Fractals and the Mandelbrot Set

Total Page:16

File Type:pdf, Size:1020Kb

Fractals and the Mandelbrot Set Fractals and the Mandelbrot Set Matt Ziemke October, 2012 Matt Ziemke Fractals and the Mandelbrot Set Outline 1. Fractals 2. Julia Fractals 3. The Mandelbrot Set 4. Properties of the Mandelbrot Set 5. Open Questions Matt Ziemke Fractals and the Mandelbrot Set What is a Fractal? "My personal feeling is that the definition of a 'fractal' should be regarded in the same way as the biologist regards the definition of 'life'." - Kenneth Falconer Common Properties 1.) Detail on an arbitrarily small scale. 2.) Too irregular to be described using traditional geometrical language. 3.) In most cases, defined in a very simple way. 4.) Often exibits some form of self-similarity. Matt Ziemke Fractals and the Mandelbrot Set The Koch Curve- 10 Iterations Matt Ziemke Fractals and the Mandelbrot Set 5-Iterations Matt Ziemke Fractals and the Mandelbrot Set The Minkowski Fractal- 5 Iterations Matt Ziemke Fractals and the Mandelbrot Set 5 Iterations Matt Ziemke Fractals and the Mandelbrot Set 5 Iterations Matt Ziemke Fractals and the Mandelbrot Set 8 Iterations Matt Ziemke Fractals and the Mandelbrot Set Heighway's Dragon Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal 1.1 Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal 1.2 Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal 1.3 Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal 1.4 Matt Ziemke Fractals and the Mandelbrot Set Matt Ziemke Fractals and the Mandelbrot Set Matt Ziemke Fractals and the Mandelbrot Set Julia Fractals 2 Step 1: Let fc : C ! C where f (z) = z + c. 1 Step 2: For each w 2 C, recursively define the sequence fwngn=0 1 where w0 = w and wn = f (wn−1): The sequence wnn=0 is referred to as the orbit of w. Step 3: "Collect" all the w 2 C whose orbit is bounded, i.e., let Kc = fw 2 C : sup jwnj ≤ M; for some M > 0g n2N and let Jc = δ(Kc ) where δ(K) is the boundary of K. Jc is called a Julia set. Matt Ziemke Fractals and the Mandelbrot Set Julia Fractals - Example Let c = 0:375 + i(0:335). Consider w = 0:1i. Then, w1 = f (w0) = f (0:1i) = (0:1i) = 0:365 + 0:335i w2 = f (w1) = f (0:365 + 0:335i) = 0:396 + 0:5796i w20 ≈ 0:014 + 0:026i 1 In fact, fwngn=0 does not converge but it is bounded by 2. So 0:1i 2 Kc . Consider x = 1. Then, x1 ≈ 1:375 + 0:335i x2 ≈ 2:153 + 1:256i x3 ≈ 3:434 + 5:745i x4 ≈ −20:843 + 39:794i x5 ≈ −1148:782 − 1658:450i So looks as though 1 2= Kc . Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal - Example, Image 1 Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal - Example, Image 2 Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal - Example, Image 3 Why the colors? Matt Ziemke Fractals and the Mandelbrot Set c=-1.145+0.25i Matt Ziemke Fractals and the Mandelbrot Set c=-0.110339+0.887262i Matt Ziemke Fractals and the Mandelbrot Set c=0.06+0.72i Matt Ziemke Fractals and the Mandelbrot Set c=-0.022803-0.672621i Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set Theorem of Julia and Fatou (1920) Every Julia set is either connected or totally disconnected. Brolin's Theorem Jc is connected if and only if the orbit of zero is bounded, i.e., if and only if 0 2 Kc . Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. A natural question to ask is... What does (n) 1 M = fc 2 C : Jc is connected g = fc 2 C : ffc (0)gn=0 is boundedg look like? Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. Matt Ziemke Fractals and the Mandelbrot Set M is a "catalog" for the connected Julia sets. Matt Ziemke Fractals and the Mandelbrot Set Interesting Facts about M 1.)If Jc is totally disconnected then Jc is homeomorphic to the Cantor set. 2.) fc : Jc ! Jc is chaotic. 3.) Julia fractals given by c-values in a given "bulb" of M are homeomorphic. 4.) M is compact. 5.) The Hausdorff dimension of δ(M) is two. Matt Ziemke Fractals and the Mandelbrot Set Open questions about M 1.) What's the area of M? (n) 1 2.) Are there any points c 2 M so that ffc (0)gn=1 is not attracted to a cycle? 3.) Is µ(δ(M)) > 0? Where µ is the Lebesgue measure. Matt Ziemke Fractals and the Mandelbrot Set.
Recommended publications
  • 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]
  • 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]
  • An Introduction to the Mandelbrot Set
    An introduction to the Mandelbrot set Bastian Fredriksson January 2015 1 Purpose and content The purpose of this paper is to introduce the reader to the very useful subject of fractals. We will focus on the Mandelbrot set and the related Julia sets. I will show some ways of visualising these sets and how to make a program that renders them. Finally, I will explain a key exchange algorithm based on what we have learnt. 2 Introduction The Mandelbrot set and the Julia sets are sets of points in the complex plane. Julia sets were first studied by the French mathematicians Pierre Fatou and Gaston Julia in the early 20th century. However, at this point in time there were no computers, and this made it practically impossible to study the structure of the set more closely, since large amount of computational power was needed. Their discoveries was left in the dark until 1961, when a Jewish-Polish math- ematician named Benoit Mandelbrot began his research at IBM. His task was to reduce the white noise that disturbed the transmission on telephony lines [3]. It seemed like the noise came in bursts, sometimes there were a lot of distur- bance, and sometimes there was no disturbance at all. Also, if you examined a period of time with a lot of noise-problems, you could still find periods without noise [4]. Could it be possible to come up with a model that explains when there is noise or not? Mandelbrot took a quite radical approach to the problem at hand, and chose to visualise the data.
    [Show full text]
  • Connectedness of Julia Sets and the Mandelbrot
    Math 207 - Spring '17 - Fran¸coisMonard 1 Lecture 20 - Introduction to complex dynamics - 3/3: Mandelbrot and friends Outline: • Recall critical points and behavior of functions nearby. • Motivate the proof on connectedness of polynomials for Julia sets. • Mandelbrot set. • The cubic family. Zoology of Fatou sets Here, we will merely state facts about the types of behavior one can expect on the Fatou set. We have seen an example before, which was the basin of attraction of a fixed points. Let Ω a connected component of the Fatou set. Since the family fF ng has normally convergent subsequences there (call accumulation function its limit), one may ask the question: how many n accumulation functions does the sequence fF gn have ? If it is a finite number, we may expect to be in the basin of attraction of an attracting/superattracting or neutral periodic cycle. In the latter case, such connected components are referred to as parabolic cycles. For example, it can be shown that a neighborhood of a neutral fixed point is split into an even number of sectors, where orbits alternatively converge or diverge away from that fixed point. Finally, if the number of accumulation n functions of the sequence fF gn is infinite (think for instance of the sequence of iterates of the map F (z) = e2iπαz with α irrational), depending on the connectedness of Ω, it may lead to a so-called Siegel disk or a Herman ring. The interested reader may find more detail on this topic in [DK], Article \Julia sets" by Linda Keen. Quasi self-similarity of Julia sets The fractal quality of Julia sets is formalized by the concept of quasi-self-similarity.
    [Show full text]
  • Furniture Design Inspired from Fractals
    169 Rania Mosaad Saad Furniture design inspired from fractals. Dr. Rania Mosaad Saad Assistant Professor, Interior Design and Furniture Department, Faculty of Applied Arts, Helwan University, Cairo, Egypt Abstract: Keywords: Fractals are a new branch of mathematics and art. But what are they really?, How Fractals can they affect on Furniture design?, How to benefit from the formation values and Mandelbrot Set properties of fractals in Furniture Design?, these were the research problem . Julia Sets This research consists of two axis .The first axe describes the most famous fractals IFS fractals were created, studies the Fractals structure, explains the most important fractal L-system fractals properties and their reflections on furniture design. The second axe applying Fractal flame functional and aesthetic values of deferent Fractals formations in furniture design Newton fractals inspired from them to achieve the research objectives. Furniture Design The research follows the descriptive methodology to describe the fractals kinds and properties, and applied methodology in furniture design inspired from fractals. Paper received 12th July 2016, Accepted 22th September 2016 , Published 15st of October 2016 nearly identical starting positions, and have real Introduction: world applications in nature and human creations. Nature is the artist's inspiring since the beginning Some architectures and interior designers turn to of creation. Despite the different artistic trends draw inspiration from the decorative formations, across different eras- ancient and modern- and the geometric and dynamic properties of fractals in artists perception of reality, but that all of these their designs which enriched the field of trends were united in the basic inspiration (the architecture and interior design, and benefited nature).
    [Show full text]