About

Polyhedra in Polyhedra is an open record book for one family of questions: how small can a polyhedron be and still hold n copies of a polyhedron? It holds the best packings found so far, each with a proof that it is valid, and it is open to better ones from anyone.

How it started

In October 2026 Yohei Nakajima packed 12 unit cubes in a cube of side 2.9315 with a soft-to-rigid method: start every cube as a ball and sharpen it into a cube under pressure. The project began by generalizing that method from cubes to all five Platonic solids, as pieces and as containers (9–11 October 2026, nine problems), with every packing certified in exact arithmetic. On 12 October it grew to every Archimedean and Catalan solid packed in itself and to regular prisms and antiprisms packed in themselves and in a cube, which needed a new soft shape (the r-rounding) for solids without an insphere and a certifier that works in any number field. The search ran in the cloud, on a laptop and on GitHub Actions until every one of the 2,145 cases had a certified packing.

The full account of the Platonic phase, including every idea that did not work, is in the final report.

People and credit

• Alejandro Zarzuelo (@alejandrozu) runs the project and the site, and is credited for the packings found by the search.
• Yohei Nakajima created the soft-to-rigid method and the original simulator, tightening and checker (soft-to-rigid-packing, MIT).
• The generalization to other solids, the search engine and the certifiers were developed with Claude (Anthropic) under Alejandro Zarzuelo's direction.
• Previously published values come from Erich Friedman's Packing Center, with packings by E. Friedman, R. Walsh, H. Lin and Y. Nakajima; published coordinates from Tencent Hunyuan's Hyra-results are used as starting points and as test cases.
• Contributors whose submissions become records are credited in the record file and on this site.

How to cite

@misc{polyhedrainpolyhedra,
  author       = {Zarzuelo, Alejandro},
  title        = {Polyhedra in Polyhedra: certified packings of n copies of a polyhedron in the smallest container},
  year         = {2026},
  howpublished = {\url{https://polyhedrainpolyhedra.wiki}},
  note         = {Records, certificates and code: \url{https://github.com/alejandrozu/platonic-packing}}
}

To cite one packing, give its record id (for example ktoinkto_n20) and the date of the record file.

License

Code and data are under the MIT license, which keeps Yohei Nakajima's copyright notice for the method, simulator, tightening and checker design.

Questions and corrections

Open an issue on GitHub. To submit a packing or propose a new category, use the contribute page.