Posts tagged 'golden ratio'

W.05 Spread Polynomials S_n(s) & Pentagonal Golden Spreads

The integer-coefficient spread polynomials S_0(s) = 0, S_1(s) = s, S_n(s) = 2(1 - 2s)S_{n-1}(s) - S_{n-2}(s) + 2s satisfy the semigroup composition law S_n(S_m(s)) = S_{nm}(s) over every commutative ring, and the fivefold polynomial S_5(s) = s(16s^2 - 20s + 5)^2 vanishes precisely at the Pentagonal Golden Spreads s = (5 -/+ sqrt(5))/8 = (3 - phi)/4 and (2 + phi)/4.

W.11 Isosceles, Equilateral & Kepler’s Golden Right Triangle

In a non-null triangle, Q_1 = Q_2 iff s_1 = s_2 (with apex spread s_3 = (Q_3/Q_1)(1 - Q_3/(4*Q_1))), and Q_1 = Q_2 = Q_3 iff s_1 = s_2 = s_3 = 3/4; in Kepler’s Golden Right Triangle with side lengths (1, sqrt(phi), phi), the nested radical vanishes in quadrance to give Q = (1, phi, phi^2) and exact golden spreads (s_1, s_2, s_3) = (2 - phi, phi - 1, 1) = (1/phi^2, 1/phi, 1).

W.19 Regular Stars, Golden Pentagons & Finite-Field Polygons

A regular n-star of lines has consecutive spread s satisfying S_n(s) = 0; for n = 5, a regular pentagon P_0..P_4 has side-to-diagonal spread alpha = (5 - sqrt(5))/8 = (3 - phi)/4, exterior/star spread beta = (5 + sqrt(5))/8 = (2 + phi)/4, and exact golden quadrance ratio Q_diag / Q_side = beta / alpha = phi^2 = phi + 1, existing rationally in F_p^2 whenever 5 and -1/5-dependent discriminants split (e.g. regular 5-stars in F_19).