{"paper":{"title":"Distances from the vertices of a regular simplex","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Bach Nguyen, Mostafa Hayajneh, Mowaffaq Hajja, Shadi Shaqaqha","submitted_at":"2016-09-21T13:34:00Z","abstract_excerpt":"If $S$ is a given regular $d$-simplex of edge length $a$ in the $d$-dimensional Euclidean space $\\mathcal{E}$, then the distances $t_1$, $\\cdots$, $t_{d+1}$ of an arbitrary point in $\\mathcal{E}$ to the vertices of $S$ are related by the elegant relation $$(d+1)\\left( a^4+t_1^4+\\cdots+t_{d+1}^4\\right)=\\left( a^2+t_1^2+\\cdots+t_{d+1}^2\\right)^2.$$ The purpose of this paper is to prove that this is essentially the only relation that exists among $t_1,\\cdots,t_{d+1}.$ The proof uses tools from analysis, algebra, and geometry."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.06552","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}