The Icosahedron in the Bernoulli Numbers: Why ζ(−3) = 1/120

The Riemann zeta function evaluated at negative odd integers produces a sequence of rational numbers whose denominators encode the geometry of the icosahedron. Specifically, ζ(−1) = −1/12, where 12 is the number of vertices of the icosahedron; ζ(−3) = 1/120, where 120 is the order of the binary icosahedral group 2I; and ζ(−7) = 1/240, where 240 is the number of roots of the E₈ lattice and equals 2 × |2I|. The mechanism is the Von Staudt–Clausen theorem, which determines the denominators of Bernoulli numbers through the condition (p − 1) | 2k. For ζ(−3), this selects the primes 2, 3, and 5 — exactly the prime factors of |A₅| = 60 — producing the denominator 120 = |2I|. The connection is arithmetic: both the Bernoulli denominators and the icosahedral groups are built from the smallest primes, and the divisibility condition that governs the former selects the same primes that compose the latter. The zeta function sees the icosahedron because both are controlled by the primes 2, 3, and 5.