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

Abstract

A Kokotsakis polyhedron is a polyhedral mesh in three-dimensional Euclidean space formed by a central n-gonal face (the base), n quadrilateral faces each sharing one edge with the base, and n triangular faces inserted between every two adjacent quadrilaterals; it is called flexible if it admits a continuous deformation that preserves the rigidity of every face. This work investigates flexible Kokotsakis polyhedra with a quadrangular base (n = 4) 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.