Quasi-symmetric nets: a constructive approach to the equimodular elliptic type of Kokotsakis polyhedra

TL;DR

This paper introduces quasi-symmetric nets, a new class of flexible Kokotsakis polyhedra with equimodular elliptic type, providing explicit constructions, algebraic characterizations, and a numerical pipeline for design and validation.

Contribution

It presents the first explicit constructions and algebraic characterization of equimodular elliptic type Kokotsakis polyhedra, along with a numerical method for their design and validation.

Findings

Every elliptic QS-net is flexible in 3D Euclidean space.

Constructed examples are non-self-intersecting and of equimodular elliptic type.

The numerical pipeline effectively verifies and visualizes the polyhedra.

Abstract

This work investigates flexible Kokotsakis polyhedra with a quadrangular base of equimodular elliptic type, filling a significant gap in the literature by providing the first explicit constructions of this type together with an explicit algebraic characterization in terms of flat and dihedral angles. A straightforwardly constructible class of polyhedra - called quasi-symmetric nets (QS-nets) - is introduced, characterized by a symmetry relation among flat angles. It is shown that every elliptic QS-net has equimodular elliptic type and is flexible in real three-dimensional Euclidean space (rather than only in complex configuration spaces), except for a few exceptional choices of dihedral angles, and that its flexion admits a closed-form parameterization. Examples are constructed that are non-self-intersecting and belong exclusively to the equimodular elliptic type. To support applications in computational geometry, a numerical pipeline is developed that searches for candidate solutions, verifies them using the explicit algebraic characterization, and constructs and visualizes the resulting polyhedra; numerical validations achieve high precision. Taken together, these results provide constructive criteria, algorithms, and validated examples for the equimodular elliptic type, enabling the design of a broad range of flexible Kokotsakis mechanisms.