

The Laplacian of v(x) = 1

2x1(x1 + 1) + x2(x2 + 1)

is 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

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SLIDE 67

Cross section

The Laplacian of v(x) = 1

2x1(x1 + 1) + 1 2x2(x2 + 1) + ⌈εx2 1⌉

is 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

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SLIDE 68

Cross section

The Laplacian of v(x) = 1

2x1(x1 + 1) + 1 2x2(x2 + 1) + ⌈εx2 1 + εx2 2⌉

is 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 3 3 3 3 3 3 4 3 3 3 3 3 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2

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SLIDE 69

Another example

We have 1

4 [ 5 2 2 4 ] ∈ ∂Γ because

v(x) = 1 8(5x2

1 + 4x1x2 + 4x2 2 + 2x1 + 4x2)

  • has Laplacian

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

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SLIDE 70

Rank-1 cones

The set Γ(Z2) is a union of downward cones {B | B ≤ A}, for a set of peaks A ∈ P.

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SLIDE 71

Periodicity

Since the matrices M(2, 0, 0) = 1 −1

  • and

M(0, 2, 0) = 1 1

  • have integer valued discrete harmonic quadratic forms

u(x) = 1

2x1(x1 + 1) − 1 2x2(x2 + 1)

and u(x) = x1x2, we see that γ is 2Z2-periodic.

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SLIDE 72

Associating a matrix to each circle

If C is a circle of radius r centered at a + bi, define AC := 1 2 a + 2 + r b b −a + 2 + r

  • .

Let A be the circle packing in the (a, b)-plane generated by the vertical lines a = 0, a = 2 and the circle (a − 1)2 + b2 = 1, repeated horizontally so it is 2Z2-periodic.

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SLIDE 73

The Apollonian structure of Γ

Theorem (L-Pegden-Smart 2013)

B ∈ Γ if and only if B ≤ AC for some C ∈ A.

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SLIDE 74

Analysis of the peaks

Theorem (L-Pegden-Smart 2013)

B ∈ Γ if and only if B ≤ AC for some C ∈ A. Proof idea: It is enough to show that each peak matrix AC lies on the boundary of Γ.

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SLIDE 75

Analysis of the peaks

Theorem (L-Pegden-Smart 2013)

B ∈ Γ if and only if B ≤ AC for some C ∈ A. Proof idea: It is enough to show that each peak matrix AC lies on the boundary of Γ. For each AC we must find vC : Z2 → Z and bC ∈ R2 such that ∆vC(x) ≤ 3 and vC(x) ≥ 1

2x · ACx + bC · x

for all x ∈ Z2. We use the recursive structure of the circle packing to construct vC and bC.

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SLIDE 76

(4,1,4)

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SLIDE 77

(9,1,6)

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SLIDE 78

(12,7,12)

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SLIDE 79

(16,1,8)

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SLIDE 80

(24,17,24)

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SLIDE 81

Curvature coordinates (Descartes 1643; Lagarias-Mallows-Wilks 2002)

If C0 has parents C1, C2, C3 and grandparent C4, then C0 = 2(C1 + C2 + C3) − C4 in curvature coordinates C = (c, cz).

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SLIDE 82
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SLIDE 83

Curvature coordinates (Descartes 1643; Lagarias-Mallows-Wilks 2002)

If C0 has parents C1, C2, C3 and grandparent C4, then C0 = 2(C1 + C2 + C3) − C4 in curvature coordinates C = (c, cz).

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SLIDE 84
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SLIDE 85
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SLIDE 86
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SLIDE 87
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SLIDE 88

Inductive tile construction

We build tiles from copies of earlier tiles, using ideas from Katherine Stange 2012 “The Sensual Apollonian Circle Packing” to keep track of the tile interfaces.

T +

1

T +

2

T +

3

T −

1

T −

2

T −

3

T4

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SLIDE 89

The magic identities

We associate an offset vector v(C, C ′) ∈ Z[i] to each pair of tangent circles. If (C0, C1, C2, C3) is a proper Descartes quadruple, then the offset vectors vij = v(Ci, Cj) satisfy v10 = v13 − iv21 v01 = iv10 v32 + v13 + v21 = 0 v2

32 = c3c2(z3 − z2)

¯ v13v21 + v13¯ v21 = −2c1 where (ci, cizi) are the curvature coordinates of the circle Ci.

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SLIDE 90

Inductive tile construction

Given tiles of the parent circles C1, C2, C3 and grandparent circle C4, arrange them using the offset vectors v14, v24, v34:

T +

1

T +

2

T +

3

T −

1

T −

2

T −

3

T4

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SLIDE 91

A topological lemma

◮ Define a tile to be a finite union of squares of the form

[x, x + 1] × [y, y + 1], x, y ∈ Z whose interior is a topological disk.

◮ Suppose T is a collection of tiles such that

◮ We suspect that T is a tiling of the plane:

C =

  • T∈T

T (disjoint union)

◮ and we can verify a lot of adjacencies between tiles; ◮ but we have no simple way to verify disjointness.

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SLIDE 92

A topological lemma

Let T be an infinite collection of tiles, and G = (T , E) be a graph with vertex set T . If the following hold, then T is a tiling of the plane.

  • 1. G is a 3-connected planar triangulation.
  • 2. G is invariant under translation by some full-rank lattice

L ⊆ Z[i], and

T∈T /L area(T) = | det L|.

  • 3. If (T1, T2) ∈ E then T1 ∩ T2 contains at least 2 integer points.
  • 4. For each face F = {T1, T2, T3} of G we can select an integer

point ρ(F) ∈ T1 ∩ T2 ∩ T3 such that for each adjacent face F ′ = {T1, T2, T4} there is a path in T1 ∩ T2 from ρ(F) to ρ(F ′).

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SLIDE 93

Extra 90◦ symmetry

For each circle C ∈ A we build a tile TC that tiles the plane. A pleasant surprise: Each tile TC has 90◦ rotational symmetry!

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SLIDE 94

An open problem: classify such tilings.

If T is a primitive, periodic, hexagonal tiling of the plane by identical 90◦ symmetric tiles, must its fundamental tile be TC for some circle C ∈ A?

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SLIDE 95

Other lattices, higher dimensions

We have described the set Γ(Z2) in terms of an Apollonian circle packing of R2. What about Γ(Zd) for d ≥ 3? In general any periodic graph G embedded in Rd has an associated set of d × d symmetric matrices Γ(G), which captures some aspect

  • f the infinitessimal geometry of 1

nG as n → ∞.

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SLIDE 96

Γ for the triangular lattice

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SLIDE 97

Thank you!

Reference: L.-Pegden-Smart, arXiv:1309.3267 The Apollonian structure of integer superharmonic matrices