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Lattices, sphere packings and Voronoi’s theorem

Abhinav Kumar

Oct 2, 2007

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What is a lattice?

Definition

A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generatedby n linearly independent vectors of Rn.

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What is a lattice?

Definition

A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generatedby n linearly independent vectors of Rn.

Examples

Z ⊂ R is a 1-dimensional lattice.

A non-example is the subgroup of R generated by 1 and √ 2.

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Square lattice

Example

Our first example in 2 dimensions is the square lattice Z2

generated by (1, 0) and (0, 1).

 q q q q q q q q q q q q q q q

 q q q q q q q q q q

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Hexagonal lattice

Example

Another example is the hexagonal lattice in R2.

It is generated by (1, 0) and (12

,√ 3

2

).

 q q q q q

 q q q q q

 q q q q q

 q q q q

 q q q q

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Eisenstein numbers

We can view R2 as C. Then the hexagonal lattice is generated by

1 and ω = e 2πi /3. This is the set of Eisenstein integers (the ring of algebraic integers in Q(ω)).

Fact

The set C/Λ has the structure of a torus, and can be identified with the complex solutions to the equation y 2 = x 3 − 1 (an elliptic curve). This generalizes to any lattice Z+ Zτ  in C.

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Eisenstein numbers

We can view R2 as C. Then the hexagonal lattice is generated by

1 and ω = e 2πi /3. This is the set of Eisenstein integers (the ring of algebraic integers in Q(ω)).

Fact

The set C/Λ has the structure of a torus, and can be identified with the complex solutions to the equation y 2 = x 3 − 1 (an elliptic curve). This generalizes to any lattice Z+ Zτ  in C.

Can you guess the equation corresponding to Z2 or Z+ Zi ?

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Eisenstein numbers

We can view R2 as C. Then the hexagonal lattice is generated by

1 and ω = e 2πi /3. This is the set of Eisenstein integers (the ring of algebraic integers in Q(ω)).

Fact

The set C/Λ has the structure of a torus, and can be identified with the complex solutions to the equation y 2 = x 3 − 1 (an elliptic curve). This generalizes to any lattice Z+ Zτ  in C.

Can you guess the equation corresponding to Z2 or Z+ Zi ?

y 2 = x 3 − x 

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An,D n

Now we go on to some lattices in general dimensions.

Definition

The lattice An = {(x 0, . . . , x n) ∈ Zn+1 | x i  = 0} is a lattice in

the n-dimensional plane {(x 0, . . . , x n) ∈ Rn+1

| x i  = 0}.

For example, the hexagonal lattice can be identified with A2 up toscaling.

DefinitionThe checkerboard lattice D n = {(x 1, . . . , x n) ∈ Zn | x i  is even }.

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E 8,E 7,E 6

Definition

The lattice E 8 in R8 is the lattice generated by D 8 and theall-halves vector (1

2, 12, 12, 12, 12, 12, 12, 12

).

Definition

E 7 is the orthogonal complement of (0, 0, 0, 0, 0, 0, 1, 1) in E 8 andE 6 is the orthogonal complement of (0, 0, 0, 0, 0, 0, 1, 1) and

(0, 0, 0, 0, 0, 1, 0, 1) in E 8.

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Dual lattices, direct sums

Definition

If Λ ⊂ Rn, its dual Λ∗ is the lattice in (Rn)∗ ∼= Rn which consistsof all v ∗ such that v ∗(u )

∈Z for all u 

∈Λ.

Definition

The orthogonal direct sum of two lattices Λ ∈ Rm and Γ ∈ Rn isthe group generated by Λ

×0n and 0m

×Γ.

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Sphere packings

Definition

A sphere packing in Rn

is a collection of spheres/balls of equal sizewhich 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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Good sphere packings

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

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Good sphere packings

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.

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Good sphere packings (contd.)

In dimension 3, the best possible way is to stack layers of thesolution in 2 dimensions.

   m   m m m   m   m m m

There are infinitely (in fact, uncountably) many ways of doing this!

In higher dimensions, we have some guesses for the densest spherepacking. But we can’t prove them :-(

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Sphere packing associated to a lattice

Fact

In general, given a lattice  Λ such that any two points of  Λ are separated by at least r (this is equivalent to saying that the 

minimal non-zero vector length m(Λ) is r), we can put balls of radius r /2 around each point of  Λ so that they don’t overlap.

The lattice packing problem asks for the largest density possible fora sphere packing arising from a lattice. In low dimensions, the best

(known) sphere packings often come from lattices.

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Density of lattices

Density of a lattice Λ in Rn is equal to

vol (B n(1))m(Λ)n

2ndet (Λ)

where m(Λ) is the minimal non-zero vector length, and det (Λ) isthe volume of the fundamental cell of Λ.

So, for a fixed dimension, we can just compare

m(Λ)n

det (Λ) =N (Λ)n/2

(disc (Λ))1/2

f

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Density of lattices

Definition

N (Λ) is the minimal non-zero norm of Λ.

The determinant of any Gram matrix of a basis of Λ is called thediscriminant of Λ. We have disc (Λ) = det (Λ)2.

The density of two lattices are the same if they are isometric orsimilar.

D l i

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Dense lattices

What are the densest lattices in every dimension?

D l i

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Dense lattices

What are the densest lattices in every dimension?

n 1 2 3 4 5 6 7 8 24

Λ A1 A2 A3 D 4 D 5 E 6 E 7 E 8 Leech

D l tti

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Dense lattices

What are the densest lattices in every dimension?

n 1 2 3 4 5 6 7 8 24

Λ A1 A2 A3 D 4 D 5 E 6 E 7 E 8 Leech

Finding the densest lattices in Rn: hard problem in general(though not as hard as finding the densest sphere packing in Rn).

One way to start: find the ones that are local maxima for thedensity function.

S f l tti

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Space of lattices

We can write the elements of a basis of Λ as column vectors of length n. Get an n × n invertible matrix (an element of  GL(n,R)).

S ace of lattices

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Space of lattices

We can write the elements of a basis of Λ as column vectors of length n. Get an n × n invertible matrix (an element of  GL(n,R)).

When are two lattices with given bases the “same”?

Space of lattices

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Space of lattices

We can write the elements of a basis of Λ as column vectors of length n. Get an n × n invertible matrix (an element of  GL(n,R)).

When are two lattices with given bases the “same”?

We can take an integral linear combination of the matrices that is

invertible (multiply by an element of  GL(n,Z) on the right), ortransform the lattice by an isometry (multiply by an orthogonaltransformation on the left).

So the space of lattices is

O (n)\GL(n,R)/GL(n,Z)

Quadratic forms

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Quadratic forms

Lattices are in correspondence with quadratic forms.

To the lattice Λ with basis v 1, . . . , v n we associate Q  : Z → R

given by Q (x 1, . . . , x n) = |x 1v 1 + · · · + x nv n|2.

In terms of the Gram matrix G  of Λ, the form equals x t Gx .

So the space of lattices is also (diffeomorphic to)

GL(n,Z)\Sym+(n,R)

where Sym+

(n,R) is the space of positive definite symmetricmatrices and the action of  A ∈ GL(n,Z) on G  is by At GA.

Local optimality: Voronoi’s theorem

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Local optimality: Voronoi s theorem

On the space of lattices up to scaling, we can ask for the localmaxima for the density function.

Voronoi’s theorem gives a necessary and sufficient condition for a

lattice to be a local maximum for density.

Local optimality: Voronoi’s theorem

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Local optimality: Voronoi s theorem

On the space of lattices up to scaling, we can ask for the localmaxima for the density function.

Voronoi’s theorem gives a necessary and sufficient condition for a

lattice to be a local maximum for density.

Theorem (Voronoi)

A lattice is a local maximum for density iff it is perfect and 

eutactic.

Perfectness and Eutaxy

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Perfectness and Eutaxy

Let S (Λ) = {u 1, . . . , u N } be the set of  minimal vectors of Λ (thoseof smallest positive norm N (Λ)).

Perfectness and Eutaxy

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Perfectness and Eutaxy

Let S (Λ) = {u 1, . . . , u N } be the set of  minimal vectors of Λ (thoseof smallest positive norm N (Λ)).

Definition

We say Λ is perfect if every quadratic form Q  : Rn

→ R thatvanishes on u 1, . . . ,u N  vanishes identically.

Perfectness and Eutaxy

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Perfectness and Eutaxy

Let S (Λ) = {u 1, . . . , u N } be the set of  minimal vectors of Λ (thoseof smallest positive norm N (Λ)).

Definition

We say Λ is perfect if every quadratic form Q  : Rn

→ R thatvanishes on u 1, . . . ,u N  vanishes identically.

Definition

We say Λ is eutactic if the norm form x  → |x |2

is a positive linearcombination of the forms Q 1(x ) = x , u 12, . . . , Q N (x ) = x , u N 2.

Perfection and Eutaxy contd.

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Perfection and Eutaxy contd.

In other words, Λ is perfect if the forms

x , u i 

2 span the space of 

quadratic forms.

Alternatively, if the rank one n × n matrices u i u t i 

span Sym(n,R).

It is eutactic if the identity matrix lies in the positive cone spannedby them.

Perfection and Eutaxy contd.

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Perfection and Eutaxy contd.

In other words, Λ is perfect if the forms

x , u i 

2 span the space of 

quadratic forms.

Alternatively, if the rank one n × n matrices u i u t i 

span Sym(n,R).

It is eutactic if the identity matrix lies in the positive cone spannedby them.

The number of (equivalence classes of) perfect forms in anydimension is finite.

Perfection and Eutaxy contd.

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y

In other words, Λ is perfect if the forms x , u i 2 span the space of 

quadratic forms.

Alternatively, if the rank one n × n matrices u i u t i 

span Sym(n,R).

It is eutactic if the identity matrix lies in the positive cone spannedby them.

The number of (equivalence classes of) perfect forms in anydimension is finite.

n 1 2 3 4 5 6 7 8

perfect 1 1 1 2 3 7 33 ≥ 10170

extreme 1 1 1 2 3 6 30

Example: square lattice

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p q

The minimal vectors are (±1, 0) and (0,±1).

The corresponding symmetric matrices are

1 00 0

and

0 00 1

.

So the square lattice is eutactic but not perfect.

Example: hexagonal lattice

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p g

The minimal vectors are (±1, 0), ±(12,√ 3

2) and ±(−1

2,√ 3

2).

The corresponding matrices are

1 00 0

,

14√ 34√ 

3

4

3

4

and

14 −√ 34

−√ 3

4

3

4

.

Therefore the hexagonal lattice is perfect and eutactic.

Spherical codes

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p

What is the maximum number of points N (θ) on a sphereS n−1 ⊂ Rn such that any two distinct points are separated by atleast an angle of θ.

Spherical codes

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What is the maximum number of points N (θ) on a sphereS n−1 ⊂ Rn such that any two distinct points are separated by atleast an angle of θ.

Conversely, given N  points on a sphere, how can you push them asfar apart as possible to maximize θ?

Spherical codes

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What is the maximum number of points N (θ) on a sphereS n−1 ⊂ Rn such that any two distinct points are separated by atleast an angle of θ.

Conversely, given N  points on a sphere, how can you push them asfar apart as possible to maximize θ?

This is sphere packing on the surface of a sphere (think aboutspherical caps of radius θ/2).

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Some examples of good spherical codes come from minimal vectors

of lattices.

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Some examples of good spherical codes come from minimal vectors

of lattices.Examples

The 240 minimal vectors of  E 8 form a code of angle π/3. In fact,it is the unique optimal code of angle π/3 (Conway, Sloane,

Levenshtein). It’s a miracle that we can prove this!

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Some examples of good spherical codes come from minimal vectors

of lattices.Examples

The 240 minimal vectors of  E 8 form a code of angle π/3. In fact,it is the unique optimal code of angle π/3 (Conway, Sloane,

Levenshtein). It’s a miracle that we can prove this!

There is a configuration of 27 points in S 5 (related to the 54minimal vectors of  E ∗6 ) which is the unique optimal code of anglecos−1(1/4). Also related to the 27 lines on a cubic surface in

algebraic geometry.

Potential energy minimization

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One way to find good spherical codes: potential energy. Put N points 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.

Potential energy minimization

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One way to find good spherical codes: potential energy. Put N points 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.

Some examples in the demo.

Questions

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1. How do we find good sphere packings in high dimensions?

Questions

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1. How do we find good sphere packings in high dimensions?

2. How do we find the best lattice in any dimension?

Questions

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1. How do we find good sphere packings in high dimensions?

2. How do we find the best lattice in any dimension?

3. How do we find all the locally optimal lattices in anydimension? How fast does their number grow?

Questions

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1. How do we find good sphere packings in high dimensions?

2. How do we find the best lattice in any dimension?

3. How do we find all the locally optimal lattices in anydimension? How fast does their number grow?

4. How do we find good spherical codes?

Questions

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1. How do we find good sphere packings in high dimensions?

2. How do we find the best lattice in any dimension?

3. How do we find all the locally optimal lattices in anydimension? How fast does their number grow?

4. How do we find good spherical codes?

5. Is there a good numerical method to find the global optimumfor any optimization problem?

Questions

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1. How do we find good sphere packings in high dimensions?

2. How do we find the best lattice in any dimension?

3. How do we find all the locally optimal lattices in anydimension? How fast does their number grow?

4. How do we find good spherical codes?

5. Is there a good numerical method to find the global optimumfor any optimization problem?

6. Can you find a way to approximate the size of the domain of 

optimality for any local optimum?

For further reading

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Conway and Sloane, Sphere packings, lattices and groups .

Martinet, Perfect lattices in Euclidean spaces .Gruber and Lekkerkerker, Geometry of Numbers .