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Universally optimal sphere packing[1]

Description

A periodic sphere packing that (weakly) minimizes all completely monotonic potentials of square Euclidean distance among all periodic packings of the same density.

Cousin

  • \(A_2\) triangular lattice— The triangular lattice is universally optimal among all lattices, but has not been proven to be optimal over all periodic packings [2].

Member of code lists

Primary Hierarchy

Classical Domain
Analog Kingdom
Analog codeECC
Sphere packing
Parents
Sphere packing
Universally optimal codeECC
Universally optimal sphere packing
Children
\(\Lambda_{24}\) Leech lattice
The \(\Lambda_{24}\) Leech lattice is universally optimal [3].
\(E_8\) Gosset lattice
The \(E_8\) Gosset lattice is universally optimal [3].

References

[1]
H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
[2]
H. L. Montgomery, “Minimal theta functions”, Glasgow Mathematical Journal 30, 75 (1988) DOI
[3]
H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, “Universal optimality of the \(E_8\) and Leech lattices and interpolation formulas”, (2022) arXiv:1902.05438
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Zoo Code ID: univ_opt_analog

Cite as:
“Universally optimal sphere packing”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/univ_opt_analog
BibTeX:
@incollection{eczoo_univ_opt_analog, title={Universally optimal sphere packing}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/univ_opt_analog} }
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Permanent link:
https://errorcorrectionzoo.org/c/univ_opt_analog

Cite as:

“Universally optimal sphere packing”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/univ_opt_analog

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/analog/univ_opt_analog.yml.