{"paper":{"title":"First Families of Regular Polygons","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"G.H. Hughes","submitted_at":"2015-03-18T19:21:14Z","abstract_excerpt":"Every regular polygon can be regarded as a member of a well-defined 'family' of related regular polygons. These families arise naturally in the study of piecewise rotations such as outer billiards. In some cases they exist on all scales and can be used to define the fractal dimension of the 'singularity set'. This is well-documented for regular N-gons such as the pentagon, octagon and dodecagon, whose algebraic complexity is 'quadratic' (EulerPhi[N]/2 = 2).\n  Recent evidence suggests that the geometry of these families is intrinsic to the 'parent' polygon and can be derived independently of an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.05536","kind":"arxiv","version":5},"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"}