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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. -
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. -
Gyã¶Rgy Kepes Papers
http://oac.cdlib.org/findaid/ark:/13030/c80r9v19 No online items Guide to the György Kepes papers M1796 Collection processed by John R. Blakinger, finding aid by Franz Kunst Department of Special Collections and University Archives 2016 Green Library 557 Escondido Mall Stanford 94305-6064 [email protected] URL: http://library.stanford.edu/spc Guide to the György Kepes M1796 1 papers M1796 Language of Material: English Contributing Institution: Department of Special Collections and University Archives Title: György Kepes papers creator: Kepes, Gyorgy Identifier/Call Number: M1796 Physical Description: 113 Linear Feet (108 boxes, 68 flat boxes, 8 cartons, 4 card boxes, 3 half-boxes, 2 map-folders, 1 tube) Date (inclusive): 1918-2010 Date (bulk): 1960-1990 Abstract: The personal papers of artist, designer, and visual theorist György Kepes. Language of Material: While most of the collection is in English, there is also a significant amount of Hungarian text, as well as printed material in German, Italian, Japanese, and other languages. Special Collections and University Archives materials are stored offsite and must be paged 36 hours in advance. Biographical / Historical Artist, designer, and visual theorist György Kepes was born in 1906 in Selyp, Hungary. Originally associated with Germany’s Bauhaus as a colleague of László Moholy-Nagy, he emigrated to the United States in 1937 to teach Light and Color at Moholy's New Bauhaus (soon to be called the Institute of Design) in Chicago. In 1944, he produced Language of Vision, a landmark book about design theory, followed by the publication of six Kepes-edited anthologies in a series called Vision + Value as well as several other books. -
Art Gallery: Global Eyes Global Gallery: Art 152 ART GALLERY ELECTRONIC ART & ANIMATION CATALOG Adamczyk Walt J
Art Gallery: Global Eyes Chair Vibeke Sorensen University at Buffalo Associate Chair Lina Yamaguchi Stanford University ELECTRONIC ART & ANIMATION CATALOG ELECTRONIC ART & ANIMATION ART GALLERY ART 151 Table of Contents 154 Art Gallery Jury & Committee 176 Joreg 199 Adrian Goya 222 Masato Takahashi 245 Andrea Polli, Joe Gilmore 270 Peter Hardie 298 Marte Newcombe 326 Mike Wong imago FACES bogs: Instrumental Aliens N. falling water Eleven Fifty Nine Elevation #2 156 Introduction to the Global Eyes Exhibition Landfill 178 Takashi Kawashima 200 Ingo Günther 223 Tamiko Thiel 246 Joseph Rabie 272 Shunsaku Hayashi 327 Michael Wright Running on Empty Takashi’s Seasons Worldprocessor.com The Travels of Mariko Horo Psychogeographical Studies Perry’s Cowboy • Animation Robotman Topography Drive (Pacific Rim) Do-Bu-Ro-Ku 179 Hyung Min Lee 224 Daria Tsoupikova 158 Vladimir Bellini 247 r e a Drought 328 Guan Hong Yeoh Bibigi (Theremin Based on Computer-Vision Rutopia 2 La grua y la jirafa (The crane and the giraffe) 203 Qinglian Guo maang (message stick) 274 Taraneh Hemami Here, There Super*Nature Technology) A digital window for watching snow scenes Most Wanted 225 Ruth G. West 159 Shunsaku Hayashi 248 Johanna Reich 300 Till Nowak 329 Solvita Zarina 180 Steve Mann ATLAS in silico Ireva 204 Yoichiro Kawaguchi De Vez En Cuando 276 Guy Hoffman Salad See - Buy - Fly CyborGLOGGER Performance of Hydrodynamics Ocean Time Bracketing Study: Stata Latin 226 Ming-Yang Yu, Po-Kuang Chen 250 Seigow Matsuoka Editorial 301 Jin Wan Park, June Seok Seo 330 Andrzej -
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. -
Computer-Generated Music Through Fractals and Genetic Theory
Ouachita Baptist University Scholarly Commons @ Ouachita Honors Theses Carl Goodson Honors Program 2000 MasterPiece: Computer-generated Music through Fractals and Genetic Theory Amanda Broyles Ouachita Baptist University Follow this and additional works at: https://scholarlycommons.obu.edu/honors_theses Part of the Artificial Intelligence and Robotics Commons, Music Commons, and the Programming Languages and Compilers Commons Recommended Citation Broyles, Amanda, "MasterPiece: Computer-generated Music through Fractals and Genetic Theory" (2000). Honors Theses. 91. https://scholarlycommons.obu.edu/honors_theses/91 This Thesis is brought to you for free and open access by the Carl Goodson Honors Program at Scholarly Commons @ Ouachita. It has been accepted for inclusion in Honors Theses by an authorized administrator of Scholarly Commons @ Ouachita. For more information, please contact [email protected]. D MasterPiece: Computer-generated ~1usic through Fractals and Genetic Theory Amanda Broyles 15 April 2000 Contents 1 Introduction 3 2 Previous Examples of Computer-generated Music 3 3 Genetic Theory 4 3.1 Genetic Algorithms . 4 3.2 Ylodified Genetic Theory . 5 4 Fractals 6 4.1 Cantor's Dust and Fractal Dimensions 6 4.2 Koch Curve . 7 4.3 The Classic Example and Evrrvday Life 7 5 MasterPiece 8 5.1 Introduction . 8 5.2 In Detail . 9 6 Analysis 14 6.1 Analysis of Purpose . 14 6.2 Analysis of a Piece 14 7 Improvements 15 8 Conclusion 16 1 A Prelude2.txt 17 B Read.cc 21 c MasterPiece.cc 24 D Write.cc 32 E Temp.txt 35 F Ternpout. txt 36 G Tempoutmid. txt 37 H Untitledl 39 2 Abstract A wide variety of computer-generated music exists. -
Analysis of the Fractal Structures for the Information Encrypting Process
INTERNATIONAL JOURNAL OF COMPUTERS Issue 4, Volume 6, 2012 Analysis of the Fractal Structures for the Information Encrypting Process Ivo Motýl, Roman Jašek, Pavel Vařacha The aim of this study was describe to using fractal geometry Abstract— This article is focused on the analysis of the fractal structures for securing information inside of the information structures for purpose of encrypting information. For this purpose system. were used principles of fractal orbits on the fractal structures. The algorithm here uses a wide range of the fractal sets and the speed of its generation. The system is based on polynomial fractal sets, II. PROBLEM SOLUTION specifically on the Mandelbrot set. In the research were used also Bird of Prey fractal, Julia sets, 4Th Degree Multibrot, Burning Ship Using of fractal geometry for the encryption is an alternative to and Water plane fractals. commonly solutions based on classical mathematical principles. For proper function of the process is necessary to Keywords—Fractal, fractal geometry, information system, ensure the following points: security, information security, authentication, protection, fractal set Generate fractal structure with appropriate parameters I. INTRODUCTION for a given application nformation systems are undoubtedly indispensable entity in Analyze the fractal structure and determine the ability Imodern society. [7] to handle a specific amount of information There are many definitions and methods for their separation use of fractal orbits to alphabet mapping and classification. These systems are located in almost all areas of human activity, such as education, health, industry, A. Principle of the Fractal Analysis in the Encryption defense and many others. Close links with the life of our Process information systems brings greater efficiency of human endeavor, which allows cooperation of man and machine. -
Fractal-Bits
fractal-bits Claude Heiland-Allen 2012{2019 Contents 1 buddhabrot/bb.c . 3 2 buddhabrot/bbcolourizelayers.c . 10 3 buddhabrot/bbrender.c . 18 4 buddhabrot/bbrenderlayers.c . 26 5 buddhabrot/bound-2.c . 33 6 buddhabrot/bound-2.gnuplot . 34 7 buddhabrot/bound.c . 35 8 buddhabrot/bs.c . 36 9 buddhabrot/bscolourizelayers.c . 37 10 buddhabrot/bsrenderlayers.c . 45 11 buddhabrot/cusp.c . 50 12 buddhabrot/expectmu.c . 51 13 buddhabrot/histogram.c . 57 14 buddhabrot/Makefile . 58 15 buddhabrot/spectrum.ppm . 59 16 buddhabrot/tip.c . 59 17 burning-ship-series-approximation/BossaNova2.cxx . 60 18 burning-ship-series-approximation/BossaNova.hs . 81 19 burning-ship-series-approximation/.gitignore . 90 20 burning-ship-series-approximation/Makefile . 90 21 .gitignore . 90 22 julia/.gitignore . 91 23 julia-iim/julia-de.c . 91 24 julia-iim/julia-iim.c . 94 25 julia-iim/julia-iim-rainbow.c . 94 26 julia-iim/julia-lsm.c . 98 27 julia-iim/Makefile . 100 28 julia/j-render.c . 100 29 mandelbrot-delta-cl/cl-extra.cc . 110 30 mandelbrot-delta-cl/delta.cl . 111 31 mandelbrot-delta-cl/Makefile . 116 32 mandelbrot-delta-cl/mandelbrot-delta-cl.cc . 117 33 mandelbrot-delta-cl/mandelbrot-delta-cl-exp.cc . 134 34 mandelbrot-delta-cl/README . 139 35 mandelbrot-delta-cl/render-library.sh . 142 36 mandelbrot-delta-cl/sft-library.txt . 142 37 mandelbrot-laurent/DM.gnuplot . 146 38 mandelbrot-laurent/Laurent.hs . 146 39 mandelbrot-laurent/Main.hs . 147 40 mandelbrot-series-approximation/args.h . 148 41 mandelbrot-series-approximation/emscripten.cpp . 150 42 mandelbrot-series-approximation/index.html . -
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. -
Fractals: an Introduction Through Symmetry
Fractals: an Introduction through Symmetry for beginners to fractals, highlights magnification symmetry and fractals/chaos connections This presentation is copyright © Gayla Chandler. All borrowed images are copyright © their owners or creators. While no part of it is in the public domain, it is placed on the web for individual viewing and presentation in classrooms, provided no profit is involved. I hope viewers enjoy this gentle approach to math education. The focus is Math-Art. It is sensory, heuristic, with the intent of imparting perspective as opposed to strict knowledge per se. Ideally, viewers new to fractals will walk away with an ability to recognize some fractals in everyday settings accompanied by a sense of how fractals affect practical aspects of our lives, looking for connections between math and nature. This personal project was put together with the input of experts from the fields of both fractals and chaos: Academic friends who provided input: Don Jones Department of Mathematics & Statistics Arizona State University Reimund Albers Center for Complex Systems & Visualization (CeVis) University of Bremen Paul Bourke Centre for Astrophysics & Supercomputing Swinburne University of Technology A fourth friend who has offered feedback, whose path I followed in putting this together, and whose influence has been tremendous prefers not to be named yet must be acknowledged. You know who you are. Thanks. ☺ First discussed will be three common types of symmetry: • Reflectional (Line or Mirror) • Rotational (N-fold) • Translational and then: the Magnification (Dilatational a.k.a. Dilational) symmetry of fractals. Reflectional (aka Line or Mirror) Symmetry A shape exhibits reflectional symmetry if the shape can be bisected by a line L, one half of the shape removed, and the missing piece replaced by a reflection of the remaining piece across L, then the resulting combination is (approximately) the same as the original.1 In simpler words, if you can fold it over and it matches up, it has reflectional symmetry. -
Algorithmic Art: Technology, Mathematics and Art
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/4359618 Algorithmic art: Technology, mathematics and art Conference Paper · July 2008 DOI: 10.1109/ITI.2008.4588386 · Source: IEEE Xplore CITATIONS READS 3 2,297 1 author: Vlatko Ceric University of Zagreb 17 PUBLICATIONS 123 CITATIONS SEE PROFILE All content following this page was uploaded by Vlatko Ceric on 29 May 2014. The user has requested enhancement of the downloaded file. Algorithmic Art: Technology, Mathematics and Art Vlatko Ceric Faculty of Economics and Business, Trg J.F.Kennedyja 6, 10000 Zagreb, Croatia [email protected] Abstract. This paper describes algorithmic art, the more narrowly I limit my field of action and i.e. visual art created on the basis of algorithms the more I surround myself with obstacles». that completely describe generation of images. So it is not quite sure whether to start painting Algorithmic art is rooted in rational approach to on a blank paper without any constraints is art, technology and application of mathematics. necessarily easier than to have a constraint that, It is based on computer technology and for example, you only use squares of two sizes. particularly on programming that make The rest of this paper is organized as follows: algorithmic art possible and that influences it In Section 2 we describe relations between art most. Therefore we first explore relations of and technology, while Section 3 presents the role technology and mathematics to art and then of computer technology in development of describe algorithmic art itself. digital art, and particularly of algorithmic art. -
Pionners of Computer Art, II RCM GALERIE
Pionners of Computer Art, II RCM GALERIE Pioneers of Computer Art Colette Bangert, Aldo Giorgini, Jean-Pierre Hebert, Desmond Paul Henry, Hervé Huitric, Ken Knowlton, Charles Mattox, Manfred Mohr, Monique Nahas, Jacques Palumbo, Edward Zajec 20 June - 31 July 2019 RCM Galerie Monique Nahas (b. 1940) and and Hervé Huitric (b. 1945) co-counded the Groupe Art et Informatique de Vincennes and were among the preeminent artists working with computers in France in the late 1960s and 1970s. Their work explores how computer algorithms can be transformed into colored images. They first worked with a CAE 510 and then an IBM 1130 writing programs in Algol and then Fortran that created random variations in constructivist color schemes. Instead of working on a palette with rules of transition, the duo considered color as a continuous variable, while treating color as a set of percentages of its basic components. Although interested in how the computer changed the rules of art, their works in the early seventies are highly influenced by pontilism, especially Georges Seurat. The duo's first work used a coded sequence of letters then printed on a plotter. R for Red, B for Blue etc. Each square was then hand painted. The couple then developed a stencil method, using the punch cards and screenprinting artwork using a plotter. The couple exhibited widely in the 1970s, including the New Tendency 5 exhibition in Zagreb. Monique Nahas and Hervé Huitric Cube, 1971 Layered perforated program cards, paint, wood, lacquer 12 x 12 x 12 inches Unique Monique Nahas and Hervé Huitric Untitled, 1971 Double serigraph 31.5 x 22 in One of two copies Signed on verso Monique Nahas and Hervé Huitric Monique Nahas and Hervé Huitric Untitled, 1971 Untitled, 1971 Computer print, hand-painted 6.2 x 7.8 in Computer print, hand-painted 9.4 x 7.4 in Unique Unique Signed on verso Signed on verso 1 1 - ' ' I I ..._ / / • / / .