Origami constraints on the initial-conditions arrangement of dark-matter caustics and streams

In a cold-dark-matter universe, cosmological structure formation proceeds in rough analogy to origami folding. Dark matter occupies a three-dimensional 'sheet' of free- fall observers, non-intersecting in six-dimensional velocity-position phase space. At early times, the sheet was flat like an origami sheet, i.e. velocities were essentially zero, but as time passes, the sheet folds up to form cosmic structure. The present paper further illustrates this analogy, and clarifies a Lagrangian definition of caustics and streams: caustics are two-dimensional surfaces in this initial sheet along which it folds, tessellating Lagrangian space into a set of three-dimensional regions, i.e. streams. The main scientific result of the paper is that streams may be colored by only two colors, with no two neighbouring streams (i.e. streams on either side of a caustic surface) colored the same. The two colors correspond to positive and negative parities of local Lagrangian volumes. This is a severe restriction on the connectivity and therefore arrangement of streams in Lagrangian space, since arbitrarily many colors can be necessary to color a general arrangement of three-dimensional regions. This stream two-colorability has consequences from graph theory, which we explain. Then, using N-body simulations, we test how these caustics correspond in Lagrangian space to the boundaries of haloes, filaments and walls. We also test how well outer caustics correspond to a Zel'dovich-approximation prediction.Comment: Clarifications and slight changes to match version accepted to MNRAS. 9 pages, 5 figure

Fun with Fonts: Algorithmic Typography

Over the past decade, we have designed six typefaces based on mathematical theorems and open problems, specifically computational geometry. These typefaces expose the general public in a unique way to intriguing results and hard problems in hinged dissections, geometric tours, origami design, computer-aided glass design, physical simulation, and protein folding. In particular, most of these typefaces include puzzle fonts, where reading the intended message requires solving a series of puzzles which illustrate the challenge of the underlying algorithmic problem.Comment: 14 pages, 12 figures. Revised paper with new glass cane font. Original version in Proceedings of the 7th International Conference on Fun with Algorithm

Design of DNA origami

The generation of arbitrary patterns and shapes at very small scales is at the heart of our effort to miniaturize circuits and is fundamental to the development of nanotechnology. Here I review a recently developed method for folding long single strands of DNA into arbitrary two-dimensional shapes using a raster fill technique - 'scaffolded DNA origami'. Shapes up to 100 nanometers in diameter can be approximated with a resolution of 6 nanometers and decorated with patterns of roughly 200 binary pixels at the same resolution. Experimentally verified by the creation of a dozen shapes and patterns, the method is easy, high yield, and lends itself well to automated design and manufacture. So far, CAD tools for scaffolded DNA origami are simple, require hand-design of the folding path, and are restricted to two dimensional designs. If the method gains wide acceptance, better CAD tools will be required

The effectiveness of origami on overall hand function after injury: A pilot controlled trial

This pilot study measured the effectiveness of using origami to improve the overall hand function of outpatients attending an NHS hand injury unit. The initiative came from one of the authors who had used origami informally in the clinical setting and observed beneficial effects. These observed effects were tested experimentally. The design was a pilot non-randomised controlled trial with 13 participants. Allocation of the seven control group members was based on patient preference. The experimental group members attended a weekly hour of origami for six weeks, in addition to their conventional rehabilitation. Hand function of all participants was measured using the Jebsen-Taylor Hand Function Test before and after the six-week period, and additional qualitative data were gathered in the form of written evaluations from patients. The quantitative data were analysed using the Mann Whitney U test or Fisher’s exact test. Themes were highlighted from the qualitative data. The results show that there was a greater difference in the total score of the experimental group using the impaired hand between pre- and post-intervention of 11.8 seconds, compared with 4.3 seconds in the control group, but this was not statistically significant at the 5% level (p=0.06). Additionally, differences in the sub-test scores show a markedly larger improvement in the experimental group. Qualitative data indicate that the experimental group experienced the origami sessions as being enjoyable and beneficial. Further research with a larger sample and randomised group allocation is recommended to verify and expand these preliminary findings

Multi-step self-guided pathways for shape-changing metamaterials

Multi-step pathways, constituted of a sequence of reconfigurations, are central to a wide variety of natural and man-made systems. Such pathways autonomously execute in self-guided processes such as protein folding and self-assembly, but require external control in macroscopic mechanical systems, provided by, e.g., actuators in robotics or manual folding in origami. Here we introduce shape-changing mechanical metamaterials, that exhibit self-guided multi-step pathways in response to global uniform compression. Their design combines strongly nonlinear mechanical elements with a multimodal architecture that allows for a sequence of topological reconfigurations, i.e., modifications of the topology caused by the formation of internal self-contacts. We realized such metamaterials by digital manufacturing, and show that the pathway and final configuration can be controlled by rational design of the nonlinear mechanical elements. We furthermore demonstrate that self-contacts suppress pathway errors. Finally, we demonstrate how hierarchical architectures allow to extend the number of distinct reconfiguration steps. Our work establishes general principles for designing mechanical pathways, opening new avenues for self-folding media, pluripotent materials, and pliable devices in, e.g., stretchable electronics and soft robotics.Comment: 16 pages, 3 main figures, 10 extended data figures. See https://youtu.be/8m1QfkMFL0I for an explanatory vide

Practical Applications of Rigid Thick Origami in Kinetic Architecture

Folding elements have already been used in architecture as either: (a) simple or negligibly thin folds such as tent-like structures; (b) thick panels with single straight hinges; or (c) flat, faceted forms that appear to have been folded. What is seldom seen is folding in more complicated patterns that also use thick panels. The more complicated crease patterns inspired from origami cannot be used interchangeably between thin and thick materials. Further, once a folding feature is designed, it must have a way to attach to the main/super structure and have a means to deploy. If design parameters and attachments can be better presented and understood, more origami patterns that are rigid and thick may be incorporated into kinetic architecture or rigid-thick origami kinetic architecture. This research creates a useful primer for understanding and designing rigid-thick origami structures by simplifying and organizing existing knowledge on rigidthick origami into a more accessible format for designers and architects without the need for deep mathematical background. It also presents a variety of design patterns which can be altered or adapted along provided guidelines, as well as propose some methods in which to attach and operate some of these designs on a superstructure through documentation of a working prototype. The hope is that more rigid-thick origami concepts will be available to allow for more practical and aesthetic design opportunities in the field of kinetic architecture