First Section

Circle Packings, Triangulations, and RigidityFor Tom Hales’ Birthday

Bob ConnellyWith Steven Gortler, Evan Solomonides, and Maria Yampolskaya.

Cornell University

June 2018

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First Section

Part I - The Packing Problem

A packing of circular disks is a collection of such disks with disjointinteriors.

The density of such a packing is the ratio of total area of the disksdivided by the area of the container that holds them. (For aninfinite packing, you take a limit.)

The Packing Problem: For a given container and collection ofdisks, find a packing of those disks with greatest density.Example: For the plane and equal radius disks the largest densityis: δ = π/

√12 = 0.9068 . . . .

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First Section

History

“Axel Thue provided the first proof that this was optimal in 1890, showing that the

hexagonal lattice is the densest of all possible circle packings, both regular and

irregular. However, his proof was considered by some to be incomplete. The first

rigorous proof is attributed to Laszlo Fejes Toth in 1940.” (From Wikipedia).

But Laszlo Fejes Toth did more: He found a whole series of packings with a range of

radii that he proposed as candidates for the most dense packings given a particular

range of ratios of the radii. For example, if there are just two radii, with a ratio of√

2− 1 = 0.414 . . . Aladar Heppes (2000) proved that the following configuration is

the most dense at δ = π(2−√

2)/2 = 0.92015 . . . .

r = 2-1 = .414..δ = 0.92015..

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First Section

Compact Packings = Triangulated Packings

The graph of a circle packing is obtained by connecting the centersof each pair of touching disks by a line segment. When that graphof a packing is a triangulation Fejes Toth called the packing acompact packing. In many cases such “compact packings” werecandidates for the most dense packings with those radii sizes. Fora range between two sizes of radii, the following are his candidatesfor the maximum density.

Most packings that come up with respect to density questions areperiodic, and so it is natural to simply assume that they areperiodic, which is the same as assuming they live on a (flat) torus.

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First Section

Fejes Toth’s packings

From “Regular Figures” by Laszlo Fejes Toth

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First Section

Fejes Toth’s density estimates

The function s above is an upper bound for the density, by AugustFlorian (1960), where q is the minimum ratio ri/rj of the circleradii.

s =πq2 + 2(1− ρ2) arcsin( q

1+q )

2q√

2q + 1.

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First Section

Koebe, Andreev, Thurston Theory

Meanwhile, Koebe, Andreev, and Thurston have the followingtheorem:

Theorem (KAT)

Given a triangulation T of a 2-manifold M with constantcurvature, there is a circle packing of M whose graph is T , whichis unique up to circle preserving maps (Moebius transformations).

Keep in mind that circle packing defines the metric, that is thelattice that defines M, up to Moebius transformations.

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First Section

Examples of KAT theory for M = torus

One way to create periodic triangulated packings is to start with apacking, and subdivide an edge as below. The yellow point is onthe edge to be subdivided for the next configuration. KAT insuresthat there is a corresponding packing with that graph. Only afundamental region is shown.

density π/ 12 = 0.9068.. π(1- 2 /2 )= 0.92015 7π/24 = 0.91629.. π/ 12 = 0.9068..

number of circles 1 2 3 4

radii 1 1: 1- 2 1: 2 :3 1

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First Section

Edge flipping

Another method is to flip an edge of the triangulation, which is where an edge is

removed and replaced as the other diagonal in the resulting quadrilateral. Any edge

can be flipped, as long as you don’t flip away a degree 3 vertex.

The packing on the right is obtained by flipping the green edge on the 4 vertex lattice

triangulation. The symmetry of the abstract triangulation and the uniqueness of the

packing implies that the symmetry persists in the metric configuration of the packing,

which implies that there are just two radii in this flipped packing. This is Laszlo Fejes

Toth’s packing #3.9 / 25

First Section

Triangulated Packings with Two Sizes of Radii

Considering Fejes Toth’s density estimates using triangulatedpackings, Tom Kennedy found all possible two sized radii packings,that were triangulated packings. He found one (or two) more thatwas not in Fejes Toth’s list. These are the following:

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First Section

Triangulated Packings with Two Sizes of Radii

Compact Packings of the Plane with Two Sizes of Discs 257

(a) r = c1 = 0.637556 · · · (b) r = c2 = 0.545151 · · ·. (c) r = c3 = 0.533296 · · ·

(d) r = c4 = 0.414214 · · ·

m

b

k

f

l

j

d geca

hi

(e) r = c5 = 0.386106 · · ·

dc b

f hgj

a i

e

(f) r = c6 = 0.349198 · · ·

(g) r = c7 = 0.280776 · · · (h) r = c8 = 0.154701 · · · (i) r = c9 = 0.101021 · · ·

Fig. 1. A compact packing is shown for each of the nine values of r which admit compact packings.

and non-negative integers l, m, n such that

lα′ + mβ + nπ/3 = 2π. (4)

For every value of r there is a trivial solution of both of these equations, namely i =j = l = m = 0, k = n = 6. In a compact packing that contains discs of both radii, theremust be at least one pair of discs of different radii that are tangent, and so there must beat least one solution of (3) other than the trivial one and at least one solution of (4) otherthan the trivial one. We start by determining when (3) has solutions.

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First Section

Problems for triangulated packings of a torus

1 What can you say about the change in density with an edgesubdivision or edge flip?

2 Any triangulation can be obtained by edge flips from atriangular lattice with the same number of vertices.

3 What is the least dense triangulated packing other than thosecoming from the triangular lattice? (Any triangulated packinghas a density strictly greater than π/

√12 if the disks are

different sizes.)

4 Florian showed that if a packing is such that the minimumradius ratios ri/rj > 0.73, then their maximum density is onlyπ/√

12, which is when the respective circles with a fixedradius are segregated into their own regions with densityπ/√

12. Can the 0.73 bound be improved?

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Problem for Tom

Is there a similar result in dimension 3? Namely is there a ratioρ < 1 such that if there is a packing of R3 by spherical balls withradii ri such the minimum ratio ri/rj ≥ ρ, then the packing hasdensity at most π/3

√2?

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First Section

Conjecture

A packing is saturated if it is not possible to move a disk toanother location while fixing the others.

Conjecture: Fix the number n of disks in a packing. Fix the radiir1, r2, . . . , rn of the disks. Suppose that there is a saturatedtriangulated packing of a torus with n circles with those radiihaving density δ. Then any other torus packing with n circles andthose radii has density at most δ.

In Kennedy’s list, any cover of packings a, b, c, d, f, g, h, havebeen shown to satisfy the conjecture, all proved by Aladar Heppes,except packing b was shown by Kennedy.

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First Section

Nazarov’s example

The condition that the packing be saturated is necessary. Fedja Nazarov pointed out

the following example.

This is a fundamental region of a triangulated packing of a torus, where the six

colored packing disks can be removed and reinserted to places shown with dashed

boundaries in the triangular regions to the right and left. The packing disks can be

freed with small motions from contact with their neighbors one at a time. At the end

of the motion, all the packing disks have room to grow and increase density.15 / 25

First Section

Part II - Jammed Packings

Fix a lattice Λ that defines a torus. Place n circular disks in thetorus with fixed radius ratios. Let them grow maintaining theradius ratios until they jam and can’t grow anymore. When thathappens, there will be at least a portion of the packing that is rigidand can only move under a global translation as in the figure below.

This is the most dense packing of 8 equal disks in a square torus, proved by Musin and

Nikitenko (2016)16 / 25

First Section

Infinitesimal Rigidity

An infinitesimal flex p′ = (p′1,p′2, . . . ,p

′n) of a tensegrity (G ,p) is a

vector p′i assigned to the i-th vertex such that

(pi − pj) · (p′i − p′j) = 0,≤ 0,≥ 0,

for a bar, cable, strut.An infinitesimal flex p′ = (p′1,p

′2, . . . ,p

′n) is trivial if it is the time

0 derivative of a one parameter family of global isometries of theambient space. If a tensegrity (G ,p) has only trivial infinitesimalflexes, then it is called infinitesimally rigid.

Theorem

Infinitesimal rigidity implies local rigidity.

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First Section

Equilibrium stress

Dual to the notion of an infinitesimal flex is the idea of anequilibrium stress, which is scalar ωij = ωji , all the same sign forpackings, associated to each edge of the packing graph, such thatfor each disk i , ∑

j

ωij(pi − pj) = 0.

The distribution of the stresses in a packing are important for thestudy of forces in granular materials, for example.

Suppose a periodic packing is rigid with a fixed lattice defining thetorus, which means that there is no deformation of the packingdisks maintaining the packing constraints other than the trivialtranslations. In this case the packing itself is called collectivelyjammed.

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First Section

The Canonical Push

Depending on the container, if p = (p1, . . . ,pn) is a configurationof centers of disks, and p′ = (p′1,p

′2, . . . ,p

′n) is an infinitesimal flex

of the centers, then for each disk center pi and t ≥ 0, definepi (t) = pi + tp′i . For each pair of i , j of touching disks,

(pi (t)−pj(t))2 = (pi−pj)2+2t(pi−pj)·(p′i−p′j)+t2(p′i−p′j)2 ≥ (pi−pj)2.

Since (pi − pj) · (p′i − p′j) ≥ 0, and the inequality is strict unlessp′i = p′j , we get the following

Theorem

For finite packings of disks in a flat torus, if the packing graph iscollectively jammed, then it is infinitesimally rigid.

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First Section

Isostatic Conjecture

Conjecture (Isostatic Conjecture)

If circular disks packed rigidly in a “container” with generic radii,then there is only a one-dimensional equilibrium stress, and thecorresponding minimal number of contacts.

Examples:

An isostatic packingwith 4 disks and 7 edges.

A non-isostatic packingwith 4 disks and 12 edges.

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The Isostatic Theorem

Theorem (Connelly, Gortler, Solomonides, and Yampolskaya, 2017)

If a collectively jammed packing P0 with n vertices in a torus T2 ischosen so that the ratio of packing disks, and torus lattice Λ0, isgeneric, then the number of contacts in P0 is 2n − 1, and thepacking graph is isostatic.

The idea is that the dimension of the space of packings with theinversive distances replacing edge lengths is only consistent withthe minimal number of contacts given by isostatic condition.

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First Section

Ren Guo’s Theorem

To prove the Isostatic Theorem we need the following which is alocal version of the KAT theory. The inversive distance betweentwo circles is defined as

σ(D1,D2) =|p1 − p2|2 − (r2

1 + r22 )

2r1r2,

where D1 and D2 are disks with corresponding radii r1 and r2.

Theorem (Ren Guo (2011))

Let T be a triangulation of a torus corresponding to a circlepacking P where σ(Di ,Dj) ≥ 0 is the inversive distance betweeneach pair of disks i , j that are an edge in the triangulation T .Then the inversive distance packings are locally determined by thevalues of σ(Di ,Dj).

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Triangulating the packing graph

This is a maximally dense packing of 3 equal disks in a squaretorus (A theorem of Will Dickinson, et. al.) with a minimalnumber of contacts, namely 5. We have inserted 4 additionaldashed edges to create a triangulation with 9 edges total.

This packing is isostatic.23 / 25

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The Isostatic proof idea

The canonical push implies that the number of contacts k for njammed disks is at least 2n − 1. (2n coordinate variables, minus 2trivial translations, plus a stress constraint is needed for rigidity.Altogether k ≥ 2n − 1 edge contact constraints are needed.)

By letting the disks grow with a fixed ratio we know that there areat least an n − 1 ratios and a 2 dimensional space of lattices, foran n + 1 dimensional space of jammed configurations, near anygiven jammed configuration.

An Euler characteristic argument shows that the total number ofedges in a triangulation of a torus with n vertices is 3n.

Ren Guo’s theorem implies that the dimension of the space ofconfigurations, with a triangulation which has k , edges havinginversive distance 0, is 3n − k . So 3n − k ≥ n + 1. That isk ≤ 2n − 1, and so k = 2n − 1.

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Generic lattice is needed

Note the lattice has to be generic as well as the radii ratios. Hereis an example for the square lattice for 2 disks with a genericradius ratio, but where the number of contacts is 4, not 3.

The slanted torus, isostatic The square torus, not isostatic

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