Arithmetic Envelopes on the Circle: An Introduction

Join every point of a circle to its image under an arithmetic map, and a curve appears that no single chord draws: an envelope. For the doubling and tripling maps these are the cardioid and the nephroid; more generally, \( x↦ kx 1 \) produces an epicycloid with k-1 cusps, which is similar to the main hyperbolic component of the Multibrot set of \( z^k+c \). In particular, the cardioid of the times table is, up to the factor -4/3, the main cardioid of the Mandelbrot set. We call such curves arithmetic envelopes and use them as a geometric toolbox. Labelling the odd integers by dyadic rationals identifies them with the Prüfer 2-group; in these coordinates multiplication by 3 is a piecewise linear circle homeomorphism, conjugate to the rotation by log23. Our main result determines the region left free by its chords, observed by Henkel and called the convexoid: it is convex and bounded by a segment of a diameter and by arcs of two epicycloids with cusps at 1/5 and -1/7. The pedal curve of the same family is heart-shaped and lies on the algebraic curves \( x=ρ\,T_5(ρ) \) and \( x=ρ\,T_7(ρ) \) with \( ρ^2=x^2+y^2 \). Lifted into space, the chords form ruled surfaces whose apparent contour is the envelope; on the number helix \( s=log_2x \), multiplication by 3 is a screw motion, and the chords wind around a helix of radius \( |cos(πlog_23)| \). In the Klein model all chords are hyperbolic geodesics, so the convexoid is hyperbolically convex. We close with connections to open problems, from Furstenberg's \( ×2 \) \( ×3 \) conjecture to the Mandelbrot set, and with applications.