Kiumars Sharifmoghaddam

arXiv2026

Rigidity and flexibility of discrete conjugate nets with flexible $3 \times 3$-subnets

Discrete conjugate nets (also known as quad-surfaces) are polyhedral surfaces made of quadrilaterals connected in the combinatorics of the square grid. A generic discrete conjugate net is rigid. The present article contains counterexamples to an erroneous statement that a non-degenerate discrete conjugate net with flexible $3 \times 3$-subnets is flexible and proves a corrected version of it under stronger non-degeneracy assumptions.

math.MG↗

arXiv2023

Generalizing rigid-foldable tubular structures of T-hedral type

We introduce an alternative way of constructing continuous flexible tubes and tubular structures based on a discrete, semi-discrete and smooth construction of surfaces known as T-hedra in the discrete case and profile-affine surfaces in the smooth setting, respectively. The geometric understanding of this method enables us to generalize discrete tubes with a rigid-foldability and to extend the construction to smooth and semi-discrete tubes with an isometric deformation. This achievement implies a unified treatment of continuous flexible structures, like surfaces and metamaterials, composed of tubes and it is the base for a deeper study of zipper tubes, and their generalization. Moreover, we discuss a potential application of the presented structures for the design of foldable bridges.

math.DG↗