# Energy minimization for lattices and periodic configurations

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Energy minimization for lattices and periodic conﬁgurations, and formal duality
Abhinav Kumar
MIT
November 14, 2011

joint work with Henry Cohn and Achill Schu¨rmann

Abhinav Kumar (MIT)

Potential

November 14, 2011 1 / 20

Sphere packings

Sphere packing problem: What is (a/the) densest sphere packing in n dimensions?
In low dimensions, the best densities known are achieved by lattice packings.

n Λ due to

123 A1 A2 A3
Gauss

45
D4 D5 KorkineZolotareﬀ

678 E6 E7 E8
Blichfeldt

24 Leech Cohn-K.

Abhinav Kumar (MIT)

Potential

November 14, 2011 2 / 20

Sphere packings

Sphere packing problem: What is (a/the) densest sphere packing in n dimensions?
In low dimensions, the best densities known are achieved by lattice packings.

n Λ due to

123 A1 A2 A3
Gauss

45
D4 D5 KorkineZolotareﬀ

678 E6 E7 E8
Blichfeldt

24 Leech Cohn-K.

Abhinav Kumar (MIT)

Potential

November 14, 2011 2 / 20

Low dimensions
n = 1: lay intervals end to end (density 1). n = 2: hexagonal or A2 arrangement [Fejes-T´oth 1940]
✎☞ ✎☞ ✎✎ ☞☞ ✎☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✍☞ ✎✌ ✍☞ ✎✌ ✍✎ ☞✌ ✍☞ ✎✌☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✍☞ ✎✌ ✍☞ ✎✌ ✍✎ ☞✌ ✍☞ ✎✌☞ ✍✌ ✍✌ ✍✍ ✌✌ ✍✌
This is the unique densest periodic packing.

Abhinav Kumar (MIT)

Potential

November 14, 2011 3 / 20

Low dimensions
n = 1: lay intervals end to end (density 1). n = 2: hexagonal or A2 arrangement [Fejes-T´oth 1940]
✎☞ ✎☞ ✎✎ ☞☞ ✎☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✍☞ ✎✌ ✍☞ ✎✌ ✍✎ ☞✌ ✍☞ ✎✌☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✍☞ ✎✌ ✍☞ ✎✌ ✍✎ ☞✌ ✍☞ ✎✌☞ ✍✌ ✍✌ ✍✍ ✌✌ ✍✌
This is the unique densest periodic packing.

Abhinav Kumar (MIT)

Potential

November 14, 2011 3 / 20

Barlow packings
n = 3 : stack layers of the solution in 2 dimensions. [Hales 1998] ✎☞ ✎☞ ✎✎ ☞☞ ✎☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✎ ✍☞ ✎ ♠☞ ✌ ✎ ✍☞ ✎ ♠✌ ✎ ✍ ☞✎ ☞ ♠☞ ✎ ✌ ✍☞ ✎ ♠☞ ✌☞ ✍✍ ✎✌ ✎ ✍✌ ☞ ✍ ✎ ♠☞ ✌ ✎ ✍✌ ✍ ✎ ☞ ♠✎ ☞ ✍ ✌✌ ✍ ☞ ✎ ♠☞ ✎ ✌ ✍✌ ☞ ♠☞ ✌ ✎✍☞ ✍ ✎✌ ✍✌ ☞ ✍ ✎✌ ✍✍ ✌ ✎ ☞✌ ✍✌ ✍ ☞ ✎✌✌ ☞ ✍✌ ✍✌ ✍✍ ✌✌ ✍✌
Uncountably many ways of doing this, the Barlow packings. Even in dimensions 5, 6, 7, densest lattices have (uncountably many) competitors.

Abhinav Kumar (MIT)

Potential

November 14, 2011 4 / 20

Barlow packings
n = 3 : stack layers of the solution in 2 dimensions. [Hales 1998] ✎☞ ✎☞ ✎✎ ☞☞ ✎☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✎ ✍☞ ✎ ♠☞ ✌ ✎ ✍☞ ✎ ♠✌ ✎ ✍ ☞✎ ☞ ♠☞ ✎ ✌ ✍☞ ✎ ♠☞ ✌☞ ✍✍ ✎✌ ✎ ✍✌ ☞ ✍ ✎ ♠☞ ✌ ✎ ✍✌ ✍ ✎ ☞ ♠✎ ☞ ✍ ✌✌ ✍ ☞ ✎ ♠☞ ✎ ✌ ✍✌ ☞ ♠☞ ✌ ✎✍☞ ✍ ✎✌ ✍✌ ☞ ✍ ✎✌ ✍✍ ✌ ✎ ☞✌ ✍✌ ✍ ☞ ✎✌✌ ☞ ✍✌ ✍✌ ✍✍ ✌✌ ✍✌
Uncountably many ways of doing this, the Barlow packings. Even in dimensions 5, 6, 7, densest lattices have (uncountably many) competitors.

Abhinav Kumar (MIT)

Potential

November 14, 2011 4 / 20

Barlow packings
n = 3 : stack layers of the solution in 2 dimensions. [Hales 1998] ✎☞ ✎☞ ✎✎ ☞☞ ✎☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✎ ✍☞ ✎ ♠☞ ✌ ✎ ✍☞ ✎ ♠✌ ✎ ✍ ☞✎ ☞ ♠☞ ✎ ✌ ✍☞ ✎ ♠☞ ✌☞ ✍✍ ✎✌ ✎ ✍✌ ☞ ✍ ✎ ♠☞ ✌ ✎ ✍✌ ✍ ✎ ☞ ♠✎ ☞ ✍ ✌✌ ✍ ☞ ✎ ♠☞ ✎ ✌ ✍✌ ☞ ♠☞ ✌ ✎✍☞ ✍ ✎✌ ✍✌ ☞ ✍ ✎✌ ✍✍ ✌ ✎ ☞✌ ✍✌ ✍ ☞ ✎✌✌ ☞ ✍✌ ✍✌ ✍✍ ✌✌ ✍✌
Uncountably many ways of doing this, the Barlow packings. Even in dimensions 5, 6, 7, densest lattices have (uncountably many) competitors.

Abhinav Kumar (MIT)

Potential

November 14, 2011 4 / 20

Root lattices
An (simplex lattice) = {x ∈ Zn+1 | xi = 0}, inside the zero-sum hyperplane {x ∈ Rn+1 | xi = 0} ∼= Rn. Dn (checkerboard lattice) = {x ∈ Zn | xi ≡ 0 (mod 2)} E8 = D8 (D8 + (1/2, . . . , 1/2)). E7 = orthogonal complement of A1 inside E8. E6 = orthogonal complement of A2 inside E8.

Abhinav Kumar (MIT)

Potential

November 14, 2011 5 / 20

High dimensions
In higher dimensions, we believe the densest sphere packings don’t come from lattices. Example In R10 the densest known is the Best packing, 40 translates of a lattice.
But do believe the densest packings can be achieved by periodic packings (Zassenhaus conjecture). Can provably come arbitrarily close for packing density. Trivial Minkowski bound implies ∃ packing with density ≥ 1/2n, but no explicit constructions known.

Abhinav Kumar (MIT)

Potential

November 14, 2011 6 / 20