{"id":"golden-ratio","domain":"geometry","type":"Constant","title":"Golden Ratio","slug":"golden-ratio","url":"/mathematics/geometry/golden-ratio/","value":"1.6180339887...","summary":"phi=(1+sqrt5)/2, satisfies phi^2=phi+1. Genuine appearances in pentagons, Fibonacci sequence, phyllotaxis. Many popular claims (pyramids, Parthenon) are exaggerated (Markowsky 1992).","formal_definition":"Root of x^2-x-1=0","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Markowsky, G. (1992). Misconceptions about the golden ratio. The College Mathematics Journal, 23(1), 2-19."},{"tier":1,"citation":"Herz-Fischler, R. (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press."},{"tier":2,"citation":"Livio, M. (2002). The Golden Ratio. Broadway Books."},{"tier":3,"citation":"Huntley, H.E. (1970). The Divine Proportion. Dover."}],"relationships":[{"type":"related_to","target":"irrational-number","label":"Irrational Number — most irrational number"},{"type":"root_of","target":"polynomial","label":"Polynomial — x²−x−1=0"},{"type":"built_on","target":"euclidean-geometry","label":"Euclidean Geometry — pentagon"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.7"}