The Cosmic Spiderweb and General Origami Tessellation Design

The cosmic web (the arrangement of matter in the universe), spider's webs, and origami tessellations are linked by their geometry (specifically, of sectional-Voronoi tessellations). This motivates origami and textile artistic representations of the cosmic web. It also relates to the scientific insights origami can bring to the cosmic web; we show results of some cosmological computer simulations, with some origami-tessellation properties. We also adapt software developed for cosmic-web research to provide an interactive tool for general origami-tessellation design.Comment: Accepted to Origami$^7$, proceedings of 7OSME, the 7th meeting of Origami, Science, Mathematics and Education, Oxford, Sep 2018. Software at https://github.com/neyrinck/sectional-tes

Toric origami manifolds and multi-fans

The notion of a toric origami manifold, which weakens the notion of a symplectic toric manifold, was introduced by Cannas da Silva-Guillemin-Pires \cite{ca-gu-pi11} and they show that toric origami manifolds bijectively correspond to origami templates via moment maps, where an origami template is a collection of Delzant polytopes with some folding data. Like a fan is associated to a Delzant polytope, a multi-fan introduced in \cite{ha-ma03} and \cite{masu99} can be associated to an oriented origami template. In this paper, we discuss their relationship and show that any simply connected compact smooth 4-manifold with a smooth action of $T^2$ can be a toric origami manifold. We also characterize products of even dimensional spheres which can be toric origami manifolds.Comment: 17 pages, 3 figure

Origami Cubes with One-DOF Rigid and Flat Foldability

Rigid origami is a branch of origami with great potential in engineering applications to deal with rigid-panel folding. One of the challenges is to compactly fold the polyhedra made from rigid facets with a single degree of freedom. In this paper, we present a new method to design origami cubes with three fundamental characteristics, rigid foldability, flat foldability and one degree of freedom (DOF). A total of four cases of crease patterns that enable origami cubes with distinct folding performances have been proposed with all possible layouts of the diagonal creases on the square facets of origami cubes. Moreover, based on the kinematic equivalence between the rigid origami and the spherical linkages, the corresponding spherical linkage loops are introduced and analysed to reveal the motion properties of the origami cubes. The newly found method can be readily utilized to design deployable structures for various engineering applications including cube-shaped cartons, small satellites, containers, etc

The topology of toric origami manifolds

A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami symplectic manifolds, studied by Cannas da Silva, Guillemin and Pires, who classified toric origami manifolds by combinatorial origami templates. In this paper, we examine the topology of toric origami manifolds that have acyclic origami template and co-orientable folding hypersurface. We prove that the cohomology is concentrated in even degrees, and that the equivariant cohomology satisfies the GKM description. Finally we show that toric origami manifolds with co-orientable folding hypersurface provide a class of examples of Masuda and Panov's torus manifolds.Comment: 20 pages, 7 figures. Minor changes from previous version, typos fixed, bibliography update

A system for constructing relatively small polyhedra from Sonob\'e modules

We develop a quite elementary graph theoretic system for designing small-size augmented origami polyhedra out of Sonob\'e modules beginning with a (convex or not) deltahedron.Comment: 4 pages. Mathematical origami method

A Universal Crease Pattern for Folding Orthogonal Shapes

We present a universal crease pattern--known in geometry as the tetrakis tiling and in origami as box pleating--that can fold into any object made up of unit cubes joined face-to-face (polycubes). More precisely, there is one universal finite crease pattern for each number n of unit cubes that need to be folded. This result contrasts previous universality results for origami, which require a different crease pattern for each target object, and confirms intuition in the origami community that box pleating is a powerful design technique.Comment: 7 pages, 4 figure

Trisections and Totally Real Origami

We introduce a trisection axiom for mathematical origami and descibe the totally real origami numbers. We also discuss the solution of Alhazen's problem and its relation to trisections.Comment: 4 figure

On origami rings

In the paper "origami rings" by Joe Buhler et al. the authors investigate the so called origami rings. Taking this paper as a starting point we find some further properties of these rings.Comment: 8 pages, 3 figure

Minimal forcing sets for 1D origami

This paper addresses the problem of finding minimum forcing sets in origami. The origami material folds flat along straight lines called creases that can be labeled as mountains or valleys. A forcing set is a subset of creases that force all the other creases to fold according to their labels. The result is a flat folding of the origami material. In this paper we develop a linear time algorithm that finds minimum forcing sets in one dimensional origami.Comment: 21 pages with a 6-page appendi

Cohomology of toric origami manifolds with acyclic proper faces

A toric origami manifold is a generalization of a symplectic toric manifold (or a toric symplectic manifold). The origami symplectic form is allowed to degenerate in a good controllable way in contrast to the usual symplectic form. It is widely known that symplectic toric manifolds are encoded by Delzant polytopes, and the cohomology and equivariant cohomology rings of a symplectic toric manifold can be described in terms of the corresponding polytope. Recently, Holm and Pires described the cohomology of a toric origami manifold $M$ in terms of the orbit space $M/T$ when $M$ is orientable and the orbit space $M/T$ is contractible. But in general the orbit space of a toric origami manifold need not be contractible. In this paper we study the topology of orientable toric origami manifolds for the wider class of examples: we require that every proper face of the orbit space is acyclic, while the orbit space itself may be arbitrary. Furthermore, we give a general description of the equivariant cohomology ring of torus manifolds with locally standard torus actions in the case when proper faces of the orbit space are acyclic and the free part of the action is a trivial torus bundle.Comment: 27 pages, 2 figure