# P2714: Tammes separation for fifteen points on the sphere

- ID: `P2714`
- Reference: `tammes-fifteen-separation`
- Page: https://theoremdb.org/statements/P2714
- Record maturity: Reviewed problem with recorded work

## Problem

Place \(15\) points on the unit sphere \(S^2\). Determine the largest possible value \(\theta_{15}\) of the minimum geodesic angle between two points.

### Problem setup

- **Definition.** Geodesic angle is the central angle arccos(x dot y), taking values in the closed interval [0,pi].
- **Convention.** Configurations are identified under orthogonal transformations and relabeling.

### What counts as a solution

- Give a fifteen-point configuration with a certified minimum angle theta and a complete contact-graph, interval branch, or spherical-code bound proving that no configuration has larger minimum angle.

## Status

With \(\alpha\) the isolated root near \(0.5926059029250737\) of \(13x^5-x^4+6x^3+2x^2-3x-1\), the checked bounds are \(\arccos(\alpha)\le\theta_{15}\le2\arccos(13/15)\), or \(53.65785012993268\ldots^\circ\) through \(59.85286973322849\ldots^\circ\); global optimality of the lower construction remains open. [2](#reference-2) [3](#reference-3) [4](#reference-4)

## Work

### Evidence for the current status

**Proposition 1 (Global optimality for fifteen points remains open).** With \(\alpha\) the isolated root near \(0.5926059029250737\) of \(13x^5-x^4+6x^3+2x^2-3x-1\), the checked bounds are \(\arccos(\alpha)\le\theta_{15}\le2\arccos(13/15)\), or \(53.65785012993268\ldots^\circ\) through \(59.85286973322849\ldots^\circ\); global optimality of the lower construction remains open.

The current spherical-code table marks proved optima with an asterisk. Its 15-point row has no asterisk. Kottwitz's primary paper describes its solutions for 15 through 90 circles as conjectured solutions. Musin and Tarasov later proved the 14-point case by enumerating irreducible contact graphs, and their proof makes no claim for 15 points.

The strongest construction checked here and the elementary cap-area bound give
\[
\arccos(\alpha)\leq\theta_{15}\leq2\arccos(13/15),
\]
or numerically
\[
53.65785012993268\ldots^\circ
\leq\theta_{15}\leq
59.85286973322849\ldots^\circ.
\]
For the upper bound, place disjoint caps of angular radius \(\theta_{15}/2\) around the points. Each cap has area \(2\pi(1-\cos(\theta_{15}/2))\). Their total area is at most \(4\pi\), so \(15(1-\cos(\theta_{15}/2))\leq2\).

### Background and intake notes

The checked coordinate set gives theta_15 at least 53.657850116 degrees. Disjoint spherical caps give the elementary upper bound 2 arccos(13/15), about 59.852870 degrees.

- Original intake status: A dated check on 2026-07-24 found a proof for the Tammes problem through fourteen points at https://arxiv.org/abs/1410.2536 and a putatively optimal fifteen-point separation of 53.6578501 degrees in Sloane's table. No proof of optimality for fifteen points was established by the checked sources.
- Enumerate possible contact graphs and combine interval coordinates with spherical stress or linear-programming bounds.
- A numerically rigid configuration can still be a local maximum. Force balance and a positive stress do not by themselves exclude a different contact graph with larger separation.
- All pairwise angles need outward-rounded interval bounds; decimal coordinates alone do not certify the incumbent endpoint.

- Recorded example: The public incumbent is the 45-coordinate file linked in provenance, interpreted as fifteen consecutive unit vectors.

### Other known results

- **Proposition 2** (supported): A 15-point code exists with minimum angle arccos(alpha), where alpha is the isolated root 0.5926059029250737... of a degree-five polynomial. [3](#reference-3)

### Runnable artifacts

- **Artifact 1** (reproduced): Exact integer comparisons certify all 105 normalized dot products and recover the 30-edge contact graph. [1](#reference-1)

### Computational notes

- The coordinate file contains fifteen vectors with squared norms between 1 and 1.0000000000000002. Direct evaluation of all 105 pairs found minimum angle 53.65785011616207 degrees; the downloaded bytes have SHA-256 d1a1d120faf0a7a69441fb8a42ea741da2209b5bd2c45819d16648a49fe0b10f. The spherical-cap area calculation independently gave theta_15<=2 arccos(13/15).

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `tammes-fifteen-separation`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>R. H. Hardin, N. J. A. Sloane, and W. D. Smith, Tables of Spherical Codes, dimension 3, 15-point coordinate file (checked 26 July 2026). The complete 15-vector coordinate file; source-file SHA-256 is recorded in the packet https://neilsloane.com/packings/dim3/pack.3.15.txt
   - Also cited at Hardin-Sloane-Smith dimension-three packing library, 15 consecutive vectors. Source-file SHA-256 recorded by the candidate audit: d1a1d120faf0a7a69441fb8a42ea741da2209b5bd2c45819d16648a49fe0b10f.
   - dataset; dataset source; Source-file SHA-256 d1a1d120faf0a7a69441fb8a42ea741da2209b5bd2c45819d16648a49fe0b10f; checked 2026-07-25
   - Source use: citation_only
   - Supplies the 15 coordinates whose pairwise products certify the stated angular-separation lower bound.
   - For Tammes separation for fifteen points on the sphere: Exact integer comparisons certify all 105 normalized dot products and recover the 30-edge contact graph.
2. <a id="reference-2"></a>D. A. Kottwitz, The densest packing of equal circles on a sphere, Acta Crystallographica Section A 47(3) (1991), 158-165. Abstract and the 15-circle result https://doi.org/10.1107/S0108767390011370
   - Also cited at Abstract and 15-point construction in Kottwitz (1991)
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Reports the conjectured 15-circle arrangement and its separation value.
3. <a id="reference-3"></a>Henry Cohn, Spherical Codes, author-maintained table, dimension 3 and 15 points (checked 26 July 2026). Dimension 3, 15 points. The entry gives cosine 0.592605902926 without an optimality asterisk and links coordinates. The table introduction states that a listed minimal polynomial certifies existence of an exact code. The plain-text polynomial table gives 13x^5-x^4+6x^3+2x^2-3x-1. https://www.spherical-codes.org/
   - Also cited at Dimension 3, 15 points; no optimality asterisk
   - reference_database; reference source; web version checked 2026-08-01; checked 2026-07-25
   - Source use: citation_only
   - Records the 15-point value and its minimal polynomial without marking the construction globally optimal.
   - Source named by the research packet.
4. <a id="reference-4"></a>Oleg R. Musin and Alexey S. Tarasov, The Tammes problem for N=14, arXiv:1410.2536v2 (2014). Musin and Tarasov, proof for N=14 by contact-graph enumeration https://arxiv.org/abs/1410.2536
   - preprint; reference source; arXiv:1410.2536v2; checked 2026-07-25
   - Source use: citation_only
   - Provides the contact-graph proof method at 14 points, a benchmark that does not settle the 15-point case.