Spreading points evenly: the max-to-min distance ratio

Place $n$ points so that the largest distance between two of them is as small as possible compared with the smallest distance. In the plane the optimal ratio is known only for $n \le 8$; beyond that, and in space, the best configurations come from numerical search, and any new configuration is checked in microseconds.

Problem

For distinct points $x_1, \dots, x_n \in \mathbb{R}^d$ let

$$R(x_1, \dots, x_n) = \frac{\max_{i < j} \|x_i - x_j\|}{\min_{i < j} \|x_i - x_j\|},$$

and let $C(d, n)$ be the minimum of $R$ over all configurations of $n$ points. Establish upper and lower bounds for $C(d, n)$, and find the optimal configurations, for $d = 2$ and $d = 3$.

Background

The question goes back to Bateman and Erdős (1951), who studied extremal point sets suggested by a lemma of Besicovitch and settled seven points in the plane. Equivalently, one asks for the smallest diameter of $n$ points with minimum distance 1, a basic packing-type question whose asymptotic answer is governed by the triangular lattice. Exact values beyond tiny cases need delicate geometric arguments, so most of the table consists of numerical candidates found by global optimization.

What is known

• Plane, exact values: $C(2,n) = 1$ for $n = 2, 3$; $C(2,4) = \sqrt{2}$; $C(2,5) = (1+\sqrt{5})/2$; $C(2,6) = 2\sin 72^\circ$; $C(2,7) = 2$ (Bateman and Erdős, 1951); $C(2,8) = (2\sin(\pi/14))^{-1}$ (Bezdek and Fodor, 1999).
• Plane, numerical: candidates for $9 \le n \le 30$ by Rechenberg (2006), Cantrell (2009) and Audet, Fournier, Hansen and Messine (2010). For $n = 16$, Georgiev, Gómez-Serrano, Tao and Wagner (2025) found $C(2,16) \le \sqrt{12.889266112}$; Berthold (2025, with the FICO Xpress global solver) lowered this to $\sqrt{12.889246743}$ and Li et al. (FM Agent, 2025) to $\sqrt{12.889230201} \approx 3.590157$.
• Space: $C(3, n) = 1$ for $n \le 4$; numerical candidates for $5 \le n \le 21$ (Cantrell, 2009). Georgiev et al. improved $n = 14$ to $C(3,14) \le \sqrt{4.165849767}$, and Berthold (2025) lowered it to $\sqrt{4.165783989} \approx 2.041025$.

What would count as progress

• A configuration with a smaller ratio than the best known for some $(d, n)$. Checking it means computing $n(n-1)/2$ distances; since optimal configurations have many equal distances, give coordinates with enough digits (or exactly) that the ratio can be confirmed in interval arithmetic.
• A proof of optimality for $n = 9$ in the plane, the smallest open case there, or for small $n$ in space; a computer-assisted proof should come with certificates that can be rerun.

References

1. P. Bateman and P. Erdős, "Geometrical extrema suggested by a lemma of Besicovitch", American Mathematical Monthly 58 (1951), 306–314. doi:10.1080/00029890.1951.11999682
2. A. Bezdek and F. Fodor, "Minimal diameter of certain sets in the plane", Journal of Combinatorial Theory, Series A 85 (1999), 105–111. doi:10.1006/jcta.1998.2889
3. C. Audet, X. Fournier, P. Hansen and F. Messine, "A note on diameters of point sets", Optimization Letters 4 (2010), 585–595. doi:10.1007/s11590-010-0185-y
4. B. Georgiev, J. Gómez-Serrano, T. Tao and A. Z. Wagner, "Mathematical exploration and discovery at scale" (2025), Problem 6.50. arXiv:2511.02864

Source

Imported from AlphaEvolve Repository of Problems, problem 50, curated by Google DeepMind. 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.