Flexible polyhedral nets in isotropic geometry

TL;DR

The paper delivers a complete classification of finitely flexible dual-convex $m\times n$ nets in isotropic 3-space, revealing exactly two fundamental classes and showing that the isotropic theory reduces to duality with deformable Euclidean nets. It proves infinitesimal flexibility aligns with Koenigs/Christoffel duality and translates finite-flexibility results via metric duality to explicit geometric conditions, including reciprocal-parallel constructions. It further extends the theory to smooth nets, establishing a parallel deformable framework and constructing a broad family of generalized smooth T-nets that are isotropically flexible. The findings illuminate deep connections between isotropic geometry and Euclidean mechanisms, enabling initialization strategies for Euclidean design algorithms and opening questions about non-convex cases, isotropic origami, and smooth deformable nets with higher-order flexibility.

Abstract

We study flexible polyhedral nets in isotropic geometry. This geometry has a degenerate metric, but there is a natural notion of flexibility. We study infinitesimal and finite flexibility, and classify all finitely flexible polyhedral nets of arbitrary size. We show that there are just two classes, in contrast to Izmestiev's rather involved classification in Euclidean geometry, for size 3x3 only. Using these nets to initialize the optimization algorithms, we turn them into approximate Euclidean mechanisms. We also explore the smooth versions of these classes.

Figures (12)

• Figure 1: Kinematic sculpture in form of a Euclidean flexible Q-net of Voss type at the 2018 Chicago design week (© Skidmore, Owings & Merrill).
• Figure 2: A few positions of an isotropic mechanism (upper row) and a Euclidean mechanism obtained by optimization (lower row) called "flower". The optimization leads to a very small change so that initial positions are very close. Images courtesy Caigui Jiang.
• Figure 3: A $2\times 2$ net. See Subsection \ref{['Subsection-dual-convex']}.
• Figure 4: Left: An admissible 4-hedral angle. The isotropic line (blue) through the vertex intersects the angle interior. Middle: Four consecutive faces $p_1,{p_2},{p_3},{p_4}$ around a vertex $F_{ij}$ of a dual-convex net. In the left and middle figures, the vertex is the same (black point), the four emanating edges are the same except for the one equipped with an arrow (whose direction is reversed), and faces of the same color lie in the same plane. The union of the faces $p_1,{p_2},{p_3},{p_4}$ is not a graph of a function $z=z(x,y)$. Right: The metric dual of the sub-net formed by $p_1,{p_2},{p_3},{p_4}$ is a convex quadrilateral ${p_1^*}{p_2^*}{p_3^*}{p_4^*}$ (see Lemma \ref{['l-dual-convex']}). The total isotropic Gaussian curvature $\Omega(F_{ij})$ concentrated at the vertex $F_{ij}$ is the oriented area of its top view $\overline{p_1^*}\,\overline{p_2^*}\,\overline{p_3^*}\,\overline{p_4^*}$. See Definitions \ref{['admissible-conv-pol--angle']} and \ref{['def-curvature']}.
• Figure 5: Dual quadrilaterals. Corresponding sides (with the same color) are parallel and non-corresponding diagonals (also with the same color) are parallel.

Theorems & Definitions (60)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• Example 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3