Septic equations are solvable by 2-fold origami

TL;DR

This paper proves that certain seventh-degree rational equations, including angle septisection, can be solved using 2-fold origami, extending previous work and providing explicit crease patterns for complex algebraic solutions.

Contribution

It demonstrates that 2-fold origami can solve seventh-degree rational equations and constructs explicit crease patterns for specific algebraic solutions.

Findings

Seventh-degree rational equations are solvable by 2-fold origami.

Exact crease patterns are provided for polynomials with Galois groups A_7 and PSL_3F_2.

Angle septisection is achievable through this origami method.

Abstract

In this paper we prove that a generic rational equation of degree $7$ is solvable by 2-fold origami. In particular we show how to septisect an arbitrary angle. This extends the work of Alperin & Lang and Nishimura on 2-fold origami. Furthermore we give exact crease patterns for folding polynomials with Galois groups $A_7$ resp. $PSL_3\mathbb{F}_2$.