$N$-Division Points of Hypocycloids

TL;DR

This paper investigates the constructibility of n-division points on hypocycloids, proving all rational hypocycloids' n-division points are constructible with straightedge and compass given a hypocycloid, and analyzing the special case of tricuspoids.

Contribution

It establishes the constructibility of n-division points for all rational hypocycloids and identifies which division points of tricuspoids are constructible without a pre-drawn hypocycloid.

Findings

All rational hypocycloids' n-division points are constructible with a pre-drawn hypocycloid.

Only 1, 2, 3, and 6-division points of a tricuspoid are constructible without a pre-drawn hypocycloid.

Constructibility depends on the specific hypocycloid and the division number.

Abstract

We show that the $n$-division points of all rational hypocycloids are constructible with an unmarked straightedge and compass for all integers $n$, given a pre-drawn hypocycloid. We also consider the question of constructibility of $n$-division points of hypocycloids without a pre-drawn hypocycloid in the case of a tricuspoid, concluding that only the $1$, $2$, $3$, and $6$-division points of a tricuspoid are constructible in this manner.