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Fractal Growth on the Surface of a Planet and in Orbit Around It
Wilfrid Laurier University Scholars Commons @ Laurier Physics and Computer Science Faculty Publications Physics and Computer Science 10-2014 Fractal Growth on the Surface of a Planet and in Orbit around It Ioannis Haranas Wilfrid Laurier University, [email protected] Ioannis Gkigkitzis East Carolina University, [email protected] Athanasios Alexiou Ionian University Follow this and additional works at: https://scholars.wlu.ca/phys_faculty Part of the Mathematics Commons, and the Physics Commons Recommended Citation Haranas, I., Gkigkitzis, I., Alexiou, A. Fractal Growth on the Surface of a Planet and in Orbit around it. Microgravity Sci. Technol. (2014) 26:313–325. DOI: 10.1007/s12217-014-9397-6 This Article is brought to you for free and open access by the Physics and Computer Science at Scholars Commons @ Laurier. It has been accepted for inclusion in Physics and Computer Science Faculty Publications by an authorized administrator of Scholars Commons @ Laurier. For more information, please contact [email protected]. 1 Fractal Growth on the Surface of a Planet and in Orbit around it 1Ioannis Haranas, 2Ioannis Gkigkitzis, 3Athanasios Alexiou 1Dept. of Physics and Astronomy, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada 2Departments of Mathematics and Biomedical Physics, East Carolina University, 124 Austin Building, East Fifth Street, Greenville, NC 27858-4353, USA 3Department of Informatics, Ionian University, Plateia Tsirigoti 7, Corfu, 49100, Greece Abstract: Fractals are defined as geometric shapes that exhibit symmetry of scale. This simply implies that fractal is a shape that it would still look the same even if somebody could zoom in on one of its parts an infinite number of times. -
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. -
An Image Cryptography Using Henon Map and Arnold Cat Map
International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 05 Issue: 04 | Apr-2018 www.irjet.net p-ISSN: 2395-0072 An Image Cryptography using Henon Map and Arnold Cat Map. Pranjali Sankhe1, Shruti Pimple2, Surabhi Singh3, Anita Lahane4 1,2,3 UG Student VIII SEM, B.E., Computer Engg., RGIT, Mumbai, India 4Assistant Professor, Department of Computer Engg., RGIT, Mumbai, India ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstract - In this digital world i.e. the transmission of non- 2. METHODOLOGY physical data that has been encoded digitally for the purpose of storage Security is a continuous process via which data can 2.1 HENON MAP be secured from several active and passive attacks. Encryption technique protects the confidentiality of a message or 1. The Henon map is a discrete time dynamic system information which is in the form of multimedia (text, image, introduces by michel henon. and video).In this paper, a new symmetric image encryption 2. The map depends on two parameters, a and b, which algorithm is proposed based on Henon’s chaotic system with for the classical Henon map have values of a = 1.4 and byte sequences applied with a novel approach of pixel shuffling b = 0.3. For the classical values the Henon map is of an image which results in an effective and efficient chaotic. For other values of a and b the map may be encryption of images. The Arnold Cat Map is a discrete system chaotic, intermittent, or converge to a periodic orbit. that stretches and folds its trajectories in phase space. Cryptography is the process of encryption and decryption of 3. -
WHAT IS a CHAOTIC ATTRACTOR? 1. Introduction J. Yorke Coined the Word 'Chaos' As Applied to Deterministic Systems. R. Devane
WHAT IS A CHAOTIC ATTRACTOR? CLARK ROBINSON Abstract. Devaney gave a mathematical definition of the term chaos, which had earlier been introduced by Yorke. We discuss issues involved in choosing the properties that characterize chaos. We also discuss how this term can be combined with the definition of an attractor. 1. Introduction J. Yorke coined the word `chaos' as applied to deterministic systems. R. Devaney gave the first mathematical definition for a map to be chaotic on the whole space where a map is defined. Since that time, there have been several different definitions of chaos which emphasize different aspects of the map. Some of these are more computable and others are more mathematical. See [9] a comparison of many of these definitions. There is probably no one best or correct definition of chaos. In this paper, we discuss what we feel is one of better mathematical definition. (It may not be as computable as some of the other definitions, e.g., the one by Alligood, Sauer, and Yorke.) Our definition is very similar to the one given by Martelli in [8] and [9]. We also combine the concepts of chaos and attractors and discuss chaotic attractors. 2. Basic definitions We start by giving the basic definitions needed to define a chaotic attractor. We give the definitions for a diffeomorphism (or map), but those for a system of differential equations are similar. The orbit of a point x∗ by F is the set O(x∗; F) = f Fi(x∗) : i 2 Z g. An invariant set for a diffeomorphism F is an set A in the domain such that F(A) = A. -
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. -
MA427 Ergodic Theory
MA427 Ergodic Theory Course Notes (2012-13) 1 Introduction 1.1 Orbits Let X be a mathematical space. For example, X could be the unit interval [0; 1], a circle, a torus, or something far more complicated like a Cantor set. Let T : X ! X be a function that maps X into itself. Let x 2 X be a point. We can repeatedly apply the map T to the point x to obtain the sequence: fx; T (x);T (T (x));T (T (T (x))); : : : ; :::g: We will often write T n(x) = T (··· (T (T (x)))) (n times). The sequence of points x; T (x);T 2(x);::: is called the orbit of x. We think of applying the map T as the passage of time. Thus we think of T (x) as where the point x has moved to after time 1, T 2(x) is where the point x has moved to after time 2, etc. Some points x 2 X return to where they started. That is, T n(x) = x for some n > 1. We say that such a point x is periodic with period n. By way of contrast, points may move move densely around the space X. (A sequence is said to be dense if (loosely speaking) it comes arbitrarily close to every point of X.) If we take two points x; y of X that start very close then their orbits will initially be close. However, it often happens that in the long term their orbits move apart and indeed become dramatically different. -
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. -
Attractors and Orbit-Flip Homoclinic Orbits for Star Flows
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141, Number 8, August 2013, Pages 2783–2791 S 0002-9939(2013)11535-2 Article electronically published on April 12, 2013 ATTRACTORS AND ORBIT-FLIP HOMOCLINIC ORBITS FOR STAR FLOWS C. A. MORALES (Communicated by Bryna Kra) Abstract. We study star flows on closed 3-manifolds and prove that they either have a finite number of attractors or can be C1 approximated by vector fields with orbit-flip homoclinic orbits. 1. Introduction The notion of attractor deserves a fundamental place in the modern theory of dynamical systems. This assertion, supported by the nowadays classical theory of turbulence [27], is enlightened by the recent Palis conjecture [24] about the abun- dance of dynamical systems with finitely many attractors absorbing most positive trajectories. If confirmed, such a conjecture means the understanding of a great part of dynamical systems in the sense of their long-term behaviour. Here we attack a problem which goes straight to the Palis conjecture: The fini- tude of the number of attractors for a given dynamical system. Such a problem has been solved positively under certain circunstances. For instance, we have the work by Lopes [16], who, based upon early works by Ma˜n´e [18] and extending pre- vious ones by Liao [15] and Pliss [25], studied the structure of the C1 structural stable diffeomorphisms and proved the finitude of attractors for such diffeomor- phisms. His work was largely extended by Ma˜n´e himself in the celebrated solution of the C1 stability conjecture [17]. On the other hand, the Japanesse researchers S. -
ORBIT EQUIVALENCE of ERGODIC GROUP ACTIONS These Are Partial Lecture Notes from a Topics Graduate Class Taught at UCSD in Fall 2
1 ORBIT EQUIVALENCE OF ERGODIC GROUP ACTIONS ADRIAN IOANA These are partial lecture notes from a topics graduate class taught at UCSD in Fall 2019. 1. Group actions, equivalence relations and orbit equivalence 1.1. Standard spaces. A Borel space (X; BX ) is a pair consisting of a topological space X and its σ-algebra of Borel sets BX . If the topology on X can be chosen to the Polish (i.e., separable, metrizable and complete), then (X; BX ) is called a standard Borel space. For simplicity, we will omit the σ-algebra and write X instead of (X; BX ). If (X; BX ) and (Y; BY ) are Borel spaces, then a map θ : X ! Y is called a Borel isomorphism if it is a bijection and θ and θ−1 are Borel measurable. Note that a Borel bijection θ is automatically a Borel isomorphism (see [Ke95, Corollary 15.2]). Definition 1.1. A standard probability space is a measure space (X; BX ; µ), where (X; BX ) is a standard Borel space and µ : BX ! [0; 1] is a σ-additive measure with µ(X) = 1. For simplicity, we will write (X; µ) instead of (X; BX ; µ). If (X; µ) and (Y; ν) are standard probability spaces, a map θ : X ! Y is called a measure preserving (m.p.) isomorphism if it is a Borel isomorphism and −1 measure preserving: θ∗µ = ν, where θ∗µ(A) = µ(θ (A)), for any A 2 BY . Example 1.2. The following are standard probability spaces: (1) ([0; 1]; λ), where [0; 1] is endowed with the Euclidean topology and its Lebegue measure λ. -
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. -
Infinitely Renormalizable Quadratic Polynomials 1
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 11, Pages 5077{5091 S 0002-9947(00)02514-9 Article electronically published on July 12, 2000 INFINITELY RENORMALIZABLE QUADRATIC POLYNOMIALS YUNPING JIANG Abstract. We prove that the Julia set of a quadratic polynomial which ad- mits an infinite sequence of unbranched, simple renormalizations with complex bounds is locally connected. The method in this study is three-dimensional puzzles. 1. Introduction Let P (z)=z2 + c be a quadratic polynomial where z is a complex variable and c is a complex parameter. The filled-in Julia set K of P is, by definition, the set of points z which remain bounded under iterations of P .TheJulia set J of P is the boundary of K. A central problem in the study of the dynamical system generated by P is to understand the topology of a Julia set J, in particular, the local connectivity for a connected Julia set. A connected set J in the complex plane is said to be locally connected if for every point p in J and every neighborhood U of p there is a neighborhood V ⊆ U such that V \ J is connected. We first give the definition of renormalizability. A quadratic-like map F : U ! V is a holomorphic, proper, degree two branched cover map, whereT U and V are ⊂ 1 −n two domains isomorphic to a disc and U V .ThenKF = n=0 F (U)and JF = @KF are the filled-in Julia set and the Julia set of F , respectively. We only consider those quadratic-like maps whose Julia sets are connected. -
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.