Figure Eights Research Articles

Flocks, Mobs, and Figure Eights: Swarming as a Lemniscatic Arch

Inspired by the natural mobbing behavior of birds, this work presents a novel, quasi-distributed swarming strategy called the Dynamic Lemniscatic Arch. It resolves the problem of producing globally-stable, evenly-spaced lemniscate (or, figure-eight) trajectories while relying on local interactions only. Such trajectories are advantageous in applications where energy consumption and mechanical strain must be minimized. Previous work in lemniscate curves has typically relied on predetermined trajectories, rather than on the emergent structure of the swarm. Furthermore, we enrich the traditional 2-dimensional lemniscate plane curve structure by forming an arch in the third dimension. This arch provides more consistent coverage in surveillance type tasks and, with minor variations in parameters, can be used to produce mobbing behavior. The technique relies on time-varying quaternion rotations linked to the positions of dynamically induced virtual agents. We provide a mathematical proof of stability, which demonstrates the swarm converges to the desired geometry. Simulations show that the strategy performs well with multiple agents and in numerous different configurations.

Hidden and unstable periodic orbits as a result of homoclinic bifurcations in the Suarez–Schopf delayed oscillator and the irregularity of ENSO

We revisit the classical Suarez–Schopf delayed oscillator. Special attention is paid to the region of linear stability in the space of parameters. By means of the theory of inertial manifolds developed in our adjacent papers, we provide analytical–numerical evidence for the existence of two-dimensional inertial manifolds in the model. This allows to suggest a complete qualitative description of the dynamics in the region of linear stability. We show that there are two subregions corresponding to the existence of hidden or self-excited attracting periodic orbits. These subregions must be separated by a curve on which homoclinic “figure eights”, bifurcating into a single one or a pair of unstable periodic orbits, should exist. We relate the observed hidden oscillations and homoclinics to the irregularity theories of ENSO and provide numerical evidence that chaotic behavior may appear if a small periodic forcing is applied to the model. We also use parameters from the Suarez–Schopf model to discover hidden and self-excited asynchronous periodic regimes in a ring array of coupled lossless transmission lines studied by J. Wu and H. Xia.

All are aromatic: A 3D globally aromatic cage containing five types of 2D aromatic macrocycles

All are aromatic: A 3D globally aromatic cage containing five types of 2D aromatic macrocycles

Learning Handwriting: Factors Affecting Pen-Movement Fluency in Beginning Writers.

Skilled handwriting of single letters is associated not only with a neat final product but also with fluent pen-movement, characterized by a smooth pen-tip velocity profile. Our study explored fluency when writing single letters in children who were just beginning to learn to handwrite, and the extent to which this was predicted by the children’s pen-control ability and by their letter knowledge. 176 Norwegian children formed letters by copying and from dictation (i.e., in response to hearing letter sounds). Performance on these tasks was assessed in terms of the counts of velocity inversions as the children produced sub-letter features that would be produced by competent handwriters as a single, smooth (ballistic) action. We found that there was considerable variation in these measures across writers, even when producing well-formed letters. Children also copied unfamiliar symbols, completed various pen-control tasks (drawing lines, circles, garlands, and figure eights), and tasks that assessed knowledge of letter sounds and shapes. After controlling for pen-control ability, pen-movement fluency was affected by letter knowledge (specifically children’s performance on a task that required selecting graphemes on the basis of their sound). This was the case when children retrieved letter forms from dictated letter sounds, but also when directly copying letters and, unexpectedly, when copying unfamiliar symbols. These findings suggest that familiarity with a letter affects movement fluency during letter production but may also point towards a more general ability to process new letter-like symbols in children with good letter knowledge.

On the dynamics of a rolling ball actuated by internal point masses

The motion of a rolling ball actuated by internal point masses that move inside the ball’s frame of reference is considered. The equations of motion are derived by applying Euler–Poincaré’s symmetry reduction method in concert with Lagrange–d’Alembert’s principle, which accounts for the presence of the nonholonomic rolling constraint. As a particular example, we consider the case when the masses move along internal rails, or trajectories, of arbitrary shape and fixed within the ball’s frame of reference. Our system of equations can treat most possible methods of actuating the rolling ball with internal moving masses encountered in the literature, such as circular motion of the masses mimicking swinging pendula or straight line motion of the masses mimicking magnets sliding inside linear tubes embedded within a solenoid. Moreover, our method can model arbitrary rail shapes and an arbitrary number of rails such as several ellipses and/or figure eights, which may be important for future designs of rolling ball robots. For further analytical study, we also reduce the system to a single differential equation when the motion is planar, that is, considering the motion of the rolling disk actuated by internal point masses, in which case we show that the results obtained from the variational derivation coincide with those obtained from Newton’s second law. Finally, the equations of motion are solved numerically, illustrating a wealth of complex behaviors exhibited by the system’s dynamics. Our results are relevant to the dynamics of nonholonomic systems containing internal degrees of freedom and to further studies of control of such systems actuated by internal masses.

Alternate Strategies for Optimal Unmanned Aerial Vehicle Thermaling

This paper investigates optimal glider trajectories for gaining altitude from thermal updraft columns (“thermaling”). Thermal updraft columns provide a natural energy resource that can be beneficial for enabling long-term autonomous soaring objectives. In this paper, computational optimal control is used to investigate possible alternatives to the common strategy of circularly orbiting the thermal to harvest potential energy. The Legendre–Gauss–Lobatto pseudospectral method is applied to a simulated glider in multiple scenarios over a range of thermal sizes, thermal intensities, and glider start conditions. Results for the Gedeon thermal model, which simulates dropoff in wind intensity with distance from the thermal core and additionally incorporates external downdrafts, are considered. These are then compared in performance to centered circular orbiting as well as two other alternate heuristic strategies suggested by the optimal control results: off-centered circular orbiting and “figure eights”. Results suggest benefits to using alternatives to centered circular orbiting when encountering small-diameter thermal updrafts. Results also support the utility of centered circular orbiting for medium- to large-diameter thermals while providing new insights into the optimal parameters of the circular orbit.

Analysing knot evidence: associating innate habits with sophisticated tying tasks

Abstract Most ligature cases feature everyday, innately-tied Overhand Knots, Half Hitches and Half Knots. These knots are the result of habitual behaviour and individual tiers demonstrate consistency, except when certain contextual factors come into play. This survey focussed on comparing the chiralities of basic knots to those of Figure Eight Knots, which occur in case evidence and require similar tying actions. It is important to note that real-world Figure Eights are oriented relative to their working ends and are therefore chiral, whereas topological Figure Eights have no ends and are amphichiral. Data summarizing the tying habits of 184 survey respondents were collected and analysed. The majority of volunteers surveyed tied common Overhand Knots and Figure Eights of equal chirality, consistently or nearly consistently, irrespective of any general learning effect. A minority tied knots of opposite chirality. The knots tied by the remaining respondents varied, and the data suggested a potentially complex pattern which may be related to previous findings. Similar but less pronounced patterns were exhibited in the Half Hitch and Half Knot data. This information could be useful when analysing case evidence and making links to suspect samples, provided cautious attention is paid to context and knot function.

Noonan’s Crown

Noonan’s Crown Neal Bowers (bio) —Fred Noonan vanished with Amelia Earhart on July 2, 1937, during an attempt to fly around the world. He was her navigator for the flight. She told me she was not afraid to die,then tried to prove it by the way she flew,full-throttle and at altitudes so highthe breathless sky became a blacker blue. I swear the stars were visible at noonas we rose over daylight and mere time,toward irrelevance of sun and moonbeyond the upper limits of our climb. Then came our mad descent, with barrel rollsand spirals, loop-the-loops and figure eights,which she completed in a death-defying stalland free fall from those unforgiving heights. She was Queen of the Air, in death supreme;I was a phantom in a mirror in a dream. 2 I was a phantom in a mirror in a dream,a blur, a no one, with a name to match, hitchhiking in the slipstream of a famethat was impossible for me to catch. [End Page 359] Perhaps she took my insignificanceas her excuse to lift me into space, then pulverize me to irrelevanceon the stones of some undiscoverable place where even insects wouldn’t come to celebratethe feast of me in all my particles, as she dispersed in ether to intoxicatethe world with her mysterious debacle. I kept my mouth shut, and I went along,because Amelia could do no wrong. 3 Because Amelia could do no wrong,she never took the blame for anythingbut found somebody else to pin it on,even if the fault was her bad piloting. So, when she crashed while taking off, she saidthe ground crew filled the fuel tanks unevenly,then settled on a blown-out tire instead,though I could tell she wanted to blame me. Because she flew by nerve more than by skill,she was a fatal crash that hadn’t happened yet,and flying with her was a terrifying thrill,like lifting up the gun in Russian roulette. My one regret, down to the very bone—I promised not to drink until the trip was done. [End Page 360] 4 I promised not to drink until the trip was done,then modified that promise with a sip or two,and quickly went from saying “just the one”to losing count and pissing on my shoeoutside a bar in some forsaken colony,where we put down for fuel and to breakthe tedium of one another’s company;of course, Amelia did not partakeof any alcoholic beveragebut stayed behind to watch over the plane,striking poses in the silver fuselageof Deco goddess and Egyptian crane.Meanwhile, while I could almost barely think,I thumped the bar for one more thumping drink. 5 I thumped the bar for one more thumping drink,then cursed myself for spilling most of it and called the bartender a goddamned finkwhen he refused to cover the deficit but cut me off, instead, showed me the door,the midnight street without a single light, and never mind the rain, the mangy curthat nipped me as I stepped into the night, my head bent low above my sodden shoes,so sorry for myself I had to laugh and sing a few lines of the Noonan Bluesinto the shearing wind and the storm’s wrath: Wanted me a new life of harmony and joy,but my new wife found Amelia too coy. [End Page 361] 6 My new wife found Amelia too coy—her hair combed as if tousled by the wind,her dungarees and neckerchief a ployto understate her sex but remind menthere was a woman underneath those clothes. “It’s just another kind of tease,” she said,“the tomboy look, the freckles, and the poseof innocence, the way she tips her headand offers up that shy smile for the lens.She pulls men in while pushing them away,which is the heart of her allure, her siren’scall across the drowning rocks where her kind plays.” Scoffing at her foolish jealousy and fears,I...

Physiological modules for generating discrete and rhythmic movements: component analysis of EMG signals.

A central question in Neuroscience is that of how the nervous system generates the spatiotemporal commands needed to realize complex gestures, such as handwriting. A key postulate is that the central nervous system (CNS) builds up complex movements from a set of simpler motor primitives or control modules. In this study we examined the control modules underlying the generation of muscle activations when performing different types of movement: discrete, point-to-point movements in eight different directions and continuous figure-eight movements in both the normal, upright orientation and rotated 90°. To test for the effects of biomechanical constraints, movements were performed in the frontal-parallel or sagittal planes, corresponding to two different nominal flexion/abduction postures of the shoulder. In all cases we measured limb kinematics and surface electromyographic activity (EMG) signals for seven different muscles acting around the shoulder. We first performed principal component analysis (PCA) of the EMG signals on a movement-by-movement basis. We found a surprisingly consistent pattern of muscle groupings across movement types and movement planes, although we could detect systematic differences between the PCs derived from movements performed in each shoulder posture and between the principal components associated with the different orientations of the figure. Unexpectedly we found no systematic differences between the figure eights and the point-to-point movements. The first three principal components could be associated with a general co-contraction of all seven muscles plus two patterns of reciprocal activation. From these results, we surmise that both “discrete-rhythmic movements” such as the figure eight, and discrete point-to-point movement may be constructed from three different fundamental modules, one regulating the impedance of the limb over the time span of the movement and two others operating to generate movement, one aligned with the vertical and the other aligned with the horizontal.

Trajectories ofListeria-type motility in two dimensions

Force generated by actin polymerization is essential in cell motility and the locomotion of organelles or bacteria such as Listeria monocytogenes. Both in vivo and in vitro experiments on actin-based motility have observed geometrical trajectories including straight lines, circles, S-shaped curves, and translating figure eights. This paper reports a phenomenological model of an actin-propelled disk in two dimensions that generates geometrical trajectories. Our model shows that when the evolutions of actin density and force per filament on the disk are strongly coupled to the disk self-rotation, it is possible for a straight trajectory to lose its stability. When the instability is due to a pitchfork bifurcation, the resulting trajectory is a circle; a straight trajectory can also lose stability through a Hopf bifurcation, and the resulting trajectory is an S-shaped curve. We also show that a half-coated disk, which mimics the distribution of functionalized proteins in Listeria, also undergoes similar symmetry-breaking bifurcations when the straight trajectory loses stability. For both a fully coated disk and a half-coated disk, when the trajectory is an S-shaped curve, the angular frequency of the disk self-rotation is different from that of the disk trajectory. However, for circular trajectories, these angular frequencies are different for a fully coated disk but the same for a half-coated disk.

Dancing Embodied Memory: The Choreography of Place in the Peruvian Andes

Dancing Embodied Memory: The Choreography of Place in the Peruvian Andes

Stationary equilibrium singularity distributions in the plane

We characterize all stationary equilibrium point singularity distributions in the plane of logarithmic type, allowing for real, imaginary or complex singularity strengths. The dynamical system follows from the assumption that each of the N singularities moves according to the flow field generated by all the others at that point. For strength vector , the dynamical system is the classical point vortex system obtained from a singular discrete representation of the vorticity field from ideal, incompressible fluid flow. When , it corresponds to a system of sources and sinks, whereas when the system consists of spiral sources and sinks discussed in Kochin et al (1964 Theoretical Hydromechanics 1 (London: Interscience)). We formulate the equilibrium problem as one in linear algebra, , , , where A is a N × N complex skew-symmetric configuration matrix which encodes the geometry of the system of interacting singularities. For an equilibrium to exist, A must have a kernel and must be an element of the nullspace of A. We prove that when N is odd, A always has a kernel, hence there is a choice of for which the system is a stationary equilibrium. When N is even, there may or may not be a non-trivial nullspace of A, depending on the relative position of the points in the plane. We provide examples of evenly and randomly distributed points on curves such as circles, figure eights, flower-petal configurations and spirals. We then show how to classify the stationary equilibria in terms of the singular spectrum of A.

ChemInform Abstract: Figure Eights, Moebius Bands, and More: Conformation and Aromaticity of Porphyrinoids

AbstractReview: 285 refs.

Figure Eights, Möbius Bands, and More: Conformation and Aromaticity of Porphyrinoids

The aromatic character of porphyrins, which has significant chemical and biological consequences, can be substantially altered by judicious modifications of the parent ring system. Expansion of the macrocycle, which is achieved by introducing additional subunits, usually increases the so-called free curvature of the ring, leading to larger angular strain. This strain is reduced by a variety of conformational changes, most notably by subunit inversion and p surface twisting. The latter effect creates a particularly convenient access to Möbius aromatic molecules, whose properties, predicted over 40 years ago, are of considerable theoretical importance. The conformational processes occurring in porphyrin analogues are often coupled to other chemical phenomena, and can thus be exploited as a means of constructing functional molecular devices. In this Review, the structural chemistry of porphyrinoids is discussed in the context of their conformational dynamics and p-electron conjugation.

A Model for the Motion of Listeria in Curved Paths

The movement of certain pathogens, such as Listeria monocytogenes, is commonly believed to be driven by actin polymerization. By hijacking the actin machinery of the host cell, these pathogens can move in the cytoplasm with a speed in the order of 0.1 μm/s. Several microscopic or macroscopic models have been proposed to account for the force generation by actin polymerization, however, a theoretical explanation for the rich trajectories traced out by the pathogens, including circles, winding S curves, translating figure eights, and serpentine shapes etc., is still lacking.

A Microscopic Formulation for the Actin-Driven Motion of Listeria in Curved Paths

Using a generalized Brownian ratchet model that accounts for the interactions of actin filaments with the surface of Listeria mediated by proteins like ActA and Arp2/3, we have developed a microscopic model for the movement of Listeria. Specifically, we show that a net torque can be generated within the comet tail, causing the bacteria to spin about its long axis, which in conjunction with spatially varying polymerization at the surface leads to motions of bacteria in curved paths that include circles, sinusoidal-like curves, translating figure eights, and serpentine shapes, as observed in recent experiments. A key ingredient in our formulation is the coupling between the motion of Listeria and the force-dependent rate of filament growth. For this reason, a numerical scheme was developed to determine the kinematic parameters of motion and stress distribution among filaments in a self-consistent manner. We find that a 5–15% variation in polymerization rates can lead to radii of curvatures of the order of 4–20 μm, measured in experiments. In a similar way, our results also show that most of the observed trajectories can be produced by a very low degree of correlation, <10%, among filament orientations. Since small fluctuations in polymerization rate, as well as filament orientation, can easily be induced by various factors, our findings here provide a reasonable explanation for why Listeria can travel along totally different paths under seemingly identical experimental conditions. Besides trajectories, stress distributions corresponding to different polymerization profiles are also presented. We have found that although some actin filaments generate propelling forces that push the bacteria forward, others can exert forces opposing the movement of Listeria, consistent with recent experimental observations.

Attentional capture by unique temporal change

The time course of attentional capture was measured in the preview search paradigm (Yantis & Joinides, 1984). Participants were first presented a circular preview display of figure eights. After one second two to four line segments of each figure eight were deleted to reveal a search display of letters, with two letters arbitrarily designated as targets. The participant's task was to press one of two keys as rapidly as possible to report target identity. Attentional capture was indexed by the relative ratio of RT slopes, taken over the display size, for targets that appear in unchanged versus unique display item locations. This is based on the assumption that when a unique item draws attention to itself, it will slow search if it happens to be one of the distractor letters. The current study points to a temporal factor, over and above the importance of new objects and relative signal speed, which is critical for attentional capture. Attention is captured by temporally unique events. The results of three experiments show that feature changes capture attention as effectively as new objects provided they occur during a period in which the display is static. Conversely, these same feature changes do not capture attention when they occur simultaneously with other display changes, such as the sudden onset of all items or the deletion of some line segments in all items. Importantly, the results show that this unique event hypothesis applies to changes in color (Experiment 1), in motion (Experiment 2) and even to the sudden appearance of new objects (Experiment 3). These findings highlight the importance of considering both space and time in studies of attentional capture.

Insights on vortex-induced, traveling waves on long risers

This paper is based on portions of a keynote presentation, which explored the boundaries of what is understood about the vibration behavior of long cylinders excited by vortex shedding. The source of data is a recent field experiment on a long flexible cylinder, densely instrumented with fiber optic strain gauges. The paper emphasis is on previously unknown or unexplained phenomena and in some cases offers provocative insights as opposed to conclusive proofs. Three particular topics are covered: (i) the occurrence of peak strain and fatigue damage rates at unexpected locations, (ii) the dominance of traveling wave rather than standing wave response, and (iii) the appearance of stable cylinder trajectories, such as figure eights and crescents in pure traveling wave regions. Potential explanations are offered and areas for promising additional research are proposed.

Complex-variable-function Gaussian beam in strongly nonlocal nonlinear media

We introduce a class of spatial complex-variable-function Gaussian (CVF Gaussian) beams, which are the product of an arbitrary analytic complex variable function and a Gaussian function, in strongly nonlocal nonlinear media (SNNM). The CVF Gaussian beam rotates generally during propagation. By choosing the input power of the beam, we can obtain a CVF Gaussian breather or a CVF Gaussian soliton. We reveal that a stable CVF Gaussian beam can exist in SNNM with different forms, including rotating dipoles, rotating elliptic doughnuts, and rotating figure eights.

Coordination of gaze and hand movements for tracking and tracing in 3D

In this study we have investigated movements in three-dimensional space. Since most studies have investigated planar movements (like ellipses, cloverleaf shapes and “figure eights”) we have compared two generalizations of the two-thirds power law to three dimensions. In particular we have tested whether the two-thirds power law could be best described by tangential velocity and curvature in a plane (compatible with the idea of planar segmentation) or whether tangential velocity and curvature should be calculated in three dimensions. We defined total curvature in three dimensions as the square root of the sum of curvature squared and torsion squared. The results demonstrate that most of the variance is explained by tangential velocity and total curvature. This indicates that all three orthogonal components of movements in 3D are equally important and that movements are truly 3D and do not reflect a concatenation of 2D planar movement segments.In addition, we have studied the coordination of eye and hand movements in 3D by measuring binocular eye movements while subjects move the finger along a curved path. The results show that the directional component and finger position almost superimpose when subjects track a target moving in 3D. However, the vergence component of gaze leads finger position by about 250msec. For drawing (tracing) the path of a visible 3D shape, the directional component of gaze leads finger position by about 225msec, and the vergence component leads finger position by about 400msec. These results are compatible with the idea that gaze leads hand position during drawing movement to assist prediction and planning of hand position in 3D space.