{"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/the-farthest-point-map-on-the-regular-1","title":"The Farthest Point Map on the Regular Dodecahedron","arxiv_id":"2104.02567","date":"2021-04-06","proceeding":null,"authors":["Richard Evan Schwartz"],"abstract":"Let $X$ be the regular dodecahedron, equipped with its intrinsic path metric. Given $p \\in X$ let $G(p)=-q$ where $q$ is the point on $X$ which maximizes the distance to $p$. (Generically, $G$ is single-valued.) We give a complete description of the map $G$ and as a consequence show that the $\\omega$-limit set of $G$ is the $1$-skeleton of a subdivision of $X$ into $180$ convex quadrilaterals. $G$ is a piecewise bi-quadratic map, and each algebraic piece is defined by a straight line construction involving a rhombus. The rhombi involved have the same shapes as the ones in the Penrose tiling. Our proof is computer-assisted but rigorous.","url_abs":"https://arxiv.org/abs/2104.02567v1","url_pdf":"https://arxiv.org/pdf/2104.02567v1.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":"the-farthest-point-map-on-the-regular-1","repo_url":"https://github.com/RichardEvanSchwartz/Dodecahedron","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}