Origami fold as algebraic graph rewriting

AbstractWe formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewriting system (O,↬), where O is the set of abstract origamis and ↬ is a binary relation on O, that models fold. An abstract origami is a structure (Π,∽,≻), where Π is a set of faces constituting an origami, and ∽ and ≻ are binary relations on Π, each representing adjacency and superposition relations between the faces.We then address representation and transformation of abstract origamis and further reasoning about the construction for computational purposes. We present a labeled hypergraph of origami and define fold as algebraic graph transformation. The algebraic graph-theoretic formalism enables us to reason about origami in two separate domains of discourse, i.e. pure combinatorial domain where symbolic computation plays the main role and geometrical domain R×R. We detail the program language for the algebraic graph rewriting and graph rewriting algorithms for the fold, and show how fold is expressed by a set of graph rewrite rules

Origami: History, folds, bases and napkins in the art of folding

Origami is an ancient art of paper folding, existing since the invention of paper. Pure traditional origami consists on performing a series of folding operations on a square sheet of paper to assemble a final paper model. This paper is basically an introduction to origami and it is composed of a brief introduction and history of this ancient art followed by three essentially demonstrative sections. The first of these sections shows the basic folds, which are the essential steps used in origami making: the valley/mountain, pleat, crimp, reverse, sink, squash, swivel, rabbit ear and petal folds are shown. The second section shows some of the standard bases. Bases are essentially incomplete origami models, used as halfway baselines to produce different finished models: the kite, fish, bird, frog, cupboard, windmill, water bomb, square and blintz bases are shown. Finally, there is a third section showing some popular napkin origami models. A final conclusion with our intentions for future work will terminate the paper.info:eu-repo/semantics/publishedVersio

On Reconfiguring Tree Linkages: Trees can Lock

It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree linkages: we show that there are trees with two simple configurations that are not connected by a motion that preserves simplicity throughout the motion. Indeed, we prove that an $N$-link tree can have $2^{\Omega(N)}$ equivalence classes of configurations.Comment: 16 pages, 6 figures Introduction reworked and references added, as the main open problem was recently close

When to Hold and When to Fold: Studies on the topology of origami and linkages

Linkages and mechanisms are pervasive in physics and engineering as models for avariety of structures and systems, from jamming to biomechanics. With the increasein physical realizations of discrete shape-changing materials, such as metamaterials,programmable materials, and self-actuating structures, an increased understandingof mechanisms and how they can be designed is crucial. At a basic level, linkagesor mechanisms can be understood to be rigid bars connected at pivots around whichthey can rotate freely. We will have a particular focus on origami-like materials, anextension to linkages with the added constraint of faces. Self-actuated versions typ-ically start flat and when exposed to an external stimulus - such as a temperaturechange or magnetic field - spontaneously fold. Since these structures fold all at once,and the number of folding patterns accessible to a given origami are exponential, theyare prone to folding to a configuration other than the desired one. Other work hassuggested methods for avoiding this misfolding, but it assumes ideal, rigid origami. Here, we expand on these models to account for the elasticity of real structures andintroduce methods for accounting for Gaussian curvature in them. We also explorehow to find and set an upper bound on minimal forcing sets, or the minimum set offolds required to force an origami, and present a graph theory algorithm for findingthem in arbitrary origami. Taken altogether, these origami studies give insight intohow the physical properties of origami influence folding and a new set of tools foravoiding misfolding. Next, we turn back to a more fundamental study of linkagesand present a new method for finding the manifold of their critical points. We thendemonstrate a design protocol that utilizes this manifold to create linkages with tun-able motions, before turning to several example structures, including the four-barlinkage and the Kane-Lubensky chain

Discrete Differential Geometry of Thin Materials for Computational Mechanics

Instead of applying numerical methods directly to governing equations, another approach to computation is to discretize the geometric structure specific to the problem first, and then compute with the discrete geometry. This structure-respecting discrete-differential-geometric (DDG) approach often leads to new algorithms that more accurately track the physically behavior of the system with less computational effort. Thin objects, such as pieces of cloth, paper, sheet metal, freeform masonry, and steel-glass structures are particularly rich in geometric structure and so are well-suited for DDG. I show how understanding the geometry of time integration and contact leads to new algorithms, with strong correctness guarantees, for simulating thin elastic objects in contact; how the performance of these algorithms can be dramatically improved without harming the geometric structure, and thus the guarantees, of the original formulation; how the geometry of static equilibrium can be used to efficiently solve design problems related to masonry or glass buildings; and how discrete developable surfaces can be used to model thin sheets undergoing isometric deformation

Active Self-Assembly of Algorithmic Shapes and Patterns in Polylogarithmic Time

We describe a computational model for studying the complexity of self-assembled structures with active molecular components. Our model captures notions of growth and movement ubiquitous in biological systems. The model is inspired by biology's fantastic ability to assemble biomolecules that form systems with complicated structure and dynamics, from molecular motors that walk on rigid tracks and proteins that dynamically alter the structure of the cell during mitosis, to embryonic development where large-scale complicated organisms efficiently grow from a single cell. Using this active self-assembly model, we show how to efficiently self-assemble shapes and patterns from simple monomers. For example, we show how to grow a line of monomers in time and number of monomer states that is merely logarithmic in the length of the line. Our main results show how to grow arbitrary connected two-dimensional geometric shapes and patterns in expected time that is polylogarithmic in the size of the shape, plus roughly the time required to run a Turing machine deciding whether or not a given pixel is in the shape. We do this while keeping the number of monomer types logarithmic in shape size, plus those monomers required by the Kolmogorov complexity of the shape or pattern. This work thus highlights the efficiency advantages of active self-assembly over passive self-assembly and motivates experimental effort to construct general-purpose active molecular self-assembly systems

Mechanologic: Designing Mechanical Devices that Compute

Despite their initial success and impact on the development of the modern computer, mechanical computers were quickly replaced once electronic computers became viable. Recently, there has been increased interest in designing devices that compute using modern and unconventional materials. In this dissertation, we investigate multiple ways to realize a mechanical device that can compute, with a main focus on designing mechanical equivalents for wires and transistors. For our first approach at designing mechanical wires, we present results on the propagation of signals in a soft mechanical wire composed of bistable elements. When we send a signal along bistable wires that do not support infinite signal propagation, we find that signals can propagate for a finite distance controlled by a penetration length for perturbations. We map out various parameters for this to occur, and present results from experiments on wires made of soft elastomers. Our second approach for designing mechanical devices that compute focuses on designing the topology of the configuration space of a linkage. By programming the configuration space through small perturbations of the bar lengths in the linkage, we are able to design a linkage that gates the propagation of a soliton in a Kane-Lubensky chain. This dissertation also includes other results related to the study of small length changes in linkages and an analysis of a version of a mechanical transistor compatible with the soft bistable wires

Field-based Coordination with the Share Operator

Field-based coordination has been proposed as a model for coordinating collective adaptive systems, promoting a view of distributed computations as functions manipulating data structures spread over space and evolving over time, called computational fields. The field calculus is a formal foundation for field computations, providing specific constructs for evolution (time) and neighbor interaction (space), which are handled by separate operators (called rep and nbr, respectively). This approach, however, intrinsically limits the speed of information propagation that can be achieved by their combined use. In this paper, we propose a new field-based coordination operator called share, which captures the space-time nature of field computations in a single operator that declaratively achieves: (i) observation of neighbors' values; (ii) reduction to a single local value; and (iii) update and converse sharing to neighbors of a local variable. We show that for an important class of self-stabilising computations, share can replace all occurrences of rep and nbr constructs. In addition to conceptual economy, use of the share operator also allows many prior field calculus algorithms to be greatly accelerated, which we validate empirically with simulations of frequently used network propagation and collection algorithms