# A conceptual breakthrough in sphere packing

@article{Cohn2016ACB, title={A conceptual breakthrough in sphere packing}, author={Henry Cohn}, journal={arXiv: Metric Geometry}, year={2016} }

This expository paper describes Viazovska's breakthrough solution of the sphere packing problem in eight dimensions, as well as its extension to twenty-four dimensions by Cohn, Kumar, Miller, Radchenko, and Viazovska.

## Figures and Tables from this paper

## 58 Citations

The sphere packing problem in dimension 24

- Mathematics
- 2016

Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and…

Packings, sausages and catastrophes

- MathematicsBeiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
- 2020

In this survey we give an overview about some of the main results on parametric densities, a concept which unifies the theory of finite (free) packings and the classical theory of infinite packings.

Towards a proof of the 24-cell conjecture

- Mathematics
- 2017

This review paper is devoted to the problems of sphere packings in 4 dimensions. The main goal is to find reasonable approaches for solutions to problems related to densest sphere packings in…

Exponential improvements for superball packing upper bounds

- Mathematics
- 2020

Abstract We prove that for all fixed p > 2 , the translative packing density of unit l p -balls in R n is at most 2 ( γ p + o ( 1 ) ) n with γ p − 1 / p . This is the first exponential improvement in…

Dense packings via lifts of codes to division rings

- Mathematics
- 2021

We obtain algorithmically effective versions of the dense lattice sphere packings constructed from orders in Q-division rings by the first author. The lattices in question are lifts of suitable codes…

Locally Optimal 2-Periodic Sphere Packings

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2020

This work generalizes Voronoi’s method to m > 1 and presents a procedure to enumerate all locally optimal 2-periodic sphere packings in any dimension, provided there are finitely many.

Perfect simulation of the Hard Disks Model by Partial Rejection Sampling

- Mathematics, Computer ScienceICALP
- 2018

A perfect simulation of the hard disks model via the partial rejection sampling method is presented, and the method extends easily to the hard spheres model in d>2 dimensions.

Eigenfunctions of the Fourier Transform with specified zeros

- Mathematics
- 2019

We give a unified description of the modular and quasi-modular functions used in Viazovska's proof of the best packing bounds in dimension 8 and the proof by Cohn, Kumar, Miller, Radchenko, and…

Eigenfunctions of the Fourier transform with specified zeros

- Mathematics
- 2021

We give a unified description of the modular and quasi-modular functions used in Viazovska's proof of the best packing bounds in dimension 8 and the proof by Cohn, Kumar, Miller, Radchenko, and…

Extreme Points of the Vandermonde Determinant on Surfaces Implicitly Determined by a Univariate Polynomial

- Mathematics
- 2017

The problem of optimising the Vandermonde determinant on a few different surfaces defined by univariate polynomials is discussed. The coordinates of the extreme points are given as roots of…

## References

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Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and…

A Note on Sphere Packings in High Dimension

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We improve on the lower bounds for the optimal density of sphere packings. In all sufficiently large dimensions the improvement is by a factor of at least 10, 000; along a sparse sequence of…

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We continue the study of the linear programming bounds for sphere packing introduced by Cohn and Elkies. We use theta series to give another proof of the principal theorem, and present some related…

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she announced her spectacular result in the paper titled ‘The sphere packing problem in dimension 8’ [10] on the arXiv-preprint server. Only one week later, on 21 March 2016, Henry Cohn, Abhinav…

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This paper constitutes the official published account of the now completed Flyspeck project and describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants.

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We study some sequences of functions of one real variable and conjecture that they converge uniformly to functions with certain positivity and growth properties. Our conjectures imply a conjecture of…

What are all the best sphere packings in low dimensions?

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We describe what may beall the best packings of nonoverlapping equal spheres in dimensionsn ≤10, where “best” means both having the highest density and not permitting any local improvement. For…

A semidefinite programming hierarchy for packing problems in discrete geometry

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This work introduces topological packing graphs as an abstraction for infinite graphs coming from packing problems in discrete geometry and shows that the hierarchy converges to the independence number.

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We prove that the Leech lattice is the unique densest lattice in R^24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore…