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

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.

@Article{Nurmatov2026_quasi_symmetric_nets,
	author   = {Nurmatov, Abdukhomid and Skopenkov, Mikhail and Rist, Florian and Klein, Jonathan and Michels, Dominik L.},
	title    = {Quasi-Symmetric Nets: A Constructive Approach to the Equimodular Elliptic Type of Kokotsakis Polyhedra},
	journal  = {arXiv},
	year     = {2025},
	month    = {12}
	doi      = {arXiv:2511.19376},
}