Quasi-Symmetric Nets: A Constructive Approach to the Equimodular Elliptic Type of Kokotsakis Polyhedra

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Metadata

A. Nurmatov, M. Skopenkov, F. Rist, J. Klein, and D. L. Michels.
Quasi-Symmetric Nets: A Constructive Approach to the Equimodular Elliptic Type of Kokotsakis Polyhedra.
Computer-Aided Design, Elsevier (2026).

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.

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