Public index · Mathematics
Triangle · heptagon · 42-gon
3, 7, 42
Unique maximal r in 1/p + 1/q + 1/r = 1/2 with p ≤ q ≤ r: (3, 7, 42). Same arithmetic as 1 − 1/2 − 1/3 − 1/7 = 1/42, the Hurwitz deficit.
An equilateral triangle, a seven-sided tile, and a forty-two-sided tile can close around a point with no gap. Forty-two is the most sides any of the three can have. Turn the same fractions over and the leftover is 1/42. That leftover is why a curved surface can have at most 84(g − 1) symmetries. 84 is twice 42. The tiles did not hire Adams.
Three tiles. One point.
What it supports
Ten integer triples p ≤ q ≤ r satisfy 1/p + 1/q + 1/r = 1/2 (OEIS A229941). Each is three regular polygons meeting in the plane. The largest r is 42: 1/3 + 1/7 + 1/42 = 1/2. Interior angles: 60° + 128.571…° + 171.429…° = 360°. Independently, the closest 1/a + 1/b + 1/c can sit under 1 is 1/2 + 1/3 + 1/7, short by 1/42. Hurwitz (1893): a compact Riemann surface of genus g ≥ 2 has at most 84(g − 1) holomorphic automorphisms, because 4(g − 1) / (1/42) = 168(g − 1) with reflections, 84(g − 1) without. Baez (2015) joins the two faces for a general reader. Euclidean closure and hyperbolic deficit share a fraction. They are not one law.
What it does not
Not Adams. Not a sky 42-gon. Not the Coxeter graph’s 42 edges (28 × 3 / 2). Plane meeting and Hurwitz bound are two readings of one arithmetic, not a proof that 42 was chosen for a joke.
Calculation
1/3 + 1/7 + 1/42 = 1/2. 1 − 1/2 − 1/3 − 1/7 = 1/42. |Aut| ≤ 84(g − 1) = 2 × 42(g − 1).
Regular n-gon interior angle (1 − 2/n)π. Three around a point: sum of reciprocals of side-counts = 1/2. Hyperbolic (2,3,7) triangle: deficit 1/42.
ExactnessExact
Form42
Unitsides
Strengthhard
When1893 · 2015
Hung12 September 2026