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

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.