Classification of flexible Kokotsakis polyhedra with quadrangular base

TL;DR

This paper provides a complete classification of flexible Kokotsakis polyhedra with quadrangular bases by analyzing dihedral angles and elliptic curve diagrams, advancing understanding of their flexibility conditions.

Contribution

It introduces a comprehensive classification framework for these polyhedra based on Euler-Chasles correspondence and elliptic curve diagrams.

Findings

A complete classification of flexible quadrangular Kokotsakis polyhedra.

Flexibility characterized by commuting diagrams of elliptic curves.

New connections between polyhedral geometry and algebraic curves.

Abstract

A Kokotsakis polyhedron with quadrangular base is a neighborhood of a quadrilateral in a quad surface. Generically, a Kokotsakis polyhedron is rigid. Up to now, several flexible classes were known, but a complete classification was missing. In the present paper, we provide such a classification. The analysis is based on the fact that the dihedral angles of a Kokotsakis polyhedron are related by an Euler-Chasles correspondence. It results in a diagram of elliptic curves covering complex projective planes. A polyhedron is flexible if and only if this diagram commutes.