Curve Stitching and Dancing Planets
Frances Herr
TL;DR
The paper investigates the designs formed by modular stitch graphs $\mathcal{M}(m,a)$—directed chords from $p$ to $ap\bmod m$ on $m$ circle points—by reframing the problem in a topological and dynamical setting. It introduces planet dances $\mathcal{P}(\alpha,\beta)$ and shows that $\mathcal{M}(m,a)$ is an $m$-sampling of $\mathcal{P}(1,a)$, linking discrete envelopes to classic trochoids such as epicycloids and hypocycloids via a torus-based viewpoint. The work develops a robust aliasing theory: samplings $\mathcal{S}_{m}(\alpha,\beta)$ coincide when $m=|\alpha a-\beta|$, explains when multiple planet dances yield the same drawings, and demonstrates how overlays occur when $m$ interacts with a divisor $b$, producing $d$ rotated copies of simpler envelopes. These results culminate in a systematic method to identify the most natural envelope for a given $\mathcal{M}(m,a)$ by studying a corresponding line on the flat torus $\mathbb{T}^2$ and its closest sampling to the origin. The framework connects discrete geometric artwork with continuous dynamical/topological structures, offering both theoretical insight and practical visualization guidance for educators and researchers. $$\mathcal{M}(m,a)=\mathcal{S}_{m}(1,a)$$ and the torus perspective provide a unified language for understanding designs, envelopes, and aliasing phenomena in modular stitch graphs.
Abstract
Curve stitching is a classic educational activity where one constructs elegant curves from a family of straight lines. We perform curve stitching around a circle to make a modular stitch graph. Take $m$ points equally spaced around a circle, choose an integer multiplier $a$, and draw a chord from point $p$ to $a p \mod m$. What design will appear as the envelope of these chords? We connect these discrete objects to a continuous-time dynamical system and apply a topological perspective to understand the answer to this question.