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The Shape of the Edge of a Leaf
, 2002
"... Leaves and flowers frequently have a characteristic rippling pattern at their edges. Recent experiments found similar patterns in torn plastic. These patterns can be reproduced by imposing metrics upon thin sheets. The goal of this paper is to discuss a collection of analytical and numerical results ..."
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Cited by 8 (2 self)
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Leaves and flowers frequently have a characteristic rippling pattern at their edges. Recent experiments found similar patterns in torn plastic. These patterns can be reproduced by imposing metrics upon thin sheets. The goal of this paper is to discuss a collection of analytical and numerical results for the shape of a sheet with a nonflat metric. First, a simple condition is found to determine when a stretched sheet folded into a cylinder loses axial symmetry, and buckles like a flower. General expressions are next found for the energy of stretched sheets, both in forms suitable for numerical investigation, and for analytical studies in the continuum. The bulk of the paper focuses upon long thin strips of material with a linear gradient in metric. In some special cases, the energyminimizing shapes of such strips can be determined analytically. Euler–Lagrange equations are found which determine the shapes in general. The paper closes with numerical investigations of these equations. KEY WORDS: thin sheets; nonflat metric. 1.
Output Devices, Computation, and the Future of Mathematical Crafts
 International Journal of Computers in Mathematical Learning
, 2002
"... As I write this sentence, I am glancing over at the color printer sitting beside my screen. In the popular jargon of the computer industry, that printer is called a "peripheral"—which, upon reflection, is a rather odd way to describe it. What, precisely, is it peripheral to? If the ..."
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Cited by 4 (2 self)
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As I write this sentence, I am glancing over at the color printer sitting beside my screen. In the popular jargon of the computer industry, that printer is called a &quot;peripheral&quot;—which, upon reflection, is a rather odd way to describe it. What, precisely, is it peripheral to? If the ultimate
Crocheting the Lorenz manifold
 The Mathematical Intelligencer
, 2004
"... You have probably seen a picture of the famous butterflyshaped Lorenz attractor — on a book cover, a conference poster, a coffee mug or a friend’s Tshirt. The Lorenz attractor is the best known image of a chaotic or strange attractor. We are concerned here with its close cousin, the twodimensiona ..."
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Cited by 3 (3 self)
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You have probably seen a picture of the famous butterflyshaped Lorenz attractor — on a book cover, a conference poster, a coffee mug or a friend’s Tshirt. The Lorenz attractor is the best known image of a chaotic or strange attractor. We are concerned here with its close cousin, the twodimensional stable manifold of the origin of the Lorenz system, which we call the Lorenz manifold for short. This surface organizes the dynamics in the threedimensional phase space of the Lorenz system. It is invariant under the flow (meaning that trajectories cannot cross it) and essentially determines how trajectories visit the two wings of the Lorenz attractor. We have been working for quite a while on the development of algorithms to compute global manifolds in vector fields and have computed the Lorenz manifold up to considerable size. Its geometry is very intriguing and we explored different ways of visualizing it on the computer [6, 9]. However, a real model of this surface was still lacking. During the Christmas break 2002/2003 Hinke was relaxing by crocheting hexagonal lace motifs when Bernd suggested: “Why don’t you crochet something useful? ” The algorithm we developed ‘grows ’ a manifold in steps. We start from a small disc in the stable eigenspace of the origin and add at each step a band of a fixed width. In other words, at any time of the calculation the computed part of the Lorenz manifold is a topological disc whose outer rim is (approximately) a level set of the geodesic distance from the origin. What we realized then and there is that the mesh generated by our algorithm can directly be interpreted as chrochet instructions! After some initial experimentation, the first model of the Lorenz manifold was 1 Osinga & Krauskopf Chrocheting the Lorenz manifold 2
Design and Analysis of an Artificial Finger Joint for Anthropomorphic Robotic Hands
"... Abstract—In order to further understand what physiological characteristics make a human hand irreplaceable for many dexterous tasks, it is necessary to develop artificial joints that are anatomically correct while sharing similar dynamic features. In this paper, we address the problem of designing a ..."
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Cited by 2 (2 self)
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Abstract—In order to further understand what physiological characteristics make a human hand irreplaceable for many dexterous tasks, it is necessary to develop artificial joints that are anatomically correct while sharing similar dynamic features. In this paper, we address the problem of designing a two degree of freedom metacarpophalangeal (MCP) joint of an index finger. The artificial MCP joint is composed of a ball joint, crocheted ligaments, and a silicon rubber sleeve which as a whole provides the functions required of a human finger joint. We quantitatively validate the efficacy of the artificial joint by comparing its dynamic characteristics with that of two human subjects ’ index fingers by analyzing their impulse response with linear regression. Design parameters of the artificial joint are varied to highlight their effect on the joint’s dynamics. A modified, secondorder model is fit which accounts for nonlinear stiffness and damping, and a higher order model is considered. Good fits are observed both in the human (R 2 = 0.97) and the artificial joint of the index finger (R 2 = 0.95). Parameter estimates of stiffness and damping for the artificial joint are found to be similar to those in the literature, indicating our new joint is a good approximation for an index finger’s MCP joint. I.
Visualizing curvature on the Lorenz manifold
 Journal of Mathematics and the Arts
, 2007
"... The Lorenz manifold is an intriguing twodimensional surface that illustrates chaotic dynamics in the wellknown Lorenz system. While it is not possible to find the Lorenz manifold as an explicit analytic solution, we have developed a method for calculating a numerical approximation that builds the ..."
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Cited by 1 (1 self)
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The Lorenz manifold is an intriguing twodimensional surface that illustrates chaotic dynamics in the wellknown Lorenz system. While it is not possible to find the Lorenz manifold as an explicit analytic solution, we have developed a method for calculating a numerical approximation that builds the surface up as successive geodesic level sets. The resulting mesh approximation can be read as crochet instructions, which means that we are able to generate a threedimensional model of the Lorenz manifold. We mount the crocheted Lorenz manifold using a stiff rod as the zaxis, and bendable wires at the outer rim and the two solutions that are perpendicular to the zaxis. The crocheted model inspired us to consider the geometrical properties of the Lorenz manifold. Specifically, we introduce a simple method to determine and visualize local curvature of a smooth surface. The colour coding according to curvature reveals a striking pattern of regions of positive and negative curvature on the Lorenz manifold. 1
1 A Compliant Biomimetic Artificial Finger for Anthropomorphic Robotic Hands via 3D Rapid Prototyping
"... Abstract—This paper presents an anthropomorphic robotic finger that is composed of three biomimetic joints whose biomechanics and dynamic properties are close to their human counterparts. By using five pneumatic cylinders, the robotic finger is actuated through a series of simplified antagonistic te ..."
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Abstract—This paper presents an anthropomorphic robotic finger that is composed of three biomimetic joints whose biomechanics and dynamic properties are close to their human counterparts. By using five pneumatic cylinders, the robotic finger is actuated through a series of simplified antagonistic tendons whose insertion points and moment arms at each joint are inherited from the anatomy of the human finger. The dynamics of the artificial finger is investigated under PID control for the tasks of set point stabilization and disturbance rejection, set point tracking and trajectory tracking. An air dynamic model is empirically derived for controlling the pneumatic system. The kinematic model of the artificial finger system is constructed with help of MuJoCo a physics engine customdeveloped to simulate the interaction between the finger’s joints and tendons. Experimental data of the tendon excursions are used to validate the efficacy of the simulation model. Index Terms—Biomimetics, compliant joint, anthropomorphic robotic hands, anatomically correct, simulation model. I.
The Kählerian geometry of quantum model order reduction with applications in the simulation of open quantum systems
, 2007
"... This article presents numerical techniques for simulating hightemperature and nonequilibrium quantum spin systems that are continuously measured and controlled. The notion of a “spin system ” is broadly conceived, to encompass test masses as the limiting case of largej spins, and in general the s ..."
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This article presents numerical techniques for simulating hightemperature and nonequilibrium quantum spin systems that are continuously measured and controlled. The notion of a “spin system ” is broadly conceived, to encompass test masses as the limiting case of largej spins, and in general the systems simulated are spatially inhomogeneous. The simulation technique has three stages: first the deliberate introduction of noise into the simulation, then the conversion of that noise into a informatically equivalent continuous measurement and control processes, and finally, projection of the trajectory onto a Kählerian statespace manifold having reduced dimensionality and possessing a Kähler potential of multilinear (i.e., productsum) functional form; these statespaces can be regarded as ruled algebraic varieties. To provide a unifying geometric context for this technique, the sectional curvature of ruled statespaces is analyzed, and proved to be nonpositive upon all sections that contain a rule. It is further shown that ruled statespaces include the Slater determinant wavefunctions of quantum chemistry as a special case and that these Slater determinant manifolds possess a KählerEinstein metric. It is suggested that the Riemannian curvature properties of ruled statespaces generically account for the fidelity, efficiency, and robustness of projective trajectory simulation on these statespaces. The resulting formalism is used to construct a positive Prepresentation for the thermal density matrix, and