A Systematic Approach to Improving the Performance of Spiral Antenna
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The Ordered Distribution of Natural Numbers on the Square Root Spiral
The Ordered Distribution of Natural Numbers on the Square Root Spiral - Harry K. Hahn - Ludwig-Erhard-Str. 10 D-76275 Et Germanytlingen, Germany ------------------------------ mathematical analysis by - Kay Schoenberger - Humboldt-University Berlin ----------------------------- 20. June 2007 Abstract : Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the “Square Root Spiral“ ( also named as “Spiral of Theodorus” or “Wurzel Spirale“ or “Einstein Spiral” ). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36 … form a highly three-symmetrical system of three spiral graphs, which divide the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. The Square Root Spiral is a geometrical structure which is based on the three basic constants: 1, sqrt2 and π (pi) , and the continuous application of the Pythagorean Theorem of the right angled triangle. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the “golden mean” ( golden section ), which behaves as a self-avoiding-walk- constant in the lattice-like structure of the square root spiral. Contents of the general section Page 1 Introduction to the Square Root Spiral 2 2 Mathematical description of the Square Root Spiral 4 3 The distribution -
Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna
Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. hal-03236909 HAL Id: hal-03236909 https://hal.archives-ouvertes.fr/hal-03236909 Preprint submitted on 26 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna Department of Applied Science and Technology Politecnico di Torino Here we consider some problems from the Finkel's solution book, concerning the length of curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate of Bernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocal or Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on the cone and the Loxodrome. The Versiera will be discussed in detail and the link of its name to the Versine function. Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881 Here we consider some of the problems propose in the Finkel's solution book, having the full title: A mathematical solution book containing systematic solutions of many of the most difficult problems, Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions. -
Archimedean Spirals ∗
Archimedean Spirals ∗ An Archimedean Spiral is a curve defined by a polar equation of the form r = θa, with special names being given for certain values of a. For example if a = 1, so r = θ, then it is called Archimedes’ Spiral. Archimede’s Spiral For a = −1, so r = 1/θ, we get the reciprocal (or hyperbolic) spiral. Reciprocal Spiral ∗This file is from the 3D-XploreMath project. You can find it on the web by searching the name. 1 √ The case a = 1/2, so r = θ, is called the Fermat (or hyperbolic) spiral. Fermat’s Spiral √ While a = −1/2, or r = 1/ θ, it is called the Lituus. Lituus In 3D-XplorMath, you can change the parameter a by going to the menu Settings → Set Parameters, and change the value of aa. You can see an animation of Archimedean spirals where the exponent a varies gradually, from the menu Animate → Morph. 2 The reason that the parabolic spiral and the hyperbolic spiral are so named is that their equations in polar coordinates, rθ = 1 and r2 = θ, respectively resembles the equations for a hyperbola (xy = 1) and parabola (x2 = y) in rectangular coordinates. The hyperbolic spiral is also called reciprocal spiral because it is the inverse curve of Archimedes’ spiral, with inversion center at the origin. The inversion curve of any Archimedean spirals with respect to a circle as center is another Archimedean spiral, scaled by the square of the radius of the circle. This is easily seen as follows. If a point P in the plane has polar coordinates (r, θ), then under inversion in the circle of radius b centered at the origin, it gets mapped to the point P 0 with polar coordinates (b2/r, θ), so that points having polar coordinates (ta, θ) are mapped to points having polar coordinates (b2t−a, θ). -
Analysis of Spiral Curves in Traditional Cultures
Forum _______________________________________________________________ Forma, 22, 133–139, 2007 Analysis of Spiral Curves in Traditional Cultures Ryuji TAKAKI1 and Nobutaka UEDA2 1Kobe Design University, Nishi-ku, Kobe, Hyogo 651-2196, Japan 2Hiroshima-Gakuin, Nishi-ku, Hiroshima, Hiroshima 733-0875, Japan *E-mail address: [email protected] *E-mail address: [email protected] (Received November 3, 2006; Accepted August 10, 2007) Keywords: Spiral, Curvature, Logarithmic Spiral, Archimedean Spiral, Vortex Abstract. A method is proposed to characterize and classify shapes of plane spirals, and it is applied to some spiral patterns from historical monuments and vortices observed in an experiment. The method is based on a relation between the length variable along a curve and the radius of its local curvature. Examples treated in this paper seem to be classified into four types, i.e. those of the Archimedean spiral, the logarithmic spiral, the elliptic vortex and the hyperbolic spiral. 1. Introduction Spiral patterns are seen in many cultures in the world from ancient ages. They give us a strong impression and remind us of energy of the nature. Human beings have been familiar to natural phenomena and natural objects with spiral motions or spiral shapes, such as swirling water flows, swirling winds, winding stems of vines and winding snakes. It is easy to imagine that powerfulness of these phenomena and objects gave people a motivation to design spiral shapes in monuments, patterns of cloths and crafts after spiral shapes observed in their daily lives. Therefore, it would be reasonable to expect that spiral patterns in different cultures have the same geometrical properties, or at least are classified into a small number of common types. -
Superconics + Spirals 17 July 2017 1 Superconics + Spirals Cye H
Superconics + Spirals 17 July 2017 Superconics + Spirals Cye H. Waldman Copyright 2017 Introduction Many spirals are based on the simple circle, although modulated by a variable radius. We have generalized the concept by allowing the circle to be replaced by any closed curve that is topologically equivalent to it. In this note we focus on the superconics for the reasons that they are at once abundant, analytical, and aesthetically pleasing. We also demonstrate how it can be applied to random closed forms. The spirals of interest are of the general form . A partial list of these spirals is given in the table below: Spiral Remarks Logarithmic b = flair coefficient m = 1, Archimedes Archimedean m = 2, Fermat (generalized) m = -1, Hyperbolic m = -2, Lituus Parabolic Cochleoid Poinsot Genesis of the superspiral (1) The idea began when someone on line was seeking to create a 3D spiral with a racetrack planiform. Our first thought was that the circle readily transforms to an ellipse, for example Not quite a racetrack, but on the right track. The rest cascades in a hurry, as This is subject to the provisions that the random form is closed and has no crossing lines. This also brought to mind Euler’s famous derivation, connecting the five most famous symbols in mathematics. 1 Superconics + Spirals 17 July 2017 In the present paper we’re concerned primarily with the superconics because the mapping is well known and analytic. More information on superconics can be found in Waldman (2016) and submitted for publication by Waldman, Chyau, & Gray (2017). Thus, we take where is a parametric variable (not to be confused with the polar or tangential angles). -
Archimedean Spirals * an Archimedean Spiral Is a Curve
Archimedean Spirals * An Archimedean Spiral is a curve defined by a polar equa- tion of the form r = θa. Special names are being given for certain values of a. For example if a = 1, so r = θ, then it is called Archimedes’ Spiral. Formulas in 3DXM: r(t) := taa, θ(t) := t, Default Morph: 1 aa 1.25. − ≤ ≤ For a = 1, so r = 1/θ, we get the− reciprocal (or Archimede’s Spiral hyperbolic) spiral: Reciprocal Spiral * This file is from the 3D-XplorMath project. Please see: http://3D-XplorMath.org/ 1 The case a = 1/2, so r = √θ, is called the Fermat (or hyperbolic) spiral. Fermat’s Spiral While a = 1/2, or r = 1/√θ, it is called the Lituus: − Lituus 2 In 3D-XplorMath, you can change the parameter a by go- ing to the menu Settings Set Parameters, and change the value of aa. You can see→an animation of Archimedean spirals where the exponent a = aa varies gradually, be- tween 1 and 1.25. See the Animate Menu, entry Morph. − The reason that the parabolic spiral and the hyperbolic spiral are so named is that their equations in polar coor- dinates, rθ = 1 and r2 = θ, respectively resembles the equations for a hyperbola (xy = 1) and parabola (x2 = y) in rectangular coordinates. The hyperbolic spiral is also called reciprocal spiral be- cause it is the inverse curve of Archimedes’ spiral, with inversion center at the origin. The inversion curve of any Archimedean spirals with re- spect to a circle as center is another Archimedean spiral, scaled by the square of the radius of the circle. -
Medium Power, Compact Periodic Spiral Antenna Jonathan O'brien University of South Florida, [email protected]
University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School January 2013 Medium Power, Compact Periodic Spiral Antenna Jonathan O'brien University of South Florida, [email protected] Follow this and additional works at: http://scholarcommons.usf.edu/etd Part of the Electrical and Computer Engineering Commons Scholar Commons Citation O'brien, Jonathan, "Medium Power, Compact Periodic Spiral Antenna" (2013). Graduate Theses and Dissertations. http://scholarcommons.usf.edu/etd/4926 This Thesis is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. Medium Power, Compact Periodic Spiral Antenna by Jonathan M. O’Brien A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering Department of Electrical Engineering College of Engineering University of South Florida Major Professor: Thomas M. Weller, Ph.D. Gokhan Mumcu, Ph.D. Huseyin Arslan, Ph.D. John Grandfield, B.S.E.E. Date of Approval: November 8, 2013 Keywords: Three Dimensional Miniaturization, Ultra-Wideband Antennas, Thermal Modeling, Miniaturization Techniques, Loop Antenna Copyright © 2013, Jonathan M. O’Brien Dedication To my family My mother who never let me quit at anything My father who taught me that life isn’t a destination Joey who made me understand more about life than I could ever explain My brother for being my best competitor in school My best sister who can always make me laugh Acknowledgments Words cannot express the gratitude and respect I have for my advisor, Prof. -
Highlights of Antenna History
~~ IEEE COMMUNICATIONS MAGAZINE HlOHLlOHTS OF ANTENNA HISTORY JACK RAMSAY A look at the major events in the development of antennas. wires. Antenna systems similar to Edison’s were used by A. E. Dolbear in 1882 when he successfully and somewhat mysteriously succeeded in transmitting code and even speech to significant ranges, allegedly by groundconduction. NINETEENTH CENTURY WIRE ANTENNAS However, in one experiment he actually flew the first kite T is not surprising that wire antennas were inaugurated antenna.About the same time, the Irish professor, in 1842 by theinventor of wire telegraphy,Joseph C. F. Fitzgerald, calculated that a loop would radiate and that Henry, Professor’ of Natural Philosophy at Princeton, a capacitance connected to a resistor would radiate at VHF NJ. By “throwing a spark” to a circuit of wire in an (undoubtedly due to radiation from the wire connecting leads). Iupper room,Henry found that thecurrent received in a In Hertz launched,processed, and received radio 1887 H. parallel circuit in a cellar 30 ft below codd.magnetize needies. waves systematically. He used a balanced or dipole antenna With a vertical wire from his study to the roof of his house, he attachedto ’ an induction coilas a transmitter, and a detected lightning flashes 7-8 mi distant. Henry also sparked one-turn loop (rectangular) containing a sparkgap as a to a telegraph wire running from his laboratory to his house, receiver. He obtained “sympathetic resonance” by tuning the and magnetized needles in a coil attached to a parailel wire dipole with sliding spheres, and the loop by adding series 220 ft away. -
APS March Meeting 2012 Boston, Massachusetts
APS March Meeting 2012 Boston, Massachusetts http://www.aps.org/meetings/march/index.cfm i Monday, February 27, 2012 8:00AM - 11:00AM — Session A51 DCMP DFD: Colloids I: Beyond Hard Spheres Boston Convention Center 154 8:00AM A51.00001 Photonic Droplets Containing Transparent Aqueous Colloidal Suspensions with Optimal Scattering Properties JIN-GYU PARK, SOFIA MAGKIRIADOU, Department of Physics, Harvard University, YOUNG- SEOK KIM, Korea Electronics Technology Institute, VINOTHAN MANOHARAN, Department of Physics, Harvard University, HARVARD UNIVERSITY TEAM, KOREA ELECTRONICS TECHNOLOGY INSTITUTE COLLABORATION — In recent years, there has been a growing interest in quasi-ordered structures that generate non-iridescent colors. Such structures have only short-range order and are isotropic, making colors invariant with viewing angle under natural lighting conditions. Our recent simulation suggests that colloidal particles with independently controlled diameter and scattering cross section can realize the structural colors with angular independence. In this presentation, we are exploiting depletion-induced assembly of colloidal particles to create isotropic structures in a milimeter-scale droplet. As a model colloidal particle, we have designed and synthesized core-shell particles with a large, low refractive index shell and a small, high refractive index core. The remarkable feature of these particles is that the total cross section for the entire core-shell particle is nearly the same as that of the core particle alone. By varying the characteristic length scales of the sub-units of such ‘photonic’ droplet we aim to tune wavelength selectivity and enhance color contrast and viewing angle. 8:12AM A51.00002 Curvature-Induced Capillary Interaction between Spherical Particles at a Liquid Interface1 , NESRIN SENBIL, CHUAN ZENG, BENNY DAVIDOVITCH, ANTHONY D. -
Dynamics of Self-Organized and Self-Assembled Structures
This page intentionally left blank DYNAMICS OF SELF-ORGANIZED AND SELF-ASSEMBLED STRUCTURES Physical and biological systems driven out of equilibrium may spontaneously evolve to form spatial structures. In some systems molecular constituents may self-assemble to produce complex ordered structures. This book describes how such pattern formation processes occur and how they can be modeled. Experimental observations are used to introduce the diverse systems and phenom- ena leading to pattern formation. The physical origins of various spatial structures are discussed, and models for their formation are constructed. In contrast to many treatments, pattern-forming processes in nonequilibrium systems are treated in a coherent fashion. The book shows how near-equilibrium and far-from-equilibrium modeling concepts are often combined to describe physical systems. This interdisciplinary book can form the basis of graduate courses in pattern forma- tion and self-assembly. It is a useful reference for graduate students and researchers in a number of disciplines, including condensed matter science, nonequilibrium sta- tistical mechanics, nonlinear dynamics, chemical biophysics, materials science, and engineering. Rashmi C. Desai is Professor Emeritus of Physics at the University of Toronto, Canada. Raymond Kapral is Professor of Chemistry at the University of Toronto, Canada. DYNAMICS OF SELF-ORGANIZED AND SELF-ASSEMBLED STRUCTURES RASHMI C. DESAI AND RAYMOND KAPRAL University of Toronto CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521883610 © R. -
Size Reduction of an Uwb Low-Profile Spiral Antenna
SIZE REDUCTION OF AN UWB LOW-PROFILE SPIRAL ANTENNA DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Brad A. Kramer, M.S., B.S. ***** The Ohio State University 2007 Dissertation Committee: Approved by Professor John L. Volakis, Adviser Adjunct Professor Chi-Chih Chen Adviser Associate Professor Fernando L. Teixeira Graduate Program in Electrical and Computer Engineering c Copyright by Brad A. Kramer 2007 ABSTRACT Fundamental physical limitations restrict antenna performance based on its elec- trical size alone. These fundamental limitations are of the utmost importance since the minimum size needed to achieve a particular figure of merit can be determined from them. In this dissertation, the physical limitations on the size reduction of a broadband antenna is examined theoretically and experimentally. This is in contrast to previous research that focused on narrowband antennas. Specifically, size reduction using antenna miniaturization techniques is considered and explored through the ap- plication of high-contrast material and reactive loading. A particular example is the miniaturization of a broadband spiral using readily available high-contrast dielectrics and a novel inductive loading technique. Using either dielectric or inductive loading, it is shown that the size can be reduced by more than a factor of two which is close to the observed theoretical limit. To enable the realization of a conformal antenna without the loss of the antenna’s broadband characteristics, a novel ground plane is introduced. The proposed ground plane consists of a traditional metallic ground plane coated with a layer of ferrite material. -
Rapid Prototyping of Bio-Inspired Dielectric Resonator Antennas for Sub-6 Ghz Applications
micromachines Article Rapid Prototyping of Bio-Inspired Dielectric Resonator Antennas for Sub-6 GHz Applications Valeria Marrocco 1,* , Vito Basile 1 , Ilaria Marasco 2,3 , Giovanni Niro 2,3, Luigi Melchiorre 2 , Antonella D’Orazio 2, Marco Grande 2 and Irene Fassi 4 1 STIIMA CNR, Institute of Intelligent Industrial Technologies and Systems for Advanced Manufacturing, National Research Council, Via P. Lembo, 38/F, 70124 Bari, Italy; [email protected] 2 Politecnico di Bari, Dipartimento di Ingegneria Elettrica e dell’Informazione, Via E. Orabona 4, 70125 Bari, Italy; [email protected] (I.M.); [email protected] (G.N.); [email protected] (L.M.); [email protected] (A.D.); [email protected] (M.G.) 3 Center for Biomolecular Nanotechnolgies, Istituto Italiano di Tecnologia (IIT), Via E. Barsanti 14, 73010 Arnesano, Italy 4 STIIMA CNR, Institute of Intelligent Industrial Technologies and Systems for Advanced Manufacturing, National Research Council, Via A. Corti, 12, 20133 Milan, Italy; [email protected] * Correspondence: [email protected] Abstract: Bio-inspired Dielectric Resonator Antennas (DRAs) are engaging more and more attention from the scientific community due to their exceptional wideband characteristic, which is especially desirable for the implementation of 5G communications. Nonetheless, since these antennas exhibit peculiar geometries in their micro-features, high dimensional accuracy must be accomplished via the selection of the most suitable fabrication process. In this study, the challenges to the manufacturing process presented by the wideband Spiral shell Dielectric Resonator Antenna (SsDRA), based on Citation: Marrocco, V.; Basile, V.; the Gielis superformula, are addressed.