{"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/a-formula-for-the-area-of-a-triangle-useless","title":"A formula for the area of a triangle: Useless, but explicitly in Deep Sets form","arxiv_id":"2503.22786","date":"2025-03-28","proceeding":null,"authors":["Connor Hainje","David W. Hogg"],"abstract":"Any permutation-invariant function of data points $\\vec{r}_i$ can be written in the form $\\rho(\\sum_i\\phi(\\vec{r}_i))$ for suitable functions $\\rho$ and $\\phi$. This form - known in the machine-learning literature as Deep Sets - also generates a map-reduce algorithm. The area of a triangle is a permutation-invariant function of the locations $\\vec{r}_i$ of the three corners $1\\leq i\\leq 3$. We find the polynomial formula for the area of a triangle that is explicitly in Deep Sets form. This project was motivated by questions about the fundamental computational complexity of $n$-point statistics in cosmology; that said, no insights of any kind were gained from these results.","url_abs":"https://arxiv.org/abs/2503.22786v1","url_pdf":"https://arxiv.org/pdf/2503.22786v1.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":"a-formula-for-the-area-of-a-triangle-useless","repo_url":"https://github.com/davidwhogg/TriangleAreas","is_official":1,"mentioned_in_paper":0,"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}