{"paper":{"title":"Cayley-Dickson Algebras and Finite Geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.RA"],"primary_cat":"math.CO","authors_text":"Frederic Holweck, Metod Saniga, Petr Pracna","submitted_at":"2014-05-27T12:46:48Z","abstract_excerpt":"Given a $2^N$-dimensional Cayley-Dickson algebra, where $3 \\leq N \\leq 6$, we first observe that the multiplication table of its imaginary units $e_a$, $1 \\leq a \\leq 2^N -1$, is encoded in the properties of the projective space PG$(N-1,2)$ if one regards these imaginary units as points and distinguished triads of them $\\{e_a, e_b, e_c\\}$, $1 \\leq a < b <c \\leq 2^N -1$ and $e_ae_b = \\pm e_c$, as lines. This projective space is seen to feature two distinct kinds of lines according as $a+b = c$ or $a+b \\neq c$. Consequently, it also exhibits (at least two) different types of points in dependence"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.6888","kind":"arxiv","version":2},"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"}