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, \( x\mapsto kx \bmod 1 \) 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 \( x=\rho\,T_5(\rho) \) and \( x=\rho\,T_7(\rho) \) with \( \rho^2=x^2+y^2 \). 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 \( |\cos(\pi\log_23)| \). 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 \( \times2 \) \( \times3 \) conjecture to the Mandelbrot set, and with applications.