[2608.14442] A kinematic explanation of the 1:3 ratio for rolling spheres and the exceptional simple Lie group of rank two Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Differential Geometry arXiv:2608.14442 (math) [Submitted on 14 Aug 2026] Title: A kinematic explanation of the 1:3 ratio for rolling spheres and the exceptional simple Lie group of rank two Authors: Jacob W. Erickson View a PDF of the paper titled A kinematic explanation of the 1:3 ratio for rolling spheres and the exceptional simple Lie group of rank two, by Jacob W. Erickson View PDF HTML (experimental) Abstract: Given a pair of (round) 2-dimensional spheres, one of which has radius three times that of the other, the Lie algebra of local infinitesimal symmetries for the distribution corresponding to rolling the spheres along each other (without letting them slip or twist) happens to be isomorphic to the split real form of the exceptional simple Lie algebra of rank 2. These exceptional local symmetries appear only for this specific 1:3 ratio of radii, however, and while there are several proofs of this result, a straightforward kinematic explanation for the seemingly miraculous appearance of an exceptional simple Lie group in this situation has long been desired. In this paper, we provide such an explanation, relating the ratio of radii to the intersections of a pair of one-parameter subgroups that can be seen from the rolling sphere perspective. The approach does not require the split-octonions, as we construct the exceptional simple Lie group directly by Tanaka prolongation to aid in visualizing the underlying geometry. Comments: 42 pages, 10 figures Subjects: Differential Geometry (math.DG) ; Representation Theory (math.RT) MSC classes: 20G41, 53C30 (Primary) 22F30 (Secondary) Cite as: arXiv:2608.14442 [math.DG] (or arXiv:2608.14442v1 [math.DG] for this version) https://doi.org/10.48550/arXiv.2608.14442 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Jacob Erickson [ view email ] [v1] Fri, 14 Aug 2026 16:26:37 UTC (9,457 KB) Full-text links: Access Paper: View a PDF of the paper titled A kinematic explanation of the 1:3 ratio for rolling spheres and the exceptional simple Lie group of rank two, by Jacob W. Erickson View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2026-08 Change to browse by: math math.RT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from