arXiv subjects
Publications and source records attributed to Elisabetta A. Matsumoto.
\emph{Magic braids} are used in leatherworking to make intricate straps and bands. They appear to be impossible to make. To determine which braids can be made with this leatherworking technique, we explore their relation to several braid groups.
A periodic tangle is a one-dimensional submanifold in $\mathbb{R}^3$ that has translational symmetry in one, two or three transverse directions. A periodic tangle can be seen as the universal cover of a link in the solid torus, the thickened torus, or the three-torus, respectively. Our goal is to study equivalence relations of such periodic tangles. Since all finite covers of a link lift to the same periodic tangle, it is necessary to prove that isotopies between different finite covers are preserved. In this paper, we show that if two links have isotopic lifts in a common finite cover, then they are isotopic. To do so, we employ techniques from 3-manifold topology to study the complements of such links.
This paper provides a construction and existence proof for a 1-parameter family of chiral unbalanced triply-periodic minimal surfaces of genus 4. We name these {\textit{gyrating H'-T} surfaces, because they are related to Schoen's H'-T surfaces in a similar way as the Gyroid is to the Primitive surface. Their chirality is manifest in a screw symmetry of order six. The two labyrinthine domains on either side of the surface are not congruent, rather one representing the quartz net (\texttt{qtz}) and the other one the dual of the quartz net (\texttt{qzd}). The family tends to the Scherk saddle tower in one limit and to the doubly periodic Scherk surface in the other. The motivation for the construction was to construct a chiral tunable unbalanced surface family, originally as a template for photonic materials. The numeric construction is based on reverse-engineering of the tubular surface of two suitably chosen dual nets, using the \textit{Surface Evolver}} to minimize area or curvature variations. The existence is proved using Weierstrass parametrizations defined on the branched torus.
In this study, we use a correspondence between two-periodic weft-knitted textiles and links in the thickened torus to study the former using link invariants. We establish a criterion to identify the set of links whose elements are realized through techniques of weft-knitting leading to new, unconventional types of weft-knitting stitch patterns. A crucial topological underpinning of these links is shown to be their correspondence with ribbon knots and links in Euclidean three-space and equivalently in the three-sphere. Using the mechanics of weft-knitting, we propose a protocol for constructing and enumerating links in the thickened torus that can be knitted as a motif of a weft-knitted textile, and we call such links \emph{swatches}. Based on our analysis of link invariants of swatches, we propose conjectures on hyperbolic structure of the link complements of swatches and their multivariable Alexander polynomials.
Fabrics are flexible thin structures made of entangled yarn or fibers, yet the topological bases of their mechanics remain poorly understood. For weft knitted fabrics, we describe how the entanglement of adjacent stitches contributes to the flexibility of the fabric. Interpreting heterogeneous stitch pairs as domain boundaries reveals that the step between pairs of neighboring stitches is responsible for direction-specific flexibility. In typical knitted fabrics, anisotropic flexibility can be attributed to latticed domain boundaries. The intersections between domain boundaries result in point defects that induce frustration that resembles the impossible Penrose stairs. We identify these by a chiral characteristic, defined summing the ascending or descending steps in a cycle surrounding the defect. Remarkably, seed fabric, a knit with high flexibility in both course and wale directions, is characterized as a racemic crystal of these chiral point defects.
A large subset of the population suffers injury or disease that causes knee pain and difficulty navigating day-to-day tasks. Off-the-shelf knee orthoses that are commonly used to treat these ailments overlook user-specific joint geometry and/or biomechanical needs. They can often be made with materials that lead to discomfort. We explore how the rich programmability of knitted fabrics can be harnessed to augment human biomechanics while promoting comfort. In this pursuit, we define \emph{ExoSkins}, a class of unpowered exoskeletons that are lightweight, comfortable, garment-like devices, and are designed based on user- and joint-specific needs. Although we foresee an expansive space for ExoSkin design (e.g., containing active materials, with sensing capabilities), in this study we focus on the interplay between knit geometry and programmable elasticity in passive orthoses as a means to augment the knee's rotational stiffness. We design geometrically-programmed ExoSkins, abbreviated \emph{G-PExos}, that capitalize on the anisotropies of four types of knitted fabric to provide high stiffness for joint torque without the need for rigid materials. Our findings indicate that G-PExos can achieve rotational stiffness of similar magnitude to off-the-shelf orthoses and can also be tuned to achieve a much broader range of rotational stiffness without sacrificing comfort to the user.
We investigate the mechanical behavior of jammed knitted fabrics, where geometric confinement leads to an initially stiff mechanical response that softens into low stiffness behavior with additional applied stress. We show that the jammed regime is distinguished by changes in yarn geometry and contact rearrangement that must occur to allow the individual stitches to stretch. These rearrangements allow for the relaxation of high residual contact forces that are present within jammed fabric, altering the low-strain response. We demonstrate that fabric jamming can be induced or reduced by changing either the constituent yarn or the fabric manufacturing parameters. Analysis of experimental samples shows that changing the "stitch size" in manufacturing affects both the in situ yarn radius and the length of yarn per stitch, both of which affect jamming.
Knitting turns yarn, a 1D material, into a 2D fabric that is flexible, durable [1], and can be patterned to adopt a wide range of 3D geometries [2]. Like other mechanical metamaterials [3], the elasticity of knitted fabrics is an emergent property of the local stitch topology and pattern that cannot solely be attributed to the yarn itself. Thus, knitting can be viewed as an additive manufacturing technique that allows for stitch-by-stitch programming of elastic properties and has applications in many fields ranging from soft robotics [4-6] and wearable electronics [7,8] to engineered tissue[9] and architected materials [10,11]. However, predicting these mechanical properties based on the stitch type remains elusive. Here we untangle the relationship between changes in stitch topology and emergent elasticity in several types of knitted fabrics. We combine experiment and simulation to construct a constitutive model for the nonlinear bulk response of these fabrics. This model serves as a basis for composite fabrics with bespoke mechanical properties, which crucially do not depend on the constituent yarn.
In this paper we give a brief overview of the geometry of involute gears, from a mathematical more than an engineering perspective. We also list some of the many variant geared mechanisms and discuss some of our 3D printed mechanisms.
We describe algorithms that produce accurate real-time interactive in-space views of the eight Thurston geometries using ray-marching. We give a theoretical framework for our algorithms, independent of the geometry involved. In addition to scenes within a geometry $X$, we also consider scenes within quotient manifolds and orbifolds $X / Γ$. We adapt the Phong lighting model to non-euclidean geometries. The most difficult part of this is the calculation of light intensity, which relates to the area density of geodesic spheres. We also give extensive practical details for each geometry.
This article presents virtual reality software designed to explore the Sol geometry. The simulation is available on 3-dimensional.space/sol.html
We describe a method of rendering real-time scenes in Nil geometry, and use this to give an expository account of some interesting geometric phenomena. You can play around with the simulation at www.3-dimensional.space/nil.html.
We describe a new family of triply-periodic minimal surfaces with hexagonal symmetry, related to the quartz (qtz) and its dual (the qzd net). We provide a solution to the period problem and provide a parametrisation of these surfaces, that are not in the regular class, by the Weierstrass-Enneper formalism. We identified this analytical description of the surface by generating an area-minimising mesh interface from a pair of dual graphs (qtz & qzd) using the generalised Voronoi construction of De Campo, Hyde and colleagues, followed by numerical identification of the flat point structure. This mechanism is not restricted to the specific pair of dual graphs, and should be applicable to a broader set of possible dual graph topologies and their corresponding minimal surfaces, if existent.
We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulation of the product of two-dimensional hyperbolic space with one-dimensional euclidean space is available at http://h2xe.hypernom.com.
We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulations of three-dimensional hyperbolic space are available at http://h3.hypernom.com.
Frustration is a powerful mechanism in condensed matter systems, driving both order and co plexity. In smectics, the frustration between macroscopic chirality and equally spaced layers generates textures characterised by a proliferation of defects. In this article, we study several different ground states of the chiral Landau-de Gennes free energy for a smectic liquid crystal. The standard theory finds the twist grain boundary (TGB) phase to be the ground state for chiral type II smectics. However, for very highly chiral systems, the hierarchical helical nanofilament (HN) phase can form and is stable over the TGB.
Exploiting elastic instability in thin films has proven a robust method for creating complex patterns and structures across a wide range of lengthscales. Even the simplest of systems, an elastic membrane with a lattice of pores, under mechanical strain, generates complex patterns featuring long-range orientational order. When we promote this system to a curved surface, in particular, a cylindrical membrane, a novel set of features, patterns and broken symmetries appears. The newfound periodicity of the cylinder allows for a novel continuous method for nanoprinting.
Riemann's minimal surfaces are a complete, embeddable, one-parameter family of minimal surfaces with translational symmetry along one direction. It's infinite number of planar ends are joined together by an array of necks, closely matching the morphology of a bicontinuous, lamellar system with pores connecting alternating layers. We demonstrate explicitly that Riemann's minimal surfaces are composed of a nonlinear sum of two oppositely-handed helicoids. This description is particularly appropriate for describing smectic liquid crystals containing two screw dislocations.