Lattices, sphere packings, spherical codes and

energy minimization

Abhinav Kumar

MIT

November 10, 2009

Sphere packings

Definition

A sphere packing in Rn is a collection of spheres/balls of equal size

which do not overlap (except for touching). The density of a spherepacking is the volume fraction of space occupied by the balls.

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Lattices

Definition

A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated

by n linearly independent vectors of Rn.

Examples: integer lattice Zn, checkerboard lattice Dn, simplex

lattice An, special root lattices E6,E7,E8, Leech lattice Λ24, and soon.

Associated sphere packing: if m(Λ) is the length of a smallestnon-zero vector of Λ, then we can put balls of radius m(Λ)/2around each point of Λ so that they don’t overlap.

Spherical codes

A spherical code C on Sn−1 is a finite subset of the sphere.

Example: The kissing configuration of a lattice Λ is the set ofminimal non-zero vectors of Λ, rescaled to the unit sphere.

The minimal angle θ(C) of the code is the smallest radial anglebetween distinct elements of C.

cos(θ(C)) = maxx ,y∈C,x 6=y

〈x , y〉.

Sphere packing problem

Problem: Find a/the densest sphere packing(s) in Rn.

In dimension 1, we can achieve density 1 by laying intervals end toend.

In dimension 2, the best possible is by using the hexagonal lattice.[Fejes Toth 1940]

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Sphere packing problem II

In dimension 3, the best possible way is to stack layers of thesolution in 2 dimensions. This is Kepler’s conjecture, now atheorem of Hales.

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There are infinitely (in fact, uncountably) many ways of doing this!These are the Barlow packings.

In higher dimensions, we have some guesses for the densest spherepacking. But no proofs yet.

Lattices packing

The packing problem for lattices asks for the densest lattice(s) inR

n for every n. This is equivalent to the determination of theHermite constant γn, which arises in the geometry of numbers.The known answers are:

n 1 2 3 4 5 6 7 8 24

Λ A1 A2 A3 D4 D5 E6 E7 E8 Leech

In low dimensions, lattices provide good candidates for the densestsphere packings. But in high dimensions, it is expected that thedensest packings will not be lattices.

Spherical code problem

The spherical code problem asks:

1 Given an angle θ0 ∈ (0, π] and a dimension n, what is themaximum number of points N for which there is a sphericalcode on Sn−1 with minimal angle at least θ0?

2 Given n and N, what is the maximum possible angle of anN-point code on Sn−1?

An optimal spherical code is one which solves the second problem.

Kissing problem

The kissing problem is the spherical code problem for angle π/3. Itcan be rephrased as: how many non-overlapping unit spheres canbe arranged around a central unit sphere in Rn?

Answers

n = 1: Two intervals around a central interval.

n = 2: Six circles around a central circle.

n = 3 (Newton-Gregory problem): Twelve spheres, but theconfiguration is not rigid or unique. Proof by Schutte and vander Waerden, 1953.

Kissing problem II

n = 4: Twenty-four, coming from kissing arrangement of theD4 lattice. Proof by Musin, 2003. Believed to be unique, butunproven.

n = 8: Kissing configuration of E8 of 240 points.

n = 24: Kissing configuration of Λ24 of 196560 points.

The last two are due to Odlyzko and Sloane and (independently)Levenshtein.

No other answers known.

LP bounds

How do we prove optimality/uniqueness of any of these?

One idea: linear programming bounds. Invented by Delsarte todeal with association schemes, binary codes etc.

Levenshtein used these to prove optimality and uniqueness of anumber of codes or families of codes.

For instance, the 240-point kissing configuration in R8 and the196560-point kissing configuration in R24 are sharp for the linearprogramming bound, and therefore optimal. This even enables oneto prove uniqueness.

Application of the linear programming bound also allowed theresolution of the lattice packing problem in 24 dimensions [Cohn-K2003].

Positive definite kernels

Fix n ≥ 2. We say f : [−1, 1] → R is a positive definite kernel if forevery code C ⊂ Sn−1, the |C | × |C | matrix

(

f (〈x , y〉))

x ,y∈Cis

positive semidefinite.

In particular,∑

x ,y∈C f (〈x , y〉) ≥ 0.

Schonberg (1930s) classified all the positive definite kernels. Heshowed that the ultraspherical or Gegenbauer polynomialsCλ

i (t), i = 0, 1, 2, . . . are PDKs and that any PDK is anon-negative linear combination of them. Here λ = n/2 − 1.

Gegenbauer polynomials

The Gegenbauer polynomials arise from representationtheory/harmonic analysis. They are given by the generatingfunction

(1 − 2tz + z2)−λ =

∞∑

i=0

Cλi (t)z i

So we have

1 C0(t) = 1

2 C1(t) = (n − 2)t

3 C2(t) = (n − 2)(nt2 − 1)/2

and so on.

Linear programming bound for codes

Theorem

Let f (t) =∑

i fiCλi (t) be a positive definite kernel (i.e. all fi ≥ 0),

such that f0 > 0 and f (t) ≤ 0 for t ∈ [−1, cos θ0]. Then any codeC with minimal angle at least θ0 has at most f (1)/f0 points.

Proof.

We have

|C |f (1) ≥∑

x ,y∈C

f (〈x , y〉)

=∑

x ,y∈C

∑

i

fiCλi (〈x , y〉)

=∑

i

fi∑

x ,y∈C

Cλi (〈x , y〉)

≥ f0|C |2.

LP bounds II

Why are these called linear programming bounds?

Write f = 1 +∑

i>0 fiCi .

Variables: fi .

Constraints: fi ≥ 0

Constraints: 1 +∑

fiCi (t) ≤ 0 for every t ∈ [−1, cos(θ0)].

Objective function: minimize 1 +∑

fiCi(1).

This is a convex optimization program, and we can approximate itby a linear program by discretizing the interval [−1, 1], andrestrinting to finitely many nonzero fi .

Potential energy minimization

One way to find good spherical codes: potential energy. Put Npoints on a sphere with a repulsive force law (e.g. electrostaticrepulsion), and let the system evolve. They will tend to separatethemselves to minimize potential energy.

For S2, this is called the Thomson problem after the physicistJ. J. Thomson, who asked it in connection with the plum-puddingmodel of the atom.

Potential energy minimization II

Definition

Let f : (0, 4] → R be a function. We define the f -potential energyof a code C ⊂ Sn−1 to be

Ef (C) =∑

x ,y∈C,x 6=y

f (|x − y |2)

Note:

1 Each pair of points counted twice, so the potential energy isdouble that of the physicists.

2 We let f be a function of squared distance, rather thandistance. This makes the formulas nicer.

Potential energy minimization III

The problem is: given n,N and a potential function f , find aspherical code of N points on Sn−1 that minimizes f -potentialenergy.

Examples of potential functions

Inverse power law: Ik(r) = 1/rk for k ∈ R>0. Fork = n/2 − 1, this gives the harmonic potential.

Gaussian: Gc(r) = exp(−cr) for some c > 0 (note that thisis Gaussian as a function of distance).

Aℓ(r) = (4 − r)ℓ for nonnegative integers ℓ.

All these functins are completely monotonic, i.e.(−1)mf (m)(r) ≥ 0.

The functions Aℓ span the cone of completely monotonic functionson (0, 4].

Universal optimality

Note: As k > 0, the potential energy minimization problem for1/rk becomes the spherical code problem (maximize the minimalangle). Similarly, the spherical code problem is a limit of theenergy minimization problems for Gaussians as well.

We say a spherical code is universally optimal if it minimizesf -potential energy (among codes of its size) for all completelymonotonic f .

There are examples of universally optimal codes, though theirexistence is very uncommon. The typical situation is that we haveone or more families of N-point configurations, each being optimalfor Aℓ in a certain range of ℓ.

Examples

N points in S1: regular N-gon.

2 points in S2: antipodal points, universally optimal.

3 points in S2: equilateral triangle, universally optimal.

4 points in S2: regular tetrahedron, universally optimal.

In general, k ≤ n + 1 points on Sn−1 ⊂ Rn always gives a regular

simplex in a k − 1 dimensional “equatorial” subspace, under anycompletely monotonic function.

Examples II

5 points in S2: two competing configurations.

Consider the configuration A of two antipodal points with threepoints on the equator forming an equilateral triangle.

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Examples III

The configuration Bθ consists of a pyramid with one point on thenorth pole, and four points in the southern hemisphere at latitudeθ = α − π/2, forming a square.

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Examples IV

For inverse power laws, some Bθ wins for steep power laws 1/rk fork > 7.524+, but A wins for smaller k.

Note that A maximizes angular distance, as does B0.

For the function Aℓ, the configuration A wins for 1 ≤ ℓ ≤ 6,whereas some Bθ wins for ℓ ≥ 7.

LP bounds for potential energy

The linear programming bounds of Delsarte for the spherical codeproblem (maximize N for a given θ), were adapted by Yudin togive LP bounds for potential energy.

Theorem (Yudin)

Let f : (0, 4] → R be any function. Suppose h : [−1, 1] → R is apolynomial such that h(t) ≤ f (2 − 2t) for all t ∈ [−1, 1], andsuppose there are nonnegative coefficients α0, . . . , αd such that

h(t) =d∑

i=0

αiCλi (t) in terms of the Gegenbauer polynomials. Then

every set of N points on Sn−1 has potential energy at leastN2α0 − Nh(1).

Examples

Known universally optimal configurations of N points on Sn−1:

n N Name2 N N-gonn n + 1 simplexn 2n cross polytope3 12 icosahedron4 120 600-cell8 240 E8 root system7 56 spherical kissing6 27 spherical kissing/Schlafli5 16 spherical kissing/Clebsch24 196560 Leech lattice minimal vectors23 4600 spherical kissing22 891 spherical kissing23 552 regular 2-graph22 275 McLaughlin21 162 Smith22 100 Higman-Sims

q q3+1q+1

(q + 1)(q3 + 1) Cameron-Goethals-Seidel

They are all sharp for LP bounds [Cohn-K, 2005].

Other conjectured universal optima

Work of Ballinger, Blekherman, Cohn, Giansiracusa, Kelly,Schurmann.

40 points in R10, inner products 1, 1/6, 0,−1/3,−1/2.

64 points in R14, inner products 1, 1/7,−1/7,−3/7.

Not sharp for linear programming bounds but perhaps stilluniversally optimal.

Other spaces

We can also define potential energy for finite subsets of othercompact metric spaces, and sometimes for infinite subsets ofnoncompact spaces.

For example, for compact two-point homogeneous spaces such asRP

n, CPn, HP

n, we can define the notion of universal optimality(for completely monotonic functions of squared chordal distance)and find examples of universally optimal codes (joint work withCohn, Elkies, Khatirinejad).

Euclidean space

Let P be a periodic point configuration in Euclidean space. Letf : R≥0 → R a sufficiently rapidly decaying function. Define thef -potential energy of a point x ∈ P to be

∑

y∈P,y 6=x

f (|x − y |2)

and the f -potential energy of P to be the (finite) average over allpoints x ∈ P of their potential energies.

The energy minimization problem asks: given n, f and a fixed pointdensity δ (usually 1), which periodic point configuration of densityδ minimizes the f -potential energy over all periodic configurationsin R

n (the number of translates N is allowed to vary).

Gradient descent

Recently, with Cohn and Schurmann, we wrote a computerprogram to carry out gradient descent on spaces of periodicconfigurations, to search for optima for the Gaussian potentialfunctions exp(−cr2), and obtained some interesting results for theminimum energy configurations observed experimentally.

In particular, we found families of formally dual periodicconfigurations, i.e. configurations whose average theta functionsare related by the generalized Jacobi formula, which replaces z by−1/z and multiplies by an appropriate constant.

The exp(−cπr2)-potential energy of P should be related to theexp(−πr2/c)-potential energy of its dual by a factor depending onthe density of P, for every c .

Formal duality

Every lattice has a formal dual (namely, its dual lattice). But wedo not know of any other formally dual configurations recordedpreviously in the literature.

Example

Let D+n be the union of Dn and its translate by the all-halves

vector. Then D+n is always formally self-dual (it is a lattice exactly

when 4 divides n).In addition, if we let D+

n (α) be the configuration obtained byscaling the last coordinate of all vectors in D+

n by α, then D+n (α)

is formally dual to D+n (1/α), so we in fact have a family of

formally dual configurations.

Inverse problem

What happens if we evolve 8 points on S2 under a 1/rk potential?

The minimum for energy is not a cube! It’s a skew cube(antiprism), where the distance between the two square facesvaries as k varies.

Similarly, 20 points on S2 don’t settle down to a regulardodecahedron under the inverse power laws or Gaussians.

Can we design a potential function which is minimized by thecube?

Can do it with potential wells, but we want a nicer function.

Inverse problem II

Theorem

Let f (r) = 1/r + r/3 − 8r2/11 + 2r3/9 − r4/50. The cube is theunique global minimum for f -potential energy among 8-pointcodes in S2.

Proof.

Linear programming bounds! We engineer f so that it’s easy tocome up with an h that works and gives a sharp bound for thecube. But note that f is in fact decreasing and convex as afunction of distance (even though not completely monotonic).

Some questions

Are there only finitely many universal optima in a givendimension?

How many local optima are there? Systematic upper/lowerbounds?

What are good ways to ”walk” the configuration space to findgood codes (other than gradient descent for potential energy)?

Is it possible to beat the D4 lattice for energy, among periodicconfigurations or among lattices?

Papers and References

Ballinger-Blekherman-Cohn-Giansiracusa-Kelly-Schurmann,Experimental study of energy-minimizing point configurationson spheres

Cohn-K, Optimality and uniqueness of the Leech lattice amonglattices

Cohn-K, Universally optimal distribution of points on spheres

Cohn-K, Algorithmic design of self-assembling structures

Cohn-K, Counterintuitive ground states in soft-core models

Cohn-K-Schurmann, Ground states and formal dualityrelations in the Gaussian core model