Related papers: Symmetry groups of origami structures

We explore the surprisingly rich energy landscape of origami-like folding planar structures. We show that the configuration space of rigid-paneled degree-4 vertices, the simplest building blocks of such systems, consists of at least two…

We investigate the graphs formed from the vertices and creases of an origami pattern that can be folded flat along all of its creases. As we show, this is possible for a tree if and only if the internal vertices of the tree all have even…

We prove several hardness results on folding origami crease patterns. Flat-folding finite crease patterns is fixed-parameter tractable in the ply of the folded pattern (how many layers overlap at any point) and the treewidth of an…

Lattices and their underlying symmetries play a central role in determining the physical properties and applications of many natural and engineered materials. By bridging the lattice geometry and rigid-folding kinematics, this study…

Rigidly and flat-foldable quadrilateral mesh origami is the class of quadrilateral mesh crease patterns with one fundamental property: the patterns can be folded from flat to fully-folded flat by a continuous one-parameter family of…

Starting with a flat sheet of paper, points can be constructed as the intersection of two folds. The set of constructible points clearly depends on which folds are admissible. In this paper, we study the situation where a fold is admissible…

Origami is the archetype of a structural material with unusual mechanical properties that arise almost exclusively from the geometry of its constituent folds and forms the basis for mechanical metamaterials with an extreme deformation…

A characterization of real numbers constructible by paper folding.

This article is concerned with an example of complex planar geometry arising from flat origami challenges. The complexity of solution algorithms is illustrated, depending on the depth of the initial analysis of the problem, starting from…

The planar ornaments are created by repeating a base unit using a combination of four primitive geometric operations: translation, rotation, reflection, and glide reflection. According to group theory, different combinations of these four…

Inspired by the allure of additive fabrication, we pose the problem of origami design from a new perspective: how can we grow a folded surface in three dimensions from a seed so that it is guaranteed to be isometric to the plane? We solve…

The use of origami in engineering has significantly expanded in recent years, spanning deployable structures across scales, folding robotics, and mechanical metamaterials. However, finding foldable paths can be a formidable task as the…

In this paper, we study how to fold a specified origami crease pattern in order to minimize the impact of paper thickness. Specifically, origami designs are often expressed by a mountain-valley pattern (plane graph of creases with relative…

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…

This paper considers an extension of origami geometry to the case of "folding" a three dimensional (3D) space along a plane. First, all possible incidence constraints between given points, lines and planes are analyzed by using the geometry…

Origami crease patterns are folding paths that transform flat sheets into spatial objects. Origami patterns with a single degree of freedom (DOF) have creases that fold simultaneously. More often, several substeps are required to…

The field of rigid origami concerns the folding of stiff, inelastic plates of material along crease lines that act like hinges and form a straight-line planar graph, called the crease pattern of the origami. Crease pattern vertices in the…

Polypolyhedra (after R. Lang) are compounds of edge-transitive 1-skeleta. There are 54 topologically different polypolyhedra, and each has icosidodecahedral, cuboctahedral, or tetrahedral symmetry, all are realizable as modular origami…

Self-folding origami, structures that are engineered flat to fold into targeted, three-dimensional shapes, have many potential engineering applications. Though significant effort in recent years has been devoted to designing fold patterns…

Symmetry is a key feature observed in nature (from flowers and leaves, to butterflies and birds) and in human-made objects (from paintings and sculptures, to manufactured objects and architectural design). Rotational, translational, and…