The Tammes problem and the table of spherical codes

How should $N$ points be placed on a sphere so that the closest pair is as far apart as possible? On the ordinary sphere this is the Tammes problem, solved only for $N \le 14$ and $N = 24$. Henry Cohn's table lists the best known configurations in dimensions 3 to 32 with up to 1024 points; any entry can be improved by a configuration that is checked in milliseconds.

Problem

A spherical code is a finite set $C$ of unit vectors in $\mathbb{R}^n$; its quality is the largest inner product between distinct points,

$$s(C) = \max_{x \ne y \in C} \langle x, y \rangle,$$

the cosine of its minimal angle. For each dimension $n$ and size $N$, the problem asks for the least possible $s(C)$ over all codes of $N$ points. In dimension 3 this is the Tammes problem, which Tammes raised in 1930 in a botanical study of the exit pores on pollen grains.

Task: find a code with a smaller maximal cosine than an entry of the table, or prove that an entry is optimal; the smallest open case of the Tammes problem is $N = 15$.

Background

Optimal codes with at most $2n$ points in $n$ dimensions are classical (Rankin, 1955), and the table omits them. Beyond that, optimality is known only in special cases: highly symmetric configurations such as the 24-cell, the 600-cell, the $E_8$ root system and the minimal vectors of the Leech lattice, and the small cases of the Tammes problem, settled by long geometric and computer-assisted arguments. The kissing problem is the special case of codes with minimal angle $60^\circ$.

What is known

• Tammes problem, proved cases: $N = 7, 8, 9$ (Schütte and van der Waerden, 1951); $N = 10, 11$ (Danzer, 1986); $N = 12$ (Fejes Tóth, 1943); $N = 13$ (Musin and Tarasov, 2012); $N = 14$ (Musin and Tarasov, 2015); $N = 24$ (Robinson, 1961). For $N = 15$ the best known code, due to Schütte and van der Waerden, has maximal cosine $0.592605902926$.
• Higher dimensions: the table marks only a few dozen of its roughly 16,000 entries as proved optimal, for example 10, 24 and 120 points in dimension 4, 240 points in dimension 8 and 196560 points in dimension 24.
• State of the table: most entries are numerically optimized, and thousands are explicitly labeled as needing more optimization. Recent additions include codes found inside shells of integer lattices (Ganzhinov and Östergård, 2024) and highly symmetric line systems (Ganzhinov, 2025).

What would count as progress

• A code that beats a table entry. Given coordinates, the maximal inner product is computed in milliseconds; small improvements should be confirmed with exact or interval arithmetic. Entries labeled as needing more optimization are the easiest targets.
• An exact description of a numerical entry (for example the minimal polynomial of its cosine) together with a proof that such a code exists.
• A proof of optimality for $N = 15$ points on the sphere, for example by extending Musin and Tarasov's computer-assisted method for $N = 13$ and $14$.

References

1. K. Schütte and B. L. van der Waerden, "Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?", Mathematische Annalen 123 (1951), 96–124. doi:10.1007/BF02054944
2. R. M. Robinson, "Arrangement of 24 points on a sphere", Mathematische Annalen 144 (1961), 17–48. doi:10.1007/BF01396539
3. L. Danzer, "Finite point-sets on $S^2$ with minimum distance as large as possible", Discrete Mathematics 60 (1986), 3–66. doi:10.1016/0012-365X(86)90002-6
4. O. R. Musin and A. S. Tarasov, "The strong thirteen spheres problem", Discrete & Computational Geometry 48 (2012), 128–141. doi:10.1007/s00454-011-9392-2
5. O. R. Musin and A. S. Tarasov, "The Tammes problem for N = 14", Experimental Mathematics 24 (2015), 460–468. doi:10.1080/10586458.2015.1022842

Source

Imported from Table of spherical codes, maintained by Henry Cohn, curated by Henry Cohn. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.