{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/space-vectors-forming-rational-angles","title":"Space vectors forming rational angles","arxiv_id":"2011.14232","date":"2020-11-28","proceeding":null,"authors":["Kiran S. Kedlaya","Alexander Kolpakov","Bjorn Poonen","Michael Rubinstein"],"abstract":"We classify all sets of nonzero vectors in $\\mathbb{R}^3$ such that the angle formed by each pair is a rational multiple of $\\pi$. The special case of four-element subsets lets us classify all tetrahedra whose dihedral angles are multiples of $\\pi$, solving a 1976 problem of Conway and Jones: there are $2$ one-parameter families and $59$ sporadic tetrahedra, all but three of which are related to either the icosidodecahedron or the $B_3$ root lattice. The proof requires the solution in roots of unity of a $W(D_6)$-symmetric polynomial equation with $105$ monomials (the previous record was $12$ monomials).","url_abs":"https://arxiv.org/abs/2011.14232v1","url_pdf":"https://arxiv.org/pdf/2011.14232v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"space-vectors-forming-rational-angles","repo_url":"https://github.com/kedlaya/tetrahedra","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}