How Much String to String a Cardioid?

TL;DR

This paper derives a simple closed-form formula for the total length of line segments in a geometric design based on residue points on a circle, facilitating easy calculation of string length needed for such constructions.

Contribution

It provides a novel closed-form expression for the sum of segment lengths in residue-based cardioid-like designs, simplifying practical implementation.

Findings

Closed-form formula for total segment length

Simplifies calculation of string length for geometric designs

Connects residue constructions with epicycloids

Abstract

A residue design is an artistic geometric construction in which we have $n$ equally-spaced points on a circle numbered 0 through $n-1$ and we join with a line segment each point $k$ to $ak$ modulo $n$ for some fixed $a\ge 2.$ The envelopes of these lines are epicycloids, like cardioids. In this note, we prove that the sum of the lengths of these line segments has a surprisingly simple closed form. In particular, if one wants to make one of these designs with string, it is easy to calculate how much string is required.