Arithmetic Envelopes on the Circle: An Introduction

We study envelopes of families of chords generated by arithmetic maps on the unit circle, which we call arithmetic envelopes. For the doubling and tripling maps these are the classical cardioid and nephroid, which are also caustics of a circular mirror; more generally, the map \( x\mapsto kx \bmod 1 \) produces an epicycloid with k-1 cusps. We then change the coordinate system on the circle by labelling the odd integers with dyadic rationals, which identifies them with the Prüfer 2-group, and we recall its group law from earlier work and describe it by a simple rule on binary digits. In these coordinates multiplication by 3 becomes a piecewise linear circle homeomorphism, conjugate to the rotation by \( \log_2 3 \), and its chords leave a region in the centre of the circle free. We prove that this region, observed by Henkel and called the convexoid, 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 \( \rho^2=x^2+y^2 \), where T_n denotes the n-th Chebyshev polynomial. We compute these and further envelopes explicitly with Gr\"obner bases and resultants, and we include code for all computations; an appendix discusses the tropicalization of the resulting curves. We close with questions on how such pictures might help to visualize problems such as Furstenberg's \( \times2 \) \( \times3 \) conjecture.