Lattices, sphere packings, spherical codes

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

Spherical codes

A spherical code C on S n−1 is a ﬁnite subset of the sphere.
Example: The kissing conﬁguration of a lattice Λ is the set of
minimal non-zero vectors of Λ, rescaled to the unit sphere.
The minimal angle θ(C) of the code is the smallest radial angle
between distinct elements of C.

cos(θ(C)) = max x, y .
x,y ∈C,x=y

Sphere packing problem

Problem: Find a/the densest sphere packing(s) in R n .
In dimension 1, we can achieve density 1 by laying intervals end to
end.
In dimension 2, the best possible is by using the hexagonal lattice.
[Fejes T´th 1940]
o

Sphere packing problem II

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

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 sphere
packing. But no proofs yet.

Lattices packing

The packing problem for lattices asks for the densest lattice(s) in
Rn for every n. This is equivalent to the determination of the
Hermite 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 densest
sphere packings. But in high dimensions, it is expected that the
densest 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 the
maximum number of points N for which there is a spherical
code on S n−1 with minimal angle at least θ0 ?
2 Given n and N, what is the maximum possible angle of an
N-point code on S n−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. It
can be rephrased as: how many non-overlapping unit spheres can
be 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 the
conﬁguration is not rigid or unique. Proof by Sch¨tte and van
u
der Waerden, 1953.

Kissing problem II

n = 4: Twenty-four, coming from kissing arrangement of the
D4 lattice. Proof by Musin, 2003. Believed to be unique, but
unproven.
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 to
deal with association schemes, binary codes etc.
Levenshtein used these to prove optimality and uniqueness of a
number of codes or families of codes.
For instance, the 240-point kissing conﬁguration in R 8 and the
196560-point kissing conﬁguration in R 24 are sharp for the linear
programming bound, and therefore optimal. This even enables one
to prove uniqueness.
Application of the linear programming bound also allowed the
resolution of the lattice packing problem in 24 dimensions [Cohn-K
2003].

Positive definite kernels

Fix n ≥ 2. We say f : [−1, 1] → R is a positive definite kernel if for
every code C ⊂ S n−1 , the |C | × |C | matrix f ( x, y ) x,y ∈C is
positive semidefinite.
In particular, x,y ∈C f ( x, y ) ≥ 0.
Sch¨nberg (1930s) classified all the positive definite kernels. He
o
showed that the ultraspherical or Gegenbauer polynomials
Ciλ (t), i = 0, 1, 2, . . . are PDKs and that any PDK is a
non-negative linear combination of them. Here λ = n/2 − 1.

Gegenbauer polynomials

The Gegenbauer polynomials arise from representation
theory/harmonic analysis. They are given by the generating
function
∞
(1 − 2tz + z 2 )−λ = Ciλ (t)z i
i =0

So we have

1 C0 (t) = 1
2 C1 (t) = (n − 2)t
3 C2 (t) = (n − 2)(nt 2 − 1)/2

and so on.

Linear programming bound for codes
Theorem
Let f (t) = i fi Ciλ (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 code
C with minimal angle at least θ0 has at most f (1)/f0 points.

Proof.
We have

|C |f (1) ≥ f ( x, y )
x,y ∈C

= fi Ciλ ( x, y )
x,y ∈C i

= fi Ciλ ( x, y )
i x,y ∈C
2
≥ f0 |C | .

LP bounds II

Why are these called linear programming bounds?
Write f = 1 + i 0 fi Ci .

Variables: fi .
Constraints: fi ≥ 0
Constraints: 1 + fi Ci (t) ≤ 0 for every t ∈ [−1, cos(θ0 )].
Objective function: minimize 1 + fi Ci (1).

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

Potential energy minimization

One way to ﬁnd good spherical codes: potential energy. Put N
points on a sphere with a repulsive force law (e.g. electrostatic
repulsion), and let the system evolve. They will tend to separate
themselves to minimize potential energy.
For S 2 , this is called the Thomson problem after the physicist
J. J. Thomson, who asked it in connection with the plum-pudding
model of the atom.

Potential energy minimization II

Deﬁnition
Let f : (0, 4] → R be a function. We deﬁne the f -potential energy
of a code C ⊂ S n−1 to be

Ef (C) = f (|x − y |2 )
x,y ∈C,x=y

Note:

1 Each pair of points counted twice, so the potential energy is
double that of the physicists.
2 We let f be a function of squared distance, rather than
distance. This makes the formulas nicer.

Potential energy minimization III

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

Examples of potential functions

Inverse power law: Ik (r ) = 1/r k for k ∈ R0 . For
k = n/2 − 1, this gives the harmonic potential.
Gaussian: Gc (r ) = exp(−cr ) for some c 0 (note that this
is Gaussian as a function of distance).
Aℓ (r ) = (4 − r )ℓ for nonnegative integers ℓ.

All these functins are completely monotonic, i.e.
(−1)m f (m) (r ) ≥ 0.
The functions Aℓ span the cone of completely monotonic functions
on (0, 4].

Universal optimality

Note: As k 0, the potential energy minimization problem for
1/r k becomes the spherical code problem (maximize the minimal
angle). Similarly, the spherical code problem is a limit of the
energy minimization problems for Gaussians as well.
We say a spherical code is universally optimal if it minimizes
f -potential energy (among codes of its size) for all completely
monotonic f .
There are examples of universally optimal codes, though their
existence is very uncommon. The typical situation is that we have
one or more families of N-point conﬁgurations, each being optimal
for Aℓ in a certain range of ℓ.

Examples

N points in S 1 : regular N-gon.
2 points in S 2 : antipodal points, universally optimal.
3 points in S 2 : equilateral triangle, universally optimal.
4 points in S 2 : regular tetrahedron, universally optimal.
In general, k ≤ n + 1 points on S n−1 ⊂ Rn always gives a regular
simplex in a k − 1 dimensional “equatorial” subspace, under any
completely monotonic function.

Examples II

5 points in S 2 : two competing conﬁgurations.
Consider the conﬁguration A of two antipodal points with three
points on the equator forming an equilateral triangle.
t

ƒ
ƒ
t h ƒ

h h
h ƒt
h
@
t @@@
@

t
t
tt

Examples III

The conﬁguration Bθ consists of a pyramid with one point on the
north pole, and four points in the southern hemisphere at latitude
θ = α − π/2, forming a square.
t
¡t
e
¡ t
e
¡ et
t¡ q et t
α t
¡ e
¡
t e
t

Examples IV

For inverse power laws, some Bθ wins for steep power laws 1/r k for
k 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 code
problem (maximize N for a given θ), were adapted by Yudin to
give LP bounds for potential energy.
Theorem (Yudin)
Let f : (0, 4] → R be any function. Suppose h : [−1, 1] → R is a
polynomial such that h(t) ≤ f (2 − 2t) for all t ∈ [−1, 1], and
suppose there are nonnegative coeﬃcients α0 , . . . , αd such that
d
h(t) = αi Ciλ (t) in terms of the Gegenbauer polynomials. Then
i =0
every set of N points on S n−1 has potential energy at least
N 2 α0 − Nh(1).

Examples

Known universally optimal conﬁgurations of N points on S n−1 :

n N Name
2 N N-gon
n n+1 simplex
n 2n cross polytope
3 12 icosahedron
4 120 600-cell
8 240 E8 root system
7 56 spherical kissing
6 27 spherical kissing/Schl¨ﬂi
a
5 16 spherical kissing/Clebsch
24 196560 Leech lattice minimal vectors
23 552 regular 2-graph
22 275 McLaughlin
21 162 Smith
22 100 Higman-Sims
3
q q +1
q+1
(q + 1)(q 3 + 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,
Sch¨rmann.
u
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 still
universally optimal.

Other spaces

We can also define potential energy for finite subsets of other
compact metric spaces, and sometimes for infinite subsets of
noncompact spaces.
For example, for compact two-point homogeneous spaces such as
RPn , CPn , HPn , 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 with
Cohn, Elkies, Khatirinejad).

Euclidean space

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

f (|x − y |2 )
y ∈P,y =x

and the f -potential energy of P to be the (finite) average over all
points x ∈ P of their potential energies.
The energy minimization problem asks: given n, f and a fixed point
density δ (usually 1), which periodic point configuration of density
δ minimizes the f -potential energy over all periodic configurations
in Rn (the number of translates N is allowed to vary).

Gradient descent

Recently, with Cohn and Sch¨rmann, we wrote a computer
u
program to carry out gradient descent on spaces of periodic
configurations, to search for optima for the Gaussian potential
functions exp(−cr 2 ), and obtained some interesting results for the
minimum energy configurations observed experimentally.
In particular, we found families of formally dual periodic
configurations, i.e. configurations whose average theta functions
are related by the generalized Jacobi formula, which replaces z by
−1/z and multiplies by an appropriate constant.
The exp(−cπr 2 )-potential energy of P should be related to the
exp(−πr 2 /c)-potential energy of its dual by a factor depending on
the density of P, for every c.

Formal duality

Every lattice has a formal dual (namely, its dual lattice). But we
do not know of any other formally dual configurations recorded
previously in the literature.
Example
+
Let Dn be the union of Dn and its translate by the all-halves
+
vector. Then Dn is always formally self-dual (it is a lattice exactly
when 4 divides n).
+
In addition, if we let Dn (α) be the configuration obtained by
+ +
scaling the last coordinate of all vectors in Dn by α, then Dn (α)
is formally dual to Dn + (1/α), so we in fact have a family of

formally dual configurations.

Lattices, sphere packings, spherical codes

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