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On the Algebraic Classification of Non-singular Flexible Kokotsakis Polyhedra

Yang Liu, Yi Ouyang, Dominik L. Michels

Abstract

Across various scientific and engineering domains, a growing interest in flexible and deployable structures is becoming evident. These structures facilitate seamless transitions between distinct states of shape and find broad applicability ranging from robotics and solar cells to meta-materials and architecture. In this contribution, we study a class of mechanisms known as Kokotsakis polyhedra with a quadrangular base. These are $3\times3$ quadrilateral meshes whose faces are rigid bodies and joined by hinges at the common edges. Compared to prior work, the quadrilateral faces do not have to be planar. In general, such meshes are not flexible, and the problem of finding and classifying the flexible ones is old, but until now largely unsolved. It appears that the tangent values of the dihedral angles between different faces are algebraically related through polynomials. Specifically, by fixing one angle as a parameter, the others can be parameterized algebraically and hence belong to an extended rational function field of the parameter. We use this approach to characterize shape restrictions resulting in flexible polyhedra.

On the Algebraic Classification of Non-singular Flexible Kokotsakis Polyhedra

Abstract

Across various scientific and engineering domains, a growing interest in flexible and deployable structures is becoming evident. These structures facilitate seamless transitions between distinct states of shape and find broad applicability ranging from robotics and solar cells to meta-materials and architecture. In this contribution, we study a class of mechanisms known as Kokotsakis polyhedra with a quadrangular base. These are quadrilateral meshes whose faces are rigid bodies and joined by hinges at the common edges. Compared to prior work, the quadrilateral faces do not have to be planar. In general, such meshes are not flexible, and the problem of finding and classifying the flexible ones is old, but until now largely unsolved. It appears that the tangent values of the dihedral angles between different faces are algebraically related through polynomials. Specifically, by fixing one angle as a parameter, the others can be parameterized algebraically and hence belong to an extended rational function field of the parameter. We use this approach to characterize shape restrictions resulting in flexible polyhedra.
Paper Structure (54 sections, 30 theorems, 158 equations, 8 figures)

This paper contains 54 sections, 30 theorems, 158 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Main approach and contributions
  3. Organization of the paper
  4. Basic setup
  5. From $\mathbb{R}^3$ to sphere
  6. Bricard equation
  7. Shape restrictions of the quads
  8. From sphere to polynomial ideals
  9. Couplings and matchings
  10. Non-singular properties
  11. Main problem (reformulated)
  12. Classification of the components and couplings
  13. Extension diagrams
  14. First application
  15. Pseudo-planar type without (anti)deltoids
  16. ...and 39 more sections

Key Result

Theorem 1.1

Every $3\times3$ non-singular flexible mesh must belong to one of the following classes:

Figures (8)

  • Figure 1: Left: Sketch of a $3\times3$ quadrilateral mesh, or equivalently, Kokotsakis polyhedron with a quadrangular base. The flexibility only relies on a smaller area marked in dashed lines; Right: A flexible mesh with fixed angles $\lambda_i', \gamma_i', \mu_i', \delta_i'$, which are well-defined by vector products and $\arccos$ function. The flexible angles $\alpha_i'$ and its complement (not shown) $\alpha_i=\pi-\alpha_i'$ are dihedral angles between the central planar panel and surrounding planar panels, which are rigorously defined in Appendix \ref{['dfnang']}.
  • Figure 2: A non-planar flexible mesh (left) and its decomposition (middle and right). $(\lambda_i', \gamma_i', \mu_i', \delta_i')$ are angles between corresponding oriented edges; $\tau_i, \zeta_i$ are fixed dihedral angles along hinges of the central tetrahedron and the attached tetrahedron respectively; $\alpha_i', \beta_i'$ are flexible dihedral angles (in planar case $\alpha_{i+1}'=\beta_i'$). All angles can be defined in a similar way (see Appendix \ref{['dfnang']}).
  • Figure 3: Left: Partial and whole spherical linkage of a mesh in Fig. \ref{['sqmesh']}, $(\lambda_i, \gamma_i, \mu_i, \delta_i,\alpha_i,\beta_i)$ and $(\lambda_i', \gamma_i', \mu_i', \delta_i',\alpha_i',\beta_i')$ are complementary to $\pi$ respectively, the gap between $\beta_1$ and $\alpha_2$ is caused by $\tau_1$ and $\zeta_1$.
  • Figure 4: A flexible mesh with central panel $V_1V_2V_3V_4$ and attached panels $V_iY_iX_{i+1}V_{i+1}$. While undetermined, one can always choose a properly folded corner panel, containing the polyline $X_iV_iY_i$, to avoid self-intersection. This image is generated by rotating $X_1$ along arc $\rho$, in which dashed lines mean the corresponding panel is beneath the central one.
  • Figure 5: This is a (conceptual) spherical image, every curve lies on a great circle. Left: A coupling $(Q_1,F_1,Q_2,0)$ and its spherical reflection $(Q_2',-F_1,Q_1',0)$. Right: According to Fig. \ref{['squads']}, the positions of adjacent quads should be upside down, so in order to get a spherical linkage, we need to flip $Q_2'$ respect to $\lambda_2'$ and $Q_1'$ respect to $\lambda_1'$.
  • ...and 3 more figures

Theorems & Definitions (87)

  • Theorem 1.1: Main Theorem
  • Remark 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 77 more