First Section

Circle Packings, Triangulations, and Rigidity For Tom Hales’ Birthday

Bob Connelly With Steven Gortler, Evan Solomonides, and Maria Yampolskaya.

Cornell University

June 2018

1 / 25 First Section Part I - The Packing Problem

A packing of circular disks is a collection of such disks with disjoint interiors. The density of such a packing is the ratio of total area of the disks divided by the area of the container that holds them. (For an infinite packing, you take a limit.) The Packing Problem: For a given container and collection of disks, find a packing of those disks with greatest density. Example: For√ the and equal radius disks the largest density is: δ = π/ 12 = 0.9068 ... .

2 / 25 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 packings, both regular and irregular. However, his proof was considered by some to be incomplete. The first rigorous proof is attributed to L´aszl´oFejes T´othin 1940.” (From Wikipedia).

But L´aszloFejes T´othdid 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 ... Al´adar Heppes (2000) proved that the following configuration is √ the most dense at δ = π(2 − 2)/2 = 0.92015 ... .

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r = 2-1 = .414.. δ = 0.92015.. First Section Compact Packings = Triangulated Packings

The graph of a is obtained by connecting the centers of each pair of touching disks by a line segment. When that graph of a packing is a triangulation Fejes T´othcalled the packing a compact packing. In many cases such “compact packings” were candidates for the most dense packings with those radii sizes. For a range between two sizes of radii, the following are his candidates for the maximum density.

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

4 / 25 First Section Fejes Toth’s packings

From “Regular Figures” by L´aszl´oFejes T´oth

5 / 25 First Section Fejes Toth’s density estimates

The function s above is an upper bound for the density, by August Florian (1960), where q is the minimum ratio ri /rj of the circle radii. πq2 + 2(1 − ρ2) arcsin( q ) s = √ 1+q . 2q 2q + 1 6 / 25 First Section Koebe, Andreev, Thurston Theory

Meanwhile, Koebe, Andreev, and Thurston have the following theorem: Theorem (KAT) Given a triangulation T of a 2- M with constant curvature, there is a circle packing of M whose graph is T , which is unique up to circle preserving maps (Moebius transformations).

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

7 / 25 First Section Examples of KAT theory for M = torus

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

radii 1 1: 1- 2 1: 2 :3 1 number of 1 2 3 4

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

8 / 25 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 T´oth’spacking #3. 9 / 25 First Section Triangulated Packings with Two Sizes of Radii

Considering Fejes T´oth’sdensity estimates using triangulated packings, Tom Kennedy found all possible two sized radii packings, that were triangulated packings. He found one (or two) more that was not in Fejes T´oth’slist. These are the following:

10 / 25 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 = = ··· = = ··· = = ···

e b d f c b a c j f h g e d g a i j i h m k l

(d) r c4 0.414214 (e) r c5 0.386106 (f) r c6 0.349198 = = ··· = = ··· = = ···

(g) r c7 0.280776 (h) r c8 0.154701 (i) r c9 0.101021 11 / 25 = = ··· = = ··· = = ···

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, there= must= be= at least= one= pair= of discs of different radii that are tangent, and so there must be at least one solution of (3) other than the trivial one and at least one solution of (4) other than the trivial one. We start by determining when (3) has solutions. First Section Problems for triangulated packings of a torus

1 What can you say about the change in density with an edge subdivision or edge flip? 2 Any triangulation can be obtained by edge flips from a triangular lattice with the same number of vertices. 3 What is the least dense triangulated packing other than those coming from the triangular lattice? (Any√ triangulated packing has a density strictly greater than π/ 12 if the disks are different sizes.) 4 Florian showed that if a packing is such that the minimum radius√ ratios ri /rj > 0.73, then their maximum density is only π/ 12, which is when the respective circles with a fixed radius√ are segregated into their own regions with density π/ 12. Can the 0.73 bound be improved?

12 / 25 First Section Problem for Tom

Is there a similar result in dimension 3? Namely is there a ratio 3 ρ < 1 such that if there is a packing of R by spherical balls with radii ri such the minimum√ ratio ri /rj ≥ ρ, then the packing has density at most π/3 2?

13 / 25 First Section Conjecture

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

Conjecture: Fix the number n of disks in a packing. Fix the radii r1, r2,..., rn of the disks. Suppose that there is a saturated triangulated packing of a torus with n circles with those radii having density δ. Then any other torus packing with n circles and those radii has density at most δ.

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

14 / 25 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 the torus with fixed radius ratios. Let them grow maintaining the radius ratios until they jam and can’t grow anymore. When that happens, there will be at least a portion of the packing that is rigid and 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

0 0 0 0 An infinitesimal flex p = (p1, p2,..., pn) of a tensegrity (G, p) is a 0 vector pi assigned to the i-th vertex such that

0 0 (pi − pj ) · (pi − pj ) = 0, ≤ 0, ≥ 0,

for a bar, cable, strut. 0 0 0 0 An infinitesimal flex p = (p1, p2,..., pn) is trivial if it is the time 0 derivative of a one parameter family of global isometries of the ambient space. If a tensegrity (G, p) has only trivial infinitesimal flexes, then it is called infinitesimally rigid. Theorem Infinitesimal rigidity implies local rigidity.

17 / 25 First Section Equilibrium stress

Dual to the notion of an infinitesimal flex is the idea of an equilibrium stress, which is scalar ωij = ωji , all the same sign for packings, associated to each edge of the packing graph, such that for each disk i, X ωij (pi − pj ) = 0. j The distribution of the stresses in a packing are important for the study of forces in granular materials, for example.

Suppose a periodic packing is rigid with a fixed lattice defining the torus, which means that there is no deformation of the packing disks maintaining the packing constraints other than the trivial translations. In this case the packing itself is called collectively jammed.

18 / 25 First Section The Canonical Push

Depending on the container, if p = (p1,..., pn) is a configuration 0 0 0 0 of centers of disks, and p = (p1, p2,..., pn) is an infinitesimal flex of the centers, then for each disk center pi and t ≥ 0, define 0 pi (t) = pi + tpi . For each pair of i, j of touching disks,

2 2 0 0 2 0 0 2 2 (pi (t)−pj (t)) = (pi −pj ) +2t(pi −pj )·(pi −pj )+t (pi −pj ) ≥ (pi −pj ) .

0 0 Since (pi − pj ) · (pi − pj ) ≥ 0, and the inequality is strict unless 0 0 pi = pj , we get the following Theorem For finite packings of disks in a flat torus, if the packing graph is collectively jammed, then it is infinitesimally rigid.

19 / 25 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 the corresponding minimal number of contacts.

Examples:

An isostatic packing A non-isostatic packing with 4 disks and 7 edges. with 4 disks and 12 edges.

20 / 25 First Section The Isostatic Theorem

Theorem (Connelly, Gortler, Solomonides, and Yampolskaya, 2017) 2 If a collectively jammed packing P0 with n vertices in a torus T is chosen so that the ratio of packing disks, and torus lattice Λ0, is generic, then the number of contacts in P0 is 2n − 1, and the packing graph is isostatic.

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

21 / 25 First Section Ren Guo’s Theorem

To prove the Isostatic Theorem we need the following which is a local version of the KAT theory. The inversive distance between two circles is defined as

2 2 2 |p1 − p2| − (r1 + r2 ) σ(D1, D2) = , 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 circle packing P where σ(Di , Dj ) ≥ 0 is the inversive distance between each pair of disks i, j that are an edge in the triangulation T . Then the inversive distance packings are locally determined by the values of σ(Di , Dj ).

22 / 25 First Section Triangulating the packing graph

This is a maximally dense packing of 3 equal disks in a square torus (A theorem of Will Dickinson, et. al.) with a minimal number of contacts, namely 5. We have inserted 4 additional dashed edges to create a triangulation with 9 edges total.

This packing is isostatic. 23 / 25 First Section The Isostatic proof idea

The canonical push implies that the number of contacts k for n jammed disks is at least 2n − 1. (2n coordinate variables, minus 2 trivial 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 are at least an n − 1 ratios and a 2 dimensional space of lattices, for an n + 1 dimensional space of jammed configurations, near any given jammed configuration.

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

Ren Guo’s theorem implies that the dimension of the space of configurations, with a triangulation which has k, edges having inversive distance 0, is 3n − k. So 3n − k ≥ n + 1. That is k ≤ 2n − 1, and so k = 2n − 1. 24 / 25 First Section Generic lattice is needed

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

The slanted torus, isostatic The square torus, not isostatic

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