Curve Stitching and Dancing Planets

TL;DR

This paper explores the geometric and topological properties of curve stitching around a circle, linking discrete constructions to continuous dynamical systems to understand the resulting envelope patterns.

Contribution

It introduces a novel connection between modular stitch graphs and continuous dynamical systems, providing a topological framework for analyzing curve stitching patterns.

Findings

Identification of envelope patterns from modular stitch graphs

Connection between discrete geometry and continuous dynamical systems

Topological analysis of curve stitching configurations

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.