On rigid origami II: Quadrilateral creased papers

Miura-ori is well-known for its capability of flatly folding a sheet of paper through a tessellated crease pattern made of repeating parallelograms. Many potential applications have been based on the Miura-ori and its primary variations. Here we are considering how to generalize the Miura-ori: what is the collection of rigid-foldable creased papers with a similar quadrilateral crease pattern as the Miura-ori? This paper reports some progress. We find some new variations of Miura-ori with less symmetry than the known rigid-foldable quadrilateral meshes. They are not necessarily developable or flat-foldable, and still only have single degree of freedom in their rigid folding motion. This article presents a classification of the new variations we discovered and explains the methods in detail.George and Lilian Schiff Foundation

Orthodiagonal anti-involutive Kokotsakis polyhedra

We study the properties of Kokotsakis polyhedra of orthodiagonal anti-involutive type. Stachel conjectured that a certain resultant connected to a polynomial system describing flexion of a Kokotsakis polyhedron must be reducible. Izmestiev \cite{izmestiev2016classification} showed that a polyhedron of the orthodiagonal anti-involutive type is the only possible candidate to disprove Stachel's conjecture. We show that the corresponding resultant is reducible, thereby confirming the conjecture. We do it in two ways: by factorization of the corresponding resultant and providing a simple geometric proof. We describe the space of parameters for which such a polyhedron exists and show that this space is non-empty. We show that a Kokotsakis polyhedron of orthodiagonal anti-involutive type is flexible and give explicit parameterizations in elementary functions and in elliptic functions of its flexion

Classification of flexible Kokotsakis polyhedra with quadrangular base

A Kokotsakis polyhedron with quadrangular base is a neighborhood of a quadrilateral in a quad surface. Generically, a Kokotsakis polyhedron is rigid. In this article we classify flexible Kokotsakis polyhedra with quadrangular bases. The analysis is based on the fact that any pair of adjacent dihedral angles of a Kokotsakis polyhedron is related by a biquadratic equation. This results in a diagram of branched covers of complex projective lines by elliptic curves. A polyhedron is flexible if and only if all repeated fiber products of coverings meet in the same Riemann surface, which is then the configuration space of the polyhedron

Combinatorics of Bricard's octahedra

We re-prove the classification of motions of an octahedron — obtained by Bricard at the beginning of the XX century — by means of combinatorial objects satisfying some elementary rules. The explanations of these rules rely on the use of a well-known creation of modern algebraic geometry, the moduli space of stable rational curves with marked points, for the description of configurations of graphs on the sphere. Once one accepts the objects and the rules, the classification becomes elementary (though not trivial) and can be enjoyed without the need of a very deep background on the topic

Theoretical characterization of a non-rigid-foldable square-twist origami for property programmability

Using non-rigid-foldable origami patterns to design mechanical metamaterials could 14 potentially offer more versatile behaviors than the rigid-foldable ones, but their applications are 15 limited by the lack of analytical framework for predicting their behavior. Here, we propose a 16 theoretical model to characterize a non-rigid-foldable square-twist origami pattern by its rigid origami 17 counterpart. Based on the experimentally observed deformation mode the square-twist, a virtual 18 crease was added in the central square to turn the non-rigid-foldable pattern to a rigid-foldable one. 19 Two possible deformation paths of the non-rigid-foldable pattern were calculated through kinematic 20 analysis of its rigid origami counterpart, and the associated energy and force were derived 21 analytically. Using the theoretical model, we for the first time discovered that the non-rigid-foldable 22 structure bifurcated to follow a low-energy deformation path, which was validated through 23 experiments. Furthermore, the mechanical properties of the structure could be programmed by the 24 geometrical parameters of the pattern and material stiffness of the creases and facets. This work thus 25 paves the way for development of non-rigid-foldable origami-based metamaterials serving for 26 mechanical, thermal, and other engineering applications