Generalized Quasi-Symmetric Nets: A Constructive Approach to the Equimodular Elliptic Type and to Flexible m × n Nets
Abstract
Quadrilateral meshes that flex with rigid planar faces are of interest to transformable architecture, rigid-foldable origami, and mechanical metamaterials. Their building blocks are flexible 3 × 3 nets, that is, Kokotsakis polyhedra with a quadrangular base, classified by Izmestiev. We study one of his types, the equimodular elliptic type, defined by three conditions on the flat angles, the third of them through Jacobi elliptic functions. We turn these conditions into equivalent algebraic ones, free of elliptic functions, which lead to the generalized quasi-symmetric nets (GQS-nets), polyhedra of this type that include the known elliptic quasi-symmetric nets. For every polyhedron of this type, we give a flexible polyhedron with the same flat angles and its flexion in closed form. From the simplest GQS-nets, we assemble arbitrarily large flexible m × n nets. Examples include nets that, at one position, form letters; further nets in the supplementary materials form digits or, seen along their rows, approximately follow the side silhouettes of a bunny and a fish. Unlike numerically computed mechanisms, these nets are exactly flexible. As an application, we design three such nets as transportable arch halls of 80 mm sandwich panels, which fold to fit through the door of a standard container or into the well of a low-bed semi-trailer, and we check them numerically under snow load and build a 1:20 model of one. Supplementary Mathematica notebooks verify the symbolic computations, a Python tool constructs the nets and checks them numerically, including for self-intersections, and further scripts recheck the halls.
Related
- A Constructive Framework for Generalized Fourier Transforms via Truncate-and-Generalized Limits
- A constructive approach to generalized principal connections
- Non-Elliptic Quadratic 𝒫𝒯-Symmetric Operators and Similarity to Self-Adjoint Operators
- Polytopic symmetric interior-penalty approximation of elliptic SPDEs driven by spatial white noise
- On generalized embeddability of a non-elliptic self-map
BibTeX
@misc{nurmatov2026generalized,
title = {{Generalized Quasi-Symmetric Nets: A Constructive Approach to the Equimodular Elliptic Type and to Flexible $m \times n$ Nets}},
author = {Abdukhomid Nurmatov and Florian Rist and Jonathan Klein and Dominik L. Michels},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02229},
eprint = {2611.02229},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02229},
}
RIS
TY - UNPB
TI - Generalized Quasi-Symmetric Nets: A Constructive Approach to the Equimodular Elliptic Type and to Flexible m × n Nets
AU - Abdukhomid Nurmatov
AU - Florian Rist
AU - Jonathan Klein
AU - Dominik L. Michels
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02229
UR - https://arxiv.org/abs/2611.02229
AB - Quadrilateral meshes that flex with rigid planar faces are of interest to transformable architecture, rigid-foldable origami, and mechanical metamaterials. Their building blocks are flexible 3 × 3 nets, that is, Kokotsakis polyhedra with a quadrangular base, classified by Izmestiev. We study one of his types, the equimodular elliptic type, defined by three conditions on the flat angles, the third of them through Jacobi elliptic functions. We turn these conditions into equivalent algebraic ones, free of elliptic functions, which lead to the generalized quasi-symmetric nets (GQS-nets), polyhedra of this type that include the known elliptic quasi-symmetric nets. For every polyhedron of this type, we give a flexible polyhedron with the same flat angles and its flexion in closed form. From the simplest GQS-nets, we assemble arbitrarily large flexible m × n nets. Examples include nets that, at one position, form letters; further nets in the supplementary materials form digits or, seen along their rows, approximately follow the side silhouettes of a bunny and a fish. Unlike numerically computed mechanisms, these nets are exactly flexible. As an application, we design three such nets as transportable arch halls of 80 mm sandwich panels, which fold to fit through the door of a standard container or into the well of a low-bed semi-trailer, and we check them numerically under snow load and build a 1:20 model of one. Supplementary Mathematica notebooks verify the symbolic computations, a Python tool constructs the nets and checks them numerically, including for self-intersections, and further scripts recheck the halls.
ER -