{"id":"platonic-solids","domain":"geometry","type":"Concept","title":"Platonic Solids","slug":"platonic-solids","url":"/mathematics/geometry/platonic-solids/","summary":"Exactly 5 regular convex 3D solids exist: tetrahedron, cube, octahedron, dodecahedron, icosahedron. Proven complete in Euclid's Elements Book XIII.","the_five":["Tetrahedron (4 triangular faces)","Cube (6 square faces)","Octahedron (8 triangular faces)","Dodecahedron (12 pentagonal faces)","Icosahedron (20 triangular faces)"],"added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Heath, T.L. (1956). Euclid: The Thirteen Books of the Elements. Vol. 3. Dover."},{"tier":1,"citation":"Coxeter, H.S.M. (1973). Regular Polytopes. 3rd ed. Dover."},{"tier":2,"citation":"Field, J.V. (1988). Kepler's Geometrical Cosmology. University of Chicago Press."},{"tier":3,"citation":"Cromwell, P.R. (1997). Polyhedra. Cambridge University Press."}],"relationships":[{"type":"proven_in","target":"euclids-elements","label":"Euclid's Elements — Book XIII"},{"type":"symmetry_described_by","target":"group","label":"Group"},{"type":"connects_to","target":"topological-space","label":"Topological Space — Euler characteristic"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"2.1"}