DOCSLIB.ORG

Mandelbrot Mandelbrot Is a Program That Explores the Dynamics of Iterating the Complex Function 2 + and a Variatio

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mandelbrot Mandelbrot Is a Program That Explores the Dynamics of Iterating the Complex Function 2 + and a Variatio Mandelbrot Mandelbrot is a program that explores the dynamics of iterating the complex function 푧2 + 푐 and a variation (the burning ship factal). It contains five modes: 2 1. Orbit, which computes and plots the iterates 푧푛+1 = 푧푛 + 푐, where 푧0 = 0 and c is a fixed complex number. 2. Zoom, which allows the user to zoom in on specified parts of the Mandelbrot set, 3. Julia, which graphs specified Julia sets associated with the Mandelbrot set, and 4. Burning Ship, which graphs the Burning Ship fractal and allows the user to zoom in on specified regions. 5. Multibrot, which graphs multibrot fractals and allows the user to zoom in on specified regions. Upon execution of Mandelbrot we need to choose a mode. Let’s begin by choosing the Orbit mode. Upon selecting that mode, three windows will appear as in Figure 1. The one in the middle is a rendering of the Mandelbrot set (the region colored in black) graphed in the complex plane where the real part ranges from -2 to 0.5 and the imaginary part ranges from -1.25 to 1.25. The window on the right displays that portion of the complex plane for which both the real and imaginary parts range from -2.5 to 2.5. A circle whose center is at the origin and with radius 2 is also displayed. The window on the left is the status window. The current value of the complex number 푐 = 푎 + 푏푖 is located in the middle window by a “+” symbol and the values of its real part a and imaginary part b are listed in the status window. Initially, c is set to zero. We can change the complex number 푐 = 푎 + 푏푖 in two ways: 1. we can input the numbers a and b in their text boxes located in the status window, or 2. we can move the cursor to move the desired location, then click the mouse. For ipads, simply touch the desired location on the screen. Figure 1 Upon pressing the “Go” button, a red dot appears at the origin in the right window and signifies the location of 푧0 = 푥0 + 푦0푖. Also a window appears at the bottom of the screen giving the real part (x) and the imaginary part (y) of the current iterate. Each time the “Next” button is pressed, the new iterate 푧푛+1 = 푥푛+1 + 푦푛+1푖 is graphed as a red dot in the right window and the location of the old iterate 푧푛 = 푥푛 + 푦푛푖 changes from red to blue. If an iterate 푧푛 = 푥푛 + 푦푛푖 ever lies outside the circle of radius 2, then the complex number 푐 = 푎 + 푏푖 is not a part of the Mandelbrot set, otherwise c is in the Mandelbrot set. Upon selecting the Zoom mode, three windows will appear: The Mandelbrot set in the middle, a blank window on the right and the status window on the left. Let us first look at the status window. At the top are text boxes which give the current intervals for the real part a and imaginary part b of the complex plane to be graphed in the right window. If, for example, if we change the left endpoint of the screen a-interval from −2 to −1, then press the “Redraw” button, the program will plot a portion of the Mandelbrot set in the right window, as in Figure 2. Figure 2 We note first of all that the “Zoom Level” is now 1. Also we note that the program, if necessary, will ‘square’ the graphing window; that is, it will redefine the screen a-interval or the screen b- interval so that they have the same length. In this example the screen a-interval was shortened to [−1, 0.5], whose length is 1.5. The program extended it to have the same midpoint but new length 2.5, the length of the longer b-interval. Thus, the program redefined the new screen a- interval to be [−1.5, 1]. Another way we can zoom in on portions of the Mandelbrot set is to use the “Zoom” button. When it is pressed, we move the cursor to one corner of the zoom window, then press “OK”. A “+” symbol appears at that location. (On ipads simply touch the location of the corner). Then we move the cursor to the diagonally opposite corner, the press “OK”. Once the “Zoom It” button is pressed, the program automatically “squares” the zoom window, increments the zoom level by one, prints the new screen a- and b-intervals in the status window, and graphs the zoomed portion of the Mandelbrot set in the right window. If the zoom level is zero, the middle window is used to define the new zoom window; otherwise the right window is used. We can go back to a previous zoom level by choosing its number in the zoom level menu. Figure 3 shows a close-up view of the small “copy” of the Mandelbrot set that appears at the extreme left in the original set. Figure 3 Now what about the colors? Here is what the program is doing. For each pixel in the zoom window, the program assigns it the appropriate complex number 푐 = 푎 + 푏푖. It then starts the 2 iterative process 푧0 = 0 and 푧푛+1 = 푧푛 + 푐. If for some n, the value of 푧푛 is such that the sum of the squares of its real and imaginary parts is greater than 4 (that is, 푧푛 lies outside the circle in the complex plane whose center is at the origin and whose radius is 2), then c is not part of the Mandelbrot set and the iteration stops and the pixel is assigned a color. Initially, if n is 5 or less the pixel is colored cyan. If n is greater than 5 but is 10 or less, the pixel is colored blue. If n is greater than 10 but is 20 or less, the pixel is colored red, and if n is greater than 20 but is 40 or less, the pixel is colored yellow. If 푧41 still lies within the circle of radius 2, there is an excellent chance that c actually lies within the Mandelbrot set, so its pixel is colored black. The cut-off values for the various colors can be edited in the status window. Figure 4 shows the same magnified portion of the Mandelbrot set as in Figure 3, but where the cut-off values for each of the colors was doubled. We press the “Redraw” button after we are finished editing the color boxes in order to get the graph. Figure 5 shows another “zoomed in” region. When at a zoom level other than level zero, all zooming is relative to the zoomed image appearing in the right hand window. Figure 6 shows an image zoomed from the right window of Figure 5. When the Julia mode is selected, the screen appears as in Figure 7. Julia sets are created by 2 generating sequences of the form 푧푛+1 = 푧푛 + 푐 where c is a fixed complex number and for various initial values 푧0. For each value of 푧0, if the sequence generated is bounded, then 푧0. is a point in the fractal. The image in the middle represents the Mandelbrot set (the black portion), Figure 4 Figure 5 Figure 6 with the real part ranging from -2.0 to 0.5and the imaginary part ranging from –1.25 to 1.25. The black portion in the right hand image depicts the Julia set corresponding to 푐 = −0.75. That c value is shown in the middle image by a “+” symbol. The c value can be changed either by typing in new values in the Selected Point text fields, or by repositioning the “+” symbol. Figure 8 shows the Julia set where c is near the boundary of the Mandelbrot set. Figure 7 The viewing window for the Julia set can be changed in two ways. First, we can enter new numbers in the Real Range and Imaginary Range text fields. No portion of the new viewing rectangle can go outside the original one. That is, the real and imaginary ranges must be Figure 8 subintervals of [−2, 2]. Also the new viewing window is automatically “squared” so that the aspect ratio remains unaltered. (This “squaring” option may extend the viewing window outside the original one.) The other way to alter the viewing window is by pressing the Zoom button. When the Zoom button is pressed, a “+” cursor appears in the middle of the Julia image. The cursor can be repositioned with the mouse and when the “OK” button is pressed one corner of the new viewing window is defined. Reposition the cursor to a new location and click the mouse to define the diagonally opposite corner. We will have the opportunity to either cancel or accept the zoomed window. As before the zoomed window is “squared” before the new image is drawn and a new zoom level is created. Figure 9 shows a zoomed portion of the Julia set of Figure 8. For certain values of c, it is important to use a higher value for the yellow color threshold, for the sequence may remain relatively stable for a long time before becoming unbounded. For example, in Figure 9, when the yellow color threshold is increased from 40 to 80, the image of Figure 10 is generated. When the Burning Ship mode is selected, the screen looks like Figure 11. Note that the center window is no longer the Mandelbrot set, but it is the “burning ship” fractal.
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]
  • Fractal (Mandelbrot and Julia) Zero-Knowledge Proof of Identity
    Journal of Computer Science 4 (5): 408-414, 2008 ISSN 1549-3636 © 2008 Science Publications Fractal (Mandelbrot and Julia) Zero-Knowledge Proof of Identity Mohammad Ahmad Alia and Azman Bin Samsudin School of Computer Sciences, University Sains Malaysia, 11800 Penang, Malaysia Abstract: We proposed a new zero-knowledge proof of identity protocol based on Mandelbrot and Julia Fractal sets. The Fractal based zero-knowledge protocol was possible because of the intrinsic connection between the Mandelbrot and Julia Fractal sets. In the proposed protocol, the private key was used as an input parameter for Mandelbrot Fractal function to generate the corresponding public key. Julia Fractal function was then used to calculate the verified value based on the existing private key and the received public key. The proposed protocol was designed to be resistant against attacks. Fractal based zero-knowledge protocol was an attractive alternative to the traditional number theory zero-knowledge protocol. Key words: Zero-knowledge, cryptography, fractal, mandelbrot fractal set and julia fractal set INTRODUCTION Zero-knowledge proof of identity system is a cryptographic protocol between two parties. Whereby, the first party wants to prove that he/she has the identity (secret word) to the second party, without revealing anything about his/her secret to the second party. Following are the three main properties of zero- knowledge proof of identity[1]: Completeness: The honest prover convinces the honest verifier that the secret statement is true. Soundness: Cheating prover can’t convince the honest verifier that a statement is true (if the statement is really false). Fig. 1: Zero-knowledge cave Zero-knowledge: Cheating verifier can’t get anything Zero-knowledge cave: Zero-Knowledge Cave is a other than prover’s public data sent from the honest well-known scenario used to describe the idea of zero- prover.
    [Show full text]
  • Rendering Hypercomplex Fractals Anthony Atella [email protected]
    Rhode Island College Digital Commons @ RIC Honors Projects Overview Honors Projects 2018 Rendering Hypercomplex Fractals Anthony Atella [email protected] Follow this and additional works at: https://digitalcommons.ric.edu/honors_projects Part of the Computer Sciences Commons, and the Other Mathematics Commons Recommended Citation Atella, Anthony, "Rendering Hypercomplex Fractals" (2018). Honors Projects Overview. 136. https://digitalcommons.ric.edu/honors_projects/136 This Honors is brought to you for free and open access by the Honors Projects at Digital Commons @ RIC. It has been accepted for inclusion in Honors Projects Overview by an authorized administrator of Digital Commons @ RIC. For more information, please contact [email protected]. Rendering Hypercomplex Fractals by Anthony Atella An Honors Project Submitted in Partial Fulfillment of the Requirements for Honors in The Department of Mathematics and Computer Science The School of Arts and Sciences Rhode Island College 2018 Abstract Fractal mathematics and geometry are useful for applications in science, engineering, and art, but acquiring the tools to explore and graph fractals can be frustrating. Tools available online have limited fractals, rendering methods, and shaders. They often fail to abstract these concepts in a reusable way. This means that multiple programs and interfaces must be learned and used to fully explore the topic. Chaos is an abstract fractal geometry rendering program created to solve this problem. This application builds off previous work done by myself and others [1] to create an extensible, abstract solution to rendering fractals. This paper covers what fractals are, how they are rendered and colored, implementation, issues that were encountered, and finally planned future improvements.
    [Show full text]
  • Generating Fractals Using Complex Functions
    Generating Fractals Using Complex Functions Avery Wengler and Eric Wasser What is a Fractal? ● Fractals are infinitely complex patterns that are self-similar across different scales. ● Created by repeating simple processes over and over in a feedback loop. ● Often represented on the complex plane as 2-dimensional images Where do we find fractals? Fractals in Nature Lungs Neurons from the Oak Tree human cortex Regardless of scale, these patterns are all formed by repeating a simple branching process. Geometric Fractals “A rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.” Mandelbrot (1983) The Sierpinski Triangle Algebraic Fractals ● Fractals created by repeatedly calculating a simple equation over and over. ● Were discovered later because computers were needed to explore them ● Examples: ○ Mandelbrot Set ○ Julia Set ○ Burning Ship Fractal Mandelbrot Set ● Benoit Mandelbrot discovered this set in 1980, shortly after the invention of the personal computer 2 ● zn+1=zn + c ● That is, a complex number c is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets. Animation based on a static number of iterations per pixel. The Mandelbrot set is the complex numbers c for which the sequence ( c, c² + c, (c²+c)² + c, ((c²+c)²+c)² + c, (((c²+c)²+c)²+c)² + c, ...) does not approach infinity. Julia Set ● Closely related to the Mandelbrot fractal ● Complementary to the Fatou Set Featherino Fractal Newton’s method for the roots of a real valued function Burning Ship Fractal z2 Mandelbrot Render Generic Mandelbrot set.
    [Show full text]
  • Iterated Function Systems, Ruelle Operators, and Invariant Projective Measures
    MATHEMATICS OF COMPUTATION Volume 75, Number 256, October 2006, Pages 1931–1970 S 0025-5718(06)01861-8 Article electronically published on May 31, 2006 ITERATED FUNCTION SYSTEMS, RUELLE OPERATORS, AND INVARIANT PROJECTIVE MEASURES DORIN ERVIN DUTKAY AND PALLE E. T. JORGENSEN Abstract. We introduce a Fourier-based harmonic analysis for a class of dis- crete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space X comes with a finite-to-one endomorphism r : X → X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in Rd, this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets B,L in Rd of the same cardinality which generate complex Hadamard matrices. Our harmonic analysis for these iterated function systems (IFS) (X, µ)is based on a Markov process on certain paths. The probabilities are determined by a weight function W on X. From W we define a transition operator RW acting on functions on X, and a corresponding class H of continuous RW - harmonic functions. The properties of the functions in H are analyzed, and they determine the spectral theory of L2(µ).ForaffineIFSsweestablish orthogonal bases in L2(µ). These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in Rd. 1. Introduction One of the reasons wavelets have found so many uses and applications is that they are especially attractive from the computational point of view.
    [Show full text]
  • Arxiv:2105.08654V1 [Math.DS] 18 May 2021
    THE DYNAMICS OF COMPLEX BOX MAPPINGS TREVOR CLARK, KOSTIANTYN DRACH, OLEG KOZLOVSKI, AND SEBASTIAN VAN STRIEN Abstract. In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is: - To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues. - To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus we call such complex box mappings dy- namically natural. Having such box mappings is the first step in tackling many problems in one-dimensional dynamics. - Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting some of these tools are - the Enhanced Nest (a nest of puzzle pieces around critical points) from [KSS1]; - the Covering Lemma (which controls the moduli of pullbacks of annuli) from [KL1]; - the QC-Criterion and the Spreading Principle from [KSS1]. The purpose of this paper is to make these tools more accessible so that they can be used as a `black box', so one does not have to redo the proofs in new settings. - To give an intuitive, but also rather detailed, outline of the proof from [KvS, KSS1] of the following results for non-renormalizable dynamically natural complex box mappings: - puzzle pieces shrink to points, - (under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate.
    [Show full text]
  • International Journal of Research in Computer Applications and Robotics Issn 2320-7345 Complex Dynamics of Multibrot Sets Fo
    INTERNATIONAL JOURNAL OF RESEARCH IN COMPUTER APPLICATIONS AND ROBOTICS Vol.2 Issue.4, Pg.: 12-22 April 2014 www.ijrcar.com INTERNATIONAL JOURNAL OF RESEARCH IN COMPUTER APPLICATIONS AND ROBOTICS ISSN 2320-7345 COMPLEX DYNAMICS OF MULTIBROT SETS FOR JUNGCK ISHIKAWA ITERATION 1 2 3 Suman Joshi , Dr. Yashwant Singh Chauhan , Dr. Priti Dimri 1 G. B. Pant Engineering College (Pauri Garhwal),[email protected] 2 G. B. Pant Engineering College (Pauri Garhwal), [email protected] 3G. B. Pant Engineering College (Pauri Garhwal), [email protected] Author Correspondence: G. B. Pant Engineering. College, Pauri Garhwal Uttarakhand, 9990423408, [email protected] Abstract The generation of fractals and study of the dynamics of polynomials is one of the emerging and interesting fields of research nowadays. We introduce in this paper the dynamics of modified multibrot function zd - z + c = 0 for d 2 and applied Jungck Ishikawa Iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. We have presented here different characteristics of Multibrot function like its trajectories, its complex dynamics and its behaviour towards Julia set are also discussed. In order to solve this function by Jungck –type iterative schemes, we write it in the form of Sz = Tz, where the function T, S are defined as Tz = zd +c and Sz= z. Only mathematical explanations are derived by applying Jungck Ishikawa Iteration for polynomials in the literature but in this paper we have generated relative Mandelbrot sets and Relative Julia sets. Keywords: Complex dynamics, Relative Superior Mandelbrot set, Relative Julia set, Jungck Ishikawa Iteration 1.
    [Show full text]
  • Sphere Tracing, Distance Fields, and Fractals Alexander Simes
    Sphere Tracing, Distance Fields, and Fractals Alexander Simes Advisor: Angus Forbes Secondary: Andrew Johnson Fall 2014 - 654108177 Figure 1: Sphere Traced images of Menger Cubes and Mandelboxes shaded by ambient occlusion approximation on the left and Blinn-Phong with shadows on the right Abstract Methods to realistically display complex surfaces which are not practical to visualize using traditional techniques are presented. Additionally an application is presented which is capable of utilizing some of these techniques in real time. Properties of these surfaces and their implications to a real time application are discussed. Table of Contents 1 Introduction 3 2 Minimal CPU Sphere Tracing Model 4 2.1 Camera Model 4 2.2 Marching with Distance Fields Introduction 5 2.3 Ambient Occlusion Approximation 6 3 Distance Fields 9 3.1 Signed Sphere 9 3.2 Unsigned Box 9 3.3 Distance Field Operations 10 4 Blinn-Phong Shadow Sphere Tracing Model 11 4.1 Scene Composition 11 4.2 Maximum Marching Iteration Limitation 12 4.3 Surface Normals 12 4.4 Blinn-Phong Shading 13 4.5 Hard Shadows 14 4.6 Translation to GPU 14 5 Menger Cube 15 5.1 Introduction 15 5.2 Iterative Definition 16 6 Mandelbox 18 6.1 Introduction 18 6.2 boxFold() 19 6.3 sphereFold() 19 6.4 Scale and Translate 20 6.5 Distance Function 20 6.6 Computational Efficiency 20 7 Conclusion 2 1 Introduction Sphere Tracing is a rendering technique for visualizing surfaces using geometric distance. Typically surfaces applicable to Sphere Tracing have no explicit geometry and are implicitly defined by a distance field.
    [Show full text]
  • How Math Makes Movies Like Doctor Strange So Otherworldly | Science News for Students
    3/13/2020 How math makes movies like Doctor Strange so otherworldly | Science News for Students MATH How math makes movies like Doctor Strange so otherworldly Patterns called fractals are inspiring filmmakers with ideas for mind-bending worlds Kaecilius, on the right, is a villain and sorcerer in Doctor Strange. He can twist and manipulate the fabric of reality. The film’s visual-effects artists used mathematical patterns, called fractals, illustrate Kaecilius’s abilities on the big screen. MARVEL STUDIOS By Stephen Ornes January 9, 2020 at 6:45 am For wild chase scenes, it’s hard to beat Doctor Strange. In this 2016 film, the fictional doctor-turned-sorcerer has to stop villains who want to destroy reality. To further complicate matters, the evildoers have unusual powers of their own. “The bad guys in the film have the power to reshape the world around them,” explains Alexis Wajsbrot. He’s a film director who lives in Paris, France. But for Doctor Strange, Wajsbrot instead served as the film’s visual-effects artist. Those bad guys make ordinary objects move and change forms. Bringing this to the big screen makes for chases that are spectacular to watch. City blocks and streets appear and disappear around the fighting foes. Adversaries clash in what’s called the “mirror dimension” — a place where the laws of nature don’t apply. Forget gravity: Skyscrapers twist and then split. Waves ripple across walls, knocking people sideways and up. At times, multiple copies of the entire city seem to appear at once, but at different sizes.
    [Show full text]
  • Understanding the Mandelbrot and Julia Set
    Understanding the Mandelbrot and Julia Set Jake Zyons Wood August 31, 2015 Introduction Fractals infiltrate the disciplinary spectra of set theory, complex algebra, generative art, computer science, chaos theory, and more. Fractals visually embody recursive structures endowing them with the ability of nigh infinite complexity. The Sierpinski Triangle, Koch Snowflake, and Dragon Curve comprise a few of the more widely recognized iterated function fractals. These recursive structures possess an intuitive geometric simplicity which makes their creation, at least at a shallow recursive depth, easy to do by hand with pencil and paper. The Mandelbrot and Julia set, on the other hand, allow no such convenience. These fractals are part of the class: escape-time fractals, and have only really entered mathematicians’ consciousness in the late 1970’s[1]. The purpose of this paper is to clearly explain the logical procedures of creating escape-time fractals. This will include reviewing the necessary math for this type of fractal, then specifically explaining the algorithms commonly used in the Mandelbrot Set as well as its variations. By the end, the careful reader should, without too much effort, feel totally at ease with the underlying principles of these fractals. What Makes The Mandelbrot Set a set? 1 Figure 1: Black and white Mandelbrot visualization The Mandelbrot Set truly is a set in the mathematica sense of the word. A set is a collection of anything with a specific property, the Mandelbrot Set, for instance, is a collection of complex numbers which all share a common property (explained in Part II). All complex numbers can in fact be labeled as either a member of the Mandelbrot set, or not.
    [Show full text]
  • Extending Mandelbox Fractals with Shape Inversions
    Extending Mandelbox Fractals with Shape Inversions Gregg Helt Genomancer, Healdsburg, CA, USA; [email protected] Abstract The Mandelbox is a recently discovered class of escape-time fractals which use a conditional combination of reflection, spherical inversion, scaling, and translation to transform points under iteration. In this paper we introduce a new extension to Mandelbox fractals which replaces spherical inversion with a more generalized shape inversion. We then explore how this technique can be used to generate new fractals in 2D, 3D, and 4D. Mandelbox Fractals The Mandelbox is a class of escape-time fractals that was first discovered by Tom Lowe in 2010 [5]. It was named the Mandelbox both as an homage to the classic Mandelbrot set fractal and due to its overall boxlike shape when visualized, as shown in Figure 1a. The interior can be rich in self-similar fractal detail as well, as shown in Figure 1b. Many modifications to the original algorithm have been developed, almost exclusively by contributors to the FractalForums online community. Although most explorations of Mandelboxes have focused on 3D versions, the algorithm can be applied to any number of dimensions. (a) (b) (c) Figure 1: Mandelbox 3D fractal examples: (a) Mandelbox exterior , (b) same Mandelbox, but zoomed in view of small section of interior , (c) Juliabox indexed by same Mandelbox. Like the Mandelbrot set and other escape-time fractals, a Mandelbox set contains all the points whose orbits under iterative transformation by a function do not escape. For a basic Mandelbox the function to apply iteratively to each point �" is defined as a composition of transformations: �#$% = ��ℎ�������1,3 �������6 �# ∗ � + �" Boxfold and Spherefold are modified reflection and spherical inversion transforms, respectively, with parameters F, H, and L which are described below.
    [Show full text]
  • New Key Exchange Protocol Based on Mandelbrot and Julia Fractal Sets Mohammad Ahmad Alia and Azman Bin Samsudin
    302 IJCSNS International Journal of Computer Science and Network Security, VOL.7 No.2, February 2007 New Key Exchange Protocol Based on Mandelbrot and Julia Fractal Sets Mohammad Ahmad Alia and Azman Bin Samsudin, School of Computer Sciences, Universiti Sains Malaysia. Summary by using two keys, a private and a public key. In most In this paper, we propose a new cryptographic key exchange cases, by exchanging the public keys, both parties can protocol based on Mandelbrot and Julia Fractal sets. The Fractal calculate a unique shared key, known only to both of them based key exchange protocol is possible because of the intrinsic [2, 3, 4]. connection between the Mandelbrot and Julia Fractal sets. In the proposed protocol, Mandelbrot Fractal function takes the chosen This paper proposed a new Fractal (Mandelbrot and private key as the input parameter and generates the corresponding public key. Julia Fractal function is then used to Julia Fractal sets) key exchange protocol as a method to calculate the shared key based on the existing private key and the securely agree on a shared key. The proposed method received public key. The proposed protocol is designed to be achieved the secure exchange protocol by creating a resistant against attacks, utilizes small key size and shared secret through the use of the Mandelbrot and Julia comparatively performs faster then the existing Diffie-Hellman Fractal sets. key exchange protocol. The proposed Fractal key exchange protocol is therefore an attractive alternative to traditional 2.1. Fractals number theory based key exchange protocols. A complex number consists of a real and an imaginary Key words: Fractals Cryptography, Key- exchange protocol, Mandelbrot component.
    [Show full text]