A new method for generalizing non-self-intersecting flexible polyhedra

TL;DR

This work broadens the landscape of embedded flexible polyhedra by introducing the base + crinkle construction, which combines rigid bases with flexible crinkles to produce non-triangulated, multi-DOF, and higher-genus polyhedra that remain non-self-intersecting. It formalizes crinkles and collar crinkles, demonstrates torus and spherical exemplars (including a two-DOF torus built from nested bases), and discusses how base geometry fixes volume while enabling diverse motions and potential applications in origami-inspired design, robotics, and metamorphic grippers. The results open pathways for topology- and symmetry-aware design of flexible mechanisms, with future work on generalizing crinkles, exploring higher genus, and applying motion-planning optimization. The study highlights practical implications for engineered morphing structures that preserve rigidity of faces while allowing large, self-avoiding deformations.

Abstract

A surface is considered flexible if it allows a continuous deformation that preserves both metric and smoothness. We introduce a novel construction method, called 'base + crinkle,' for generating a broad class of non-self-intersecting flexible closed polyhedral surfaces (i.e. flexible polyhedra). These flexible polyhedra can be non-triangulated, exhibit multiple kinematic degrees of freedom, and possess topologies beyond the sphere. The geometric result provides fresh insights into the geometry of origami and the design of engineering mechanisms, such as sealed-chamber robotics and distortion-free metamorphic grippers.

Figures (4)

• Figure 1: Two new flexible polyhedra generated using the 'base + crinkle' method. (a) A non-triangulated, topologically spherical flexible polyhedron exhibiting a single kinematic degree of freedom. It achieves a wider folding range (29.2$^\circ$) compared to the Steffen’s polyhedron (27$^\circ$, Figure \ref{['fig: Steffen']}) and, notably, contains no unfolded edges --- all dihedral angles vary continuously during its motion. (b) A non-triangulated, topologically toroidal flexible polyhedron with two kinematic degrees of freedom. The construction begins with two nested 'flappy birds,' whose folding motions are independent and non-interfering under appropriate parameter adjustments, ensuring no self-intersection occurs. To achieve toroidal topology, two triangles from the inner 'flappy bird' are removed, while the fixed tunnels and the triangulated openings on the outer surface together form the handle. Further details of the construction are provided in Section \ref{['section: result and method']}.
• Figure 2: (a) A crinkle $ABCDEF$, obtained by removing two triangular faces from Bricard’s first type of self-intersecting flexible octahedron. (b) A net of (a), with all edge lengths labeled. During the folding process from (b) to (a), the vertex pairs $A_1, A_2$ and $C_1, C_2$ are respectively identified with vertices $A$ and $C$. Visually, vertex $E$ 'pops in', while vertex $F$ 'pops out' in the resulting three-dimensional configuration (a). (c) A base designed to support the assembly of crinkles, which leads to the construction of Steffen’s polyhedron. Vertices $1\text{-}2\text{-}3\text{-}4$ define a fixed tetrahedral chamber. The distance between vertices $1$ and $3$ is labelled. Triangle $2\text{-}4\text{-}5$, with fixed edge lengths, can rotate freely about edge $2\text{-}4$. By attaching two crinkles to the two spatial quadrilaterals $4\text{-}5\text{-}2\text{-}3$ and $2\text{-}5\text{-}4\text{-}1$ in opposing configurations ('pop in' vs. 'pop out'), we obtain Steffen’s polyhedron (d). The motion of Steffen’s polyhedron closely resembles the motion of the base.
• Figure 3: (a) A collar crinkle $ABCDEFGHIJ$, obtained by adding a row of rectangles to the crinkle generated from Bricard’s second type of self-intersecting flexible octahedron. (b) A net of (a), with all edge lengths labelled. During the folding process from (b) to (a), the vertex pairs $A_1, A_2$, $C_1, C_2$, $D_1, D_2$, and $F_1, F_2$ are respectively identified with vertices $A$, $C$, $D$, and $F$. Visually, vertex $G$ and $J$ 'pop in', while vertex $H$ and $I$ 'pop out' in the resulting three-dimensional configuration (a). (c) A base designed to support the assembly of a collar crinkle and two crinkles, which leads to the construction of Figure \ref{['fig: bird']}(a). Vertices $2\text{-}3\text{-}5\text{-}6\text{-}7$ define a fixed right rectangular pyramid. The distance between vertices $2$ and $7$ is labelled. Triangles $1\text{-}2\text{-}6$ and $3\text{-}4\text{-}5$, with fixed edge lengths, can rotate freely about edge $2\text{-}6$ and $3\text{-}5$. By attaching two crinkles to the two spatial quadrilaterals $6\text{-}7\text{-}2\text{-}1$ and $5\text{-}7\text{-}3\text{-}4$ in aligned configurations ('pop in' vs. 'pop in'), and attaching a collar crinkle to the hexagon $3\text{-}4\text{-}5\text{-}6\text{-}1\text{-}2$, we obtain the 'flappy bird' Figure \ref{['fig: bird']}(a). The motion of the flappy bird closely resembles the motion of the base. (d) is the right-side view of (c).
• Figure 4: Further examples generated using the 'base + crinkle' method, showing the form-finding potential. (a) shows the base of a polyhedral torus whose full construction is provided in Figure \ref{['fig: bird']}(b). (b) A topologically spherical flexible polyhedron formed by joining two copies of 'flappy birds' using two fixed trapezoids and one fixed rectangle, with two triangular faces removed. This polyhedron possesses two kinematic degrees of freedom and demonstrates potential applications in crawlable robotics with constant enclosed volume while keeping all faces perfectly rigid.