Generalizations of Bricard octahedra

TL;DR

This paper introduces five new classes of flexible polyhedra derived from Bricard octahedra, featuring self-intersections, zero volume, and non-constant dihedral angles, expanding understanding of flexible polyhedral structures.

Contribution

It generalizes Bricard octahedra to create new flexible polyhedra with unique properties, including genus 0 and indefinite size, broadening the scope of flexible polyhedral research.

Findings

Polyhedra are flexible and self-intersecting.

They have zero oriented volume during flexion.

Smallest examples are dodecahedra with eight vertices.

Abstract

We construct five types of polyhedra by generalizing the description of Bricard octahedra and applying the generalizations to polyhedral suspensions. The resulting polyhedra are flexible, are of genus 0, exhibit self-intersections, have zero oriented volume, have dihedral angles all of which are non-constant during flexion and are of indefinite size, the smallest of which are dodecahedra with eight vertexes.