The Coin Rotation Paradox: Curvature Adds One Revolution — E8 Intelligence Research

Abstract

FINDING: The coin rotation paradox resolves to N = (R+r)/r revolutions for a circle of radius r rolling externally around a fixed circle of radius R — the +1 term being the hidden rotation from the path's curvature, not from spin alone. | MATH: For external rolling: revolutions = (R+r)/r = R/r + 1. For internal rolling (coin inside a ring): revolutions = (R−r)/r = R/r − 1. The general epicycloid/hypocycloid parametric form: x(θ) = (R+r)cosθ − r·cos((R+r)/r·θ), y(θ) = (R+r)sinθ − r·sin((R+r)/r·θ). The ratio R/r = k (integer) gives k cusps for the epicycloid. The curvature of the path traced by the rolling circle's center is 1/(R+r) — this curvature contributes exactly one full rotation per orbit, independent of r. | CONNECTION: The +1 term is the holonomy of parallel transport around a closed curve on a curved surface — the Gaussian curvature integrated over the enclosed area equals 2π for a sphere-like topology. The ratio R/r = 2 gives the nephroid (2 cusps); R/r = 3 gives the deltoid' Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com