# Division of an angle into equal parts and construction of regular polygons by multi-fold origami

## TL;DR

This paper explores how multi-fold origami can be used to divide angles into equal parts and construct regular polygons, extending classical geometric methods by allowing multiple simultaneous folds.

## Contribution

It introduces new conditions under which arbitrary angles can be m-sected and regular polygons constructed using multi-fold origami, based on prime factor constraints.

## Key findings

- Any angle can be m-sected if the largest prime factor of m is ≤ n+2.
- Regular m-gons can be constructed if the largest prime factor of φ(m) is ≤ n+2.
- Multi-fold origami extends classical geometric constructibility.

## Abstract

This article analyses geometric constructions by origami when up to $n$ simultaneous folds may be done at each step. It shows that any arbitrary angle can be $m$-sected if the largest prime factor of $m$ is $p\le n+2$. Also, the regular $m$-gon can be constructed if the largest prime factor of $\phi(m)$ is $q\le n+2$, where $\phi$ is Euler's totient function.