{"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/towards-a-geometric-understanding-of-the-4","title":"Towards a Geometric Understanding of the 4-Dimensional Point Groups","arxiv_id":"2205.04965","date":"2022-05-10","proceeding":null,"authors":["Laith Rastanawi","Günter Rote"],"abstract":"We classify the finite groups of orthogonal transformations in 4-space, and we study these groups from the viewpoint of their geometric action, using polar orbit polytopes. For one type of groups (the toroidal groups), we develop a new classification based on their action on an invariant torus, while we rely on classic results for the remaining groups. As a tool, we develop a convenient parameterization of the oriented great circles on the 3-sphere, which leads to (oriented) Hopf fibrations in a natural way.","url_abs":"https://arxiv.org/abs/2205.04965v1","url_pdf":"https://arxiv.org/pdf/2205.04965v1.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":"towards-a-geometric-understanding-of-the-4","repo_url":"https://github.com/laisrast/point-groups","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}