{"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/190412761","title":"The graphs behind Reuleaux polyhedra","arxiv_id":"1904.12761","date":"2019-04-29","proceeding":null,"authors":["Luis Montejano","Eric Pauli","Miguel Raggi","Edgardo Roldán-Pensado"],"abstract":"This work is about graphs arising from Reuleaux polyhedra. Such graphs must necessarily be planar, $3$-connected and strongly self-dual. We study the question of when these conditions are sufficient. If $G$ is any such a graph with isomorphism $\\tau : G \\to G^*$ (where $G^*$ is the unique dual graph), a metric mapping is a map $\\eta : V(G) \\to \\mathbb R^3$ such that the diameter of $\\eta(G)$ is $1$ and for every pair of vertices $(u,v)$ such that $u\\in \\tau(v)$ we have dist$(\\eta(u),\\eta(v)) = 1$. If $\\eta$ is injective, it is called a metric embedding. Note that a metric embedding gives rise to a Reuleaux Polyhedra. Our contributions are twofold: Firstly, we prove that any planar, $3$-connected, strongly self-dual graph has a metric mapping by proving that the chromatic number of the diameter graph (whose vertices are $V(G)$ and whose edges are pairs $(u,v)$ such that $u\\in \\tau(v)$) is at most $4$, which means there exists a metric mapping to the tetrahedron. Furthermore, we use the Lov\\'asz neighborhood-complex theorem in algebraic topology to prove that the chromatic number of the diameter graph is exactly $4$. Secondly, we develop algorithms that allow us to obtain every such graph with up to $14$ vertices. Furthermore, we numerically construct metric embeddings for every such graph. From the theorem and this computational evidence we conjecture that every such graph is realizable as a Reuleaux polyhedron in $\\mathbb R^3$. In previous work the first and last authors described a method to construct a constant-width body from a Reuleaux polyhedron. So in essence, we also construct hundreds of new examples of constant-width bodies. This is related to a problem of V\\'azsonyi, and also to a problem of Blaschke-Lebesgue.","url_abs":"http://arxiv.org/abs/1904.12761v1","url_pdf":"http://arxiv.org/pdf/1904.12761v1.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":"190412761","repo_url":"https://github.com/mraggi/ReuleauxPolyhedra","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}