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

A. Nurmatov, M. Skopenkov, F. Rist, J. Klein, D. L. Michels

TL;DR

This work provides the first explicit constructive realizations of Kokotsakis polyhedra of the equimodular elliptic type via the novel QS-net class, establishing that elliptic QS-nets are flexible in real 3D space with a closed-form flexion. It offers a precise algebraic characterization linking flat and dihedral angles through a structured parametric system, and develops a robust numerical-constructive pipeline to generate, verify, and visualize these mechanisms with high precision. The results enable direct CAD/inverse-design use, including closed-form and numerically discovered examples, and demonstrate exclusivity to the equimodular elliptic class even under boundary-strip switching. Together, these contributions close a gap in Izmestiev’s classification and provide practical tools for designing flexible Kokotsakis mechanisms for architecture, robotics, and deployable systems.

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.

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

TL;DR

This work provides the first explicit constructive realizations of Kokotsakis polyhedra of the equimodular elliptic type via the novel QS-net class, establishing that elliptic QS-nets are flexible in real 3D space with a closed-form flexion. It offers a precise algebraic characterization linking flat and dihedral angles through a structured parametric system, and develops a robust numerical-constructive pipeline to generate, verify, and visualize these mechanisms with high precision. The results enable direct CAD/inverse-design use, including closed-form and numerically discovered examples, and demonstrate exclusivity to the equimodular elliptic class even under boundary-strip switching. Together, these contributions close a gap in Izmestiev’s classification and provide practical tools for designing flexible Kokotsakis mechanisms for architecture, robotics, and deployable systems.

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.

Paper Structure

This paper contains 8 sections, 13 theorems, 39 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction and Prior Work
  2. Notation and Main Results
  3. Proofs
  4. Algorithms and Examples
  5. Closed-form Examples
  6. Numerical Examples
  7. Discussion
  8. Conclusion

Key Result

Theorem 1

Any elliptic QS-net has equimodular elliptic type and its flat angles satisfy Conversely, any $\alpha_i$, $\beta_i$, $\gamma_i$, $\delta_i$ satisfying eq:QS, eq:N.0 and eq:M.0.a are flat angles of some flexible elliptic QS-net with a flexion given by where $t$ varies in some interval, the signs in $\pm$ agree, and

Figures (11)

  • Figure 1: A QS-net is a particular Kokotsakis polyhedron with quadrangular base. The flat angles of the same color are equal. Such a net is generically flexible.
  • Figure 2: Illustration of a $3 \times 3$ net.
  • Figure 3: Notation for the vertices, flat and dihedral angles of a $3 \times 3$ net without corners.
  • Figure 4: A polyhedron with some of the angles prescribed.
  • Figure 5: The QS-net in Example \ref{['ex:ex1']} at different values of parameter $t$. The sign "$+$" is chosen in each $\pm$ in \ref{['eq:E.1']}.
  • ...and 6 more figures

Theorems & Definitions (36)

  • Definition 1
  • Theorem 1
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • proof
  • ...and 26 more