DOCSLIB.ORG

The Mysterious Crop Circles

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Mysterious Crop Circles George and Iris Owen 55 Charles St. West, #1201 Toronto, Ontario, Canada M5S 2W9 THE MYSTERIOUS CROP CIRCLES. A New Horizons Note. Copyright: New Horizons Research Foundation. December 1991. INTRODUCTION. About ten years ago a fascinating, strange, and unusual phenomenon made its appearance. It seemed to have little relationship to any other known phenomena, and it didn't seem to belong under any category. However, a number of claims have been made over the years which attempt to categorise the phenomena as belonging to some one particular belief or category. We refer to the phenomenon known as "crop circles". The discipline concerning the study of these circles is known as "cerealogy". This paper is an attempt to draw together some of the information about crop circles, and arrive at an understanding of what is known, or believed, about them, today. The phenomenon has become widely known during the last ten years, and in fact, in its present form, it has apparently only been occurring during the last ten years. It takes the form of large patterns of perfectly bent corn being formed in fields of growing corn. Initially, these were large beautifully formed circles, hence the name; but, more recently, the patterns formed have changed, increasing in complexity from year to year; the patterns that appeared last year, for instance, were not only extremely complex, but they were indeed beautiful. Most of these patterns in the cornfields have appeared in a certain part of England, namely the south western area, in the vicinity of Stonehenge, and near the other various ancient monuments of that region. However, they have also been reported in other parts of England, in the Midlands and eastern regions, and in other countries, notably Canada, Australia, Germany, New Zealand, France, The United States of America, and one has been reported in Sweden. Although the phenomenon has caused so much excitement in recent years, stories of simple circles appearing in fields of growing corn go back many centuries. There are legends dating from the Middle Ages that talk of circles being formed in the fields overnight, and these were attributed to fairies dancing through the corn, or it was said mowing devils came in the night and cut the corn in'rings. One of us (IMO) grew up on an arable farm in eastern England and remembers such simple circles being found in the cornfields in the 1920's and 1930s when harvest time came around. Local farmers attributed them to weather disturbances, the general theory being that they were caused by miniature whirlwinds. This was before there were many aircraft in the sky, and as the circles were usually situated right in the middle of the cornfields, they were not usually discovered until the machines cutting the corn reached them. They were invisible from the sides of the fields, and so it was impossible to know exactly when they were formed. It goes without saying that nobody walks through a field of growing corn, it is so easy to damage it. In these early circles it was sometimes obvious that the circle had been formed earlier in the growing season, as the com was lying flat, but the ears had continued to grow, and were putting out green shoots. The present rash of phenomena started with just such circles, a simple round, some twenty or more feet in diameter, with the corn laying down, all in one direction. It was laid perfectly smoothly, as if it had been swept downwards by some strange force. Typically the roots were not damaged, and the stalks were not broken, only bent. The corn continued to grow in this position. The circles usually appeared overnight, and were quickly noticed, not only by pilots of aircraft overflying the fields, but also because in many cases they appeared in fields overlooked by hilly country. However, this simple pattern was not to remain for long. During succeeding years the patterns grew more and more elaborate, first taking the form of double and triple rings around the original circle, and then acquiring further extensions that in many cases looked like hieroglyphics, or some primitive attempt at communication. Recent circles were also considerably larger, some being as many as hundreds of feet in diameter, and occupying in their design some thousands of feet of cornfield. For many farmers these formations proved more than a minor problem. Not only were large tracts of growing corn badly damaged, and in some cases ruined, but the rest of the farmer's crops were usually trampled by the rush of the sightseers and investigators who came in the wake of such sightings. Many farmers had their whole crops destroyed. If, as has been suggested, some of these circles are caused by hoaxers and pranksters, it is a very irresponsible course of action indeed, and one would think that the farmers who suffer would have a claim for compensation. THE PHENOMENA. This current rash of phenomena started in 1980, when a large number of circles were found in the south western part of England. They appeared to be cut with almost surgical precision in fields of growing wheat and barley. The earlier circles were always reported as being in cornfields, but of recent years the patterns have been found in fields of rape, among growing root crops, such as sugar beet, in hayfmelds, in sugar cane in Australia, and there are even a few reports of the formations occurring on the sides of dusty roads, and one is reported as happening in a field of prickly thistles. An interesting feature of many of these patterns is that even after the crop has been harvested, the place where the circle has been is still obvious, as if the ground surface were in some way changed. It seems, in these cases, as if something more than just the bending of the corn took place. However, all traces of the patterns disappears when winter comes and the soil is prepared for next year's crop. Also, there are no reports, as far as we are aware, of a circle, or pattern, reappearing on the same spot the following year, or of the crop being in any way affected the next year. If the soil had been affected by either chemicals, or burning, one would expect subsequent crops to show evidence of this. As we have said in the Introduction, originally these patterns took the form of simple circles, some small, and others huge in diameter, but during the succeeding years the patterns took on the forms of oblongs, squares, rectangles, and triangles, and other geometrical patterns. Circles would be connected, or there would be a large circle with several small ones grouped around, or a series of rings would surround an inner circle. Later still the patterns became even more complicated, and some took on insect-like forms, others produced squiggly patterns, likened to brain patterns, and others resembled a child's scribbling. We have attached a number of examples to this paper. During the ten years from 1980 - 1990, almost every conceivable kind of pattern emerged. In all the reported cases the corn is bent in one direction only, it is not trampled haphazardly, it is not crumpled, it lays straight. The stalks, although bent, are not broken, and the corn continues to grow in the ground. Corn continues to grow and ripen when the stalks are bent, if the roots are not pulled out and the stalks not broken. The corn will continue to ripen, and when ripe, because of the contact with the ground,the wheat in the ears will sprout green shoots. The loss to the farmer lies in the fact that he is unable to reap the harvest. A further word of explanation. In all the published pictures there are straight lines running the length of the fields. These straight lines are furrows made by the wheels of the tractors that sowed the seed in the springtime. The seed does not fall in these furrows, and so these lines are left when the corn grows. Because modern machinery is often very large, and the vehicles have wide wheels these furrows can be wide enough for persons to walk in without damaging the corn. In the days when horses:pulled the seeding machinery these furrows did not exist. If one deviates from the tracks, however, the corn will be damaged. Growing corn is very heavy in the ears, and it is easy to damage it, and the damage is obvious. This is a factor to take into consideration if one is looking at the possibility of these patterns being the result of hoaxes or practical jokes. The patterns usually have occurred during the months of July and August, when the crops are ripening. However, in 1991, the weather in England in the springtime was cold and wet, and so the harvest was later than usual. However, in many cases, the circles started forming at the usual time, appearing in the young, still green, corn. This would cause one to wonder if the time of year was a factor, and not necessarily the ripeness of the corn. This last year, 1991. saw some very varied and complicated pattern formations indeed. The earlier ones were labelled 'insectograms' by the investigators, as they resembled some weird kind of insect. About a dozen 'insectograms' appeared in the counties of Wiltshire and Hampshire during the early 1991 'season'. As the summer went on further strange shapes and patterns appeared. A report in the Fortean Times giving the types of formations during that summer lists 'dumb-bells', (consisting of two or sometimes three plain or ringed circles joined by a narrow corridor)} haloes were sometimes added to these shapes, or perhaps rectangular 'coffins' or other appendages.
Load more

Recommended publications
  • On the Mandelbrot Set for I2 = ±1 and Imaginary Higgs Fields
    Journal of Advances in Applied Mathematics, Vol. 6, No. 2, April 2021 https://dx.doi.org/10.22606/jaam.2021.62001 27 On the Mandelbrot Set for i2 = ±1 and Imaginary Higgs Fields Jonathan Blackledge Stokes Professor, Science Foundation Ireland. Distinguished Professor, Centre for Advanced Studies, Warsaw University of Technology, Poland. Visiting Professor, Faculty of Arts, Science and Technology, Wrexham Glyndwr University of Wales, UK. Professor Extraordinaire, Faculty of Natural Sciences, University of Western Cape, South Africa. Honorary Professor, School of Electrical and Electronic Engineering, Technological University Dublin, Ireland. Honorary Professor, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, South Africa. Email: [email protected] Abstract We consider the consequence√ of breaking with a fundamental result in complex analysis by letting i2 = ±1 where i = −1 is the basic unit of all imaginary numbers. An analysis of the Mandelbrot set for this case shows that a demarcation between a Fractal and a Euclidean object is possible based on i2 = −1 and i2 = +1, respectively. Further, we consider the transient behaviour associated with the two cases to produce a range of non-standard sets in which a Fractal geometric structure is transformed into a Euclidean object. In the case of the Mandelbrot set, the Euclidean object is a square whose properties are investigate. Coupled with the associated Julia sets and other complex plane mappings, this approach provides the potential to generate a wide range of new semi-fractal structures which are visually interesting and may be of artistic merit. In this context, we present a mathematical paradox which explores the idea that i2 = ±1.
    [Show full text]
  • On the Numerical Construction Of
    ON THE NUMERICAL CONSTRUCTION OF HYPERBOLIC STRUCTURES FOR COMPLEX DYNAMICAL SYSTEMS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Jennifer Suzanne Lynch Hruska August 2002 c 2002 Jennifer Suzanne Lynch Hruska ALL RIGHTS RESERVED ON THE NUMERICAL CONSTRUCTION OF HYPERBOLIC STRUCTURES FOR COMPLEX DYNAMICAL SYSTEMS Jennifer Suzanne Lynch Hruska, Ph.D. Cornell University 2002 Our main interest is using a computer to rigorously study -pseudo orbits for polynomial diffeomorphisms of C2. Periodic -pseudo orbits form the -chain re- current set, R. The intersection ∩>0R is the chain recurrent set, R. This set is of fundamental importance in dynamical systems. Due to the theoretical and practical difficulties involved in the study of C2, computers will presumably play a role in such efforts. Our aim is to use computers not only for inspiration, but to perform rigorous mathematical proofs. In this dissertation, we develop a computer program, called Hypatia, which locates R, sorts points into components according to their -dynamics, and inves- tigates the property of hyperbolicity on R. The output is either “yes”, in which case the computation proves hyperbolicity, or “not for this ”, in which case infor- mation is provided on numerical or dynamical obstructions. A diffeomorphism f is hyperbolic on a set X if for each x there is a splitting of the tangent bundle of x into an unstable and a stable direction, with the unstable (stable) direction expanded by f (f −1). A diffeomorphism is hyperbolic if it is hyperbolic on its chain recurrent set.
    [Show full text]
  • Fast Visualisation and Interactive Design of Deterministic Fractals
    Computational Aesthetics in Graphics, Visualization, and Imaging (2008) P. Brown, D. W. Cunningham, V. Interrante, and J. McCormack (Editors) Fast Visualisation and Interactive Design of Deterministic Fractals Sven Banisch1 & Mateu Sbert2 1Faculty of Media, Bauhaus–University Weimar, D-99421 Weimar (GERMANY) 2Department of Informàtica i Matemàtica Aplicada, University of Girona, 17071 Girona (SPAIN) Abstract This paper describes an interactive software tool for the visualisation and the design of artistic fractal images. The software (called AttractOrAnalyst) implements a fast algorithm for the visualisation of basins of attraction of iterated function systems, many of which show fractal properties. It also presents an intuitive technique for fractal shape exploration. Interactive visualisation of fractals allows that parameter changes can be applied at run time. This enables real-time fractal animation. Moreover, an extended analysis of the discrete dynamical systems used to generate the fractal is possible. For a fast exploration of different fractal shapes, a procedure for the automatic generation of bifurcation sets, the generalizations of the Mandelbrot set, is implemented. This technique helps greatly in the design of fractal images. A number of application examples proves the usefulness of the approach, and the paper shows that, put into an interactive context, new applications of these fascinating objects become possible. The images presented show that the developed tool can be very useful for artistic work. 1. Introduction an interactive application was not really feasible till the ex- ploitation of the capabilities of new, fast graphics hardware. The developments in the research of dynamical systems and its attractors, chaos and fractals has already led some peo- The development of an interactive fractal visualisation ple to declare that god was a mathematician, because mathe- method, is the main objective of this paper.
    [Show full text]
  • Arxiv:1808.10408V1 [Math.DS] 30 Aug 2018 Set Md ⊂ C Is Defined As the Set of C Such That the Julia Set J(Fc) Is Con- Nected
    ON THE INHOMOGENEITY OF THE MANDELBROT SET YUSHENG LUO Abstract. We will show the Mandelbrot set M is locally conformally inhomogeneous: the only conformal map f defined in an open set U intersecting @M and satisfying f(U \ @M) ⊂ @M is the identity map. The proof uses the study of local conformal symmetries of the Julia sets of polynomials: we will show in many cases, the dynamics can be recovered from the local conformal structure of the Julia sets. 1. Introduction Given a monic polynomial f(z), the filled Julia set is n K = K(f) = fz 2 C :(f (z))n2N is boundedg 2 and the Julia set is J = @K. For quadratic family fc(z) = z + c, the Mandelbrot set M can be defined as the subset in the parameter plane such that the Julia set Jc is connected, M = fc 2 C : Jc is connectedg Let A be a compact subset of C. We call an orientation preserving home- omorphism H : U −! V a local conformal symmetry of A if H is conformal, U is connected and H sends U \ A onto V \ A. We say a local conformal symmetry is trivial if it is the identity map or U \ A = ;. In this paper, we will show Theorem 1.1. The boundary of the Mandelbrot set @M admits no non- trivial local conformal symmetries. d Remark 1.2. For a fixed d ≥ 2, and let fc(z) = z + c. The Multibrot arXiv:1808.10408v1 [math.DS] 30 Aug 2018 set Md ⊂ C is defined as the set of c such that the Julia set J(fc) is con- nected.
    [Show full text]
  • Quaternion Julia Set Shape Optimization
    Eurographics Symposium on Geometry Processing 2015 Volume 34 (2015), Number 5 Mirela Ben-Chen and Ligang Liu (Guest Editors) Quaternion Julia Set Shape Optimization Theodore Kim Media Arts and Technology Program University of California, Santa Barbara (a) (b) (c) Figure 1: (a) The original Bunny. (b) Julia set of the 331-root rational map found by our optimization. (c) Highly intricate surface obtained by translating the roots by 1.41 in the z direction. Image is high-resolution; please zoom in to see details. Abstract We present the first 3D algorithm capable of answering the question: what would a Mandelbrot-like set in the shape of a bunny look like? More concretely, can we find an iterated quaternion rational map whose potential field contains an isocontour with a desired shape? We show that it is possible to answer this question by casting it as a shape optimization that discovers novel, highly complex shapes. The problem can be written as an energy minimization, the optimization can be made practical by using an efficient method for gradient evaluation, and convergence can be accelerated by using a variety of multi-resolution strategies. The resulting shapes are not in- variant under common operations such as translation, and instead undergo intricate, non-linear transformations. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations 1. Introduction variants [LLC∗10], and similar methods such as hypertextur- ing [EMP∗02]. These methods are widely employed to add The problem of robustly generating smooth geometry with details once the overall geometry has been finalized, e.g.
    [Show full text]
  • Laplacians on Julia Sets of Rational Maps
    LAPLACIANS ON JULIA SETS OF RATIONAL MAPS MALTE S. HASSLER, HUA QIU, AND ROBERT S. STRICHARTZ Abstract. The study of Julia sets gives a new and natural way to look at fractals. When mathematicians investigated the special class of Misiurewicz's rational maps, they found out that there is a Julia set which is homeomorphic to a well known fractal, the Sierpinski gasket. In this paper, we apply the method of Kigami to give rise to a new construction of Laplacians on the Sierpinski gasket like Julia sets with a dynamically invariant property. 1. Introduction Recall that the familiar Sierpinski gasket (SG) is generated in the following way. Starting with a triangle one divides it into four copies, removes the central one, and repeats the iterated process. One aspect of the study of this fractal stems from the analytical construction of a Laplacian developed by Jun Kigami in 1989 [10]. Since then, the analysis on SG has been extensively investigated from various viewpoints. The theory has been extended to some other fractals [11], too. And for some of them one could say they are more invented like SG than discovered. A standard reference to this topic is the book of Robert S. Strichartz [15]. When mathematicians studied the behaviour of polynomial maps under itera- tion, at first, they did not think about fractals. Given a starting point z0 2 C and a polynomial P (z) they wanted to know, whether the sequence of iterations P (z0);P (P (z0))::: converges. They named the set of points that show this behaviour the filled Julia set and its boundary the Julia set after the French mathematician Gaston Julia.
    [Show full text]
  • Self-Similarity of the Mandelbrot Set
    Self-similarity of the Mandelbrot set Wolf Jung, Gesamtschule Aachen-Brand, Aachen, Germany. 1. Introduction 2. Asymptotic similarity 3. Homeomorphisms 4. Quasi-conformal surgery The images are made with Mandel, a program available from www.mndynamics.com. Talk given at Perspectives of modern complex analysis, Bedlewo, July 2014. 1a Dynamics and Julia sets 2 n Iteration of fc(z) = z + c . Filled Julia set Kc = fz 2 C j fc (z) 6! 1g . Kc is invariant under fc(z) . n fc (z) is asymptotically linear at repelling n-periodic points. 1b The Mandelbrot set Parameter plane of quadratic polynomials 2 fc(z) = z + c . Mandelbrot set M = fc 2 C j c 2 Kcg . Two kinds of self-similarity: • Convergence of rescaled subsets: geometry is asymptotically linear. • Homeomorphisms between subsets: non-linear, maybe quasiconformal. 2. Asymptotic similarity 134 Subsets M ⊃ Mn ! fag ⊂ @M with 2a Misiurewicz points asymptotic model: 'n(Mn) ! Y in 2b Multiple scales Hausdorff-Chabauty distance. 2c The Fibonacci parameter 2d Elephant and dragon Or convergence of subsets of different 2e Siegel parameters Julia sets: c 2 M , K ⊂ K with n n n cn 2f Feigenbaum doubling asymptotics n(Kn) ! Z. 2g Local similarity 2h Embedded Julia sets Maybe Z = λY . 2a Misiurewicz points n Misiurewicz point a with multiplier %a. Blowing up both planes with %a gives λ(M − a) ≈ (Ka − a) asymptotically. Classical result by Tan Lei. Error bound by Rivera-Letelier. Generalization to non-hyperbolic semi-hyperbolic a. Proof by Kawahira with Zalcman Lemma. - 2b Multiple scales (1) −n Centers cn ! a with cn ∼ a + K%a and small Mandelbrot sets of diameter −2n j%aj .
    [Show full text]
  • Fractal Geometry: the Mandelbrot and Julia Sets
    Fractal Geometry: The Mandelbrot and Julia Sets Stephanie Avalos-Bock July, 2009 1 Introduction The Mandelbrot set is a set of values c ∈ C with certain important proper- ties. We will examine the formal definition of the set as well as many of its interesting, strange, and beautiful properties. The Mandelbrot set is most well known outside of mathematics as a set of beautiful images of fractals; this is partially thanks to the work of Heinz-Otto Peitgen and Peter Richter. The Mandelbrot set is relevant to the fields of complex dynamics and chaos theory, as well as the study of fractals. Here we will delve into the mathematics behind the phenomenon of the Mandelbrot set. 2 Formal Definitions First we will give some definitions of the Mandelbrot set as a whole, and then of some related concepts and characteristics. Definition 2.1. First we must define a set of complex quadratic polynomials 2 Pc : z → z + c. The Mandelbrot set M is the set of all c-values such that the sequence 0,Pc(0),Pc(Pc(0)),... does not escape to infinity, z, c ∈ C. 2.1 Alternate definitions of the Mandelbrot set n Definition 2.2. The Mandelbrot set is the set M = {c ∈ C|∃s ∈ R, ∀n ∈ N, |Pc (0)| ≤ s} n where Pc (z) is the nth iterate of Pc(z). (Note that the same result is achieved by replacing s with 2 because the Mandelbrot set falls entirely within a circle of radius 2 and centered at the origin on the complex plane.) Definition 2.3.
    [Show full text]
  • An Introduction to Fatou and Julia Sets
    AN INTRODUCTION TO JULIA AND FATOU SETS SCOTT SUTHERLAND ABSTRACT. We give an elementary introduction to the holomorphic dynamics of mappings on the Riemann sphere, with a focus on Julia and Fatou sets. Our main emphasis is on the dynamics of polynomials, especially quadratic polynomials. 1. INTRODUCTION In this note, we briefly introduce some of the main elementary ideas in holomorphic dynamics. This is by no means a comprehensive coverage. We do not discuss the Mandelbrot set, despite its inherent relevance to the subject; the Mandelbrot set is covered in companion articles by Devaney [Dev13, Dev14]. For a more detailed and and comprehensive introduction to this subject, the reader is refered to the book [Mil06] by John Milnor, the survey article [Bla84] by Blanchard, the text by Devaney [Dev89], or any of several other introductory texts ([Bea91], [CG93], etc.) Holomorphic dynamics is the study of the iterates of a holomorphic map f on a complex man- ifold. Classically, this manifold is one of the complex plane C, the punctured plane C∗, or the Riemann sphere Cb = C [ f1g. In these notes, we shall primarily restrict our attention to the last case, where f is a rational map (in fact, most examples will be drawn from quadratic polynomials f(z) = z2 + c, with z and c 2 C). Based on the behavior of the point z under iteration of f, the Riemann sphere Cb is partitioned into two sets • The Fatou set Ff (or merely F), on which the dynamics are tame, and • The Julia set Jf (or J ), where there is sensitive dependence on initial conditions and the dynamics are chaotic.
    [Show full text]
  • Contents 5 Introduction to Complex Dynamics
    Dynamics, Chaos, and Fractals (part 5): Introduction to Complex Dynamics (by Evan Dummit, 2015, v. 1.02) Contents 5 Introduction to Complex Dynamics 1 5.1 Dynamical Properties of Complex-Valued Functions . 1 5.1.1 Complex Arithmetic . 2 5.1.2 The Complex Derivative and Holomorphic Functions . 3 5.1.3 Attracting, Repelling, and Neutral Fixed Points and Cycles . 6 5.1.4 Orbits of Linear Maps on C ..................................... 9 5.1.5 Some Properties of Orbits of Quadratic Maps on C ........................ 10 5.2 Julia Sets For Polynomials . 11 5.2.1 Denition of the Julia Set and Basic Examples . 12 5.2.2 Computing the Filled Julia Set: Escape-Time Plotting . 13 5.2.3 Computing the Julia Set: Backwards Iteration . 16 5.3 Additional Properties of Julia Sets, The Mandelbrot Set . 18 5.3.1 General Julia Sets . 18 5.3.2 Julia Sets and Cantor Sets . 19 5.3.3 Critical Orbits and the Fundamental Dichotomy . 21 5.3.4 The Mandelbrot Set . 22 5.3.5 Attracting Orbits and Bulbs of the Mandelbrot Set . 23 5 Introduction to Complex Dynamics In this chapter, our goal is to discuss complex dynamical systems in the complex plane. As we will discuss, much of the general theory of real-valued dynamical systems will carry over to the complex setting, but there are also a number of new behaviors that will occur. We begin with a brief review of the complex numbers and dene the complex derivative d of a complex-valued C dz function, and then turn our attention to the basic study of complex dynamical systems: xed points and periodic points, attracting and repelling xed points and cycles, behaviors of neutral xed points, attracting basins, and orbit analysis.
    [Show full text]
  • The Sets of Julia and Mandelbrot for Multi- Dimensional Case of Logistic Mapping
    Central Asian Problems of Modern Science and Education Volume 2020 Issue 3 Central Asian Problems of Modern Article 4 Science and Education 2020-3 11-19-2020 THE SETS OF JULIA AND MANDELBROT FOR MULTI- DIMENSIONAL CASE OF LOGISTIC MAPPING R.N. Ganikhodzhaev National University of Uzbekistan named after Mirzo Ulugbek Tashkent, Uzbekistan, [email protected] Sh.J Seytov National University of Uzbekistan named after Mirzo Ulugbek Tashkent, Uzbekistan, [email protected] I.N Obidjonov Urgench State University, [email protected] L.O Sadullayev Urgench State University, [email protected] Follow this and additional works at: https://uzjournals.edu.uz/capmse Part of the Physical Sciences and Mathematics Commons Recommended Citation Ganikhodzhaev, R.N.; Seytov, Sh.J; Obidjonov, I.N; and Sadullayev, L.O (2020) "THE SETS OF JULIA AND MANDELBROT FOR MULTI-DIMENSIONAL CASE OF LOGISTIC MAPPING," Central Asian Problems of Modern Science and Education: Vol. 2020 : Iss. 3 , Article 4. Available at: https://uzjournals.edu.uz/capmse/vol2020/iss3/4 This Article is brought to you for free and open access by 2030 Uzbekistan Research Online. It has been accepted for inclusion in Central Asian Problems of Modern Science and Education by an authorized editor of 2030 Uzbekistan Research Online. For more information, please contact [email protected]. UDK 51 (075) The sets of Julia and Mandelbrot for multi-dimensional case of logistic mapping Ganikhodzhaev R.N. National University of Uzbekistan named after Mirzo Ulugbek Tashkent, Uzbekistan e-mail: [email protected] Seytov Sh.J. National University of Uzbekistan named after Mirzo Ulugbek Tashkent, Uzbekistan e-mail: [email protected] Obidjonov I.N Urgench State University, Urgench, Uzbekistan e-mail: [email protected] Sadullayev L.O.
    [Show full text]
  • Generation of Julia and Mandelbrot Sets Via Fixed Points
    S S symmetry Article Generation of Julia and Mandelbrot Sets via Fixed Points Mujahid Abbas 1,2, Hira Iqbal 3 and Manuel De la Sen 4,* 1 Department of Mathematics, Government College University, Lahore 54000, Pakistan; [email protected] 2 Department of Medical Research, China Medical University No. 91, Hsueh-Shih Road, Taichung 400, Taiwan 3 Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, Lahore 54000, Pakistan; [email protected] 4 Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644, Bilbao, Barrio Sarriena, 48940 Leioa, Spain * Correspondence: [email protected] Received: 27 November 2019; Accepted: 26 December 2019; Published: 2 January 2020 Abstract: The aim of this paper is to present an application of a fixed point iterative process in generation of fractals namely Julia and Mandelbrot sets for the complex polynomials of the form T(x) = xn + mx + r where m, r 2 C and n ≥ 2. Fractals represent the phenomena of expanding or unfolding symmetries which exhibit similar patterns displayed at every scale. We prove some escape time results for the generation of Julia and Mandelbrot sets using a Picard Ishikawa type iterative process. A visualization of the Julia and Mandelbrot sets for certain complex polynomials is presented and their graphical behaviour is examined. We also discuss the effects of parameters on the color variation and shape of fractals. Keywords: iteration; fixed points; fractals MSC: Primary: 47H10; Secondary:47J25 1. Introduction Fixed point theory provides a suitable framework to investigate various nonlinear phenomena arising in the applied sciences including complex graphics, geometry, biology and physics [1–4].
    [Show full text]