DOCSLIB.ORG

Mandelbulb 3D Tutorial: Kleinian Spine and Starfish with Joskn-Kleinifs by Persistent Aura (

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mandelbulb 3D Tutorial: Kleinian Spine and Starfish with Joskn-Kleinifs by Persistent Aura ( Mandelbulb 3D Tutorial: Kleinian Spine and Starfish with JosKn-KleinIFS By Persistent Aura (http://persistentaura.deviantart.com/) The JosKn-KleinIFS is a fantastic formula to work with. It is remarkably simple to achieve great results on its own without the need for complicated formula combinations. In this tutorial we will be learning how to create spine- and starfish-like kleinian structures using the JosKn-KleinIFS formula in Mandelbulb 3D. All credit goes to Jos Leys for creating the code, with the help of Knighty, and LucaGN (aka Dark- Beam) for making the Mandelbulb3D formula. First of all it is important to add the JosKn-KleinIFS formula to your M3Formulas folder in the Mandelbulb 3D directory if you haven't already (...\Mandelbulb 3D v1.9.1\M3Formulas). You can find the latest version of the formula here: http://www.fractalforums.com/mandelbulb- 3d/custom-formulas-and-transforms-release-t17106/ (download the .zip file at the bottom of the first post). Once we have added the formula to the program we'll open up our Mandelbulb 3D software. Let's go to the formula tab and select the JosKn-KleinIFS formula in the first formula slot (replace the default Integer Power formula). We'll immediately want to set the Max. iterations to 1 as this will dramatically improve the rendering time. We'll be working in sphere inversion mode. So go ahead and change the Options parameter to 4. We'll also be resetting both the position and rotation parameters. At this point our image should look something like this: To start tweaking the fractal I recommend opening the navigator window and clicking the sideways pointing arrow in the bottom right. This will allow us to tweak the formula parameters using sliders and seeing the results in (quasi) real-time. Using the sliders and seeing what happens will really help you get a better feel for the fractal and will give you a better understanding of which parts of the fractal a certain parameter affects. Now let's start by raising the KleinR parameter to 2. We'll leave invCx at 1 for now but we'll raise invCy from 0.5 to 1. Be sure to play around with sliders and take a good look at what happens when manipulating these parameters (you can set the intensity of the adjustments at the top). You might want to reorient the camera a little bit to get a better view of the fractal. Now we'll start tweaking some very crucial parameters for these types of fractals, BoxSzX and BoxSzY. Tweaking the BoxSzX slider you will see the fractal folding in and out of itself. Especially when lowering the value, we start seeing some more spine-like features. Setting the BoxSzX value to 0.5 will give us a good looking result. Again, we may want to change our camera view to get a better view of the fractal. As we're working with a vertical object, I also decided to change the aspect ratio of the image from 4:3 to 3:4 (you could of course just rotate the fractal instead). Now let's take a look at the BoxSzY parameter. This one really changes the 'texture' of the spheres. Raising the value seems to make the spheres look more empty and clean. Lowering it gives us some more interesting, rougher looking results. Once past the 0.85 mark we seem to lose the "clean" part of the spheres almost completely. 0.8 looks like a good value to me. Another important parameter we haven't looked at is KleinI. Playing around with the slider you can see the "vertebrae" of the spine rotating. After fiddling around a bit I ended up with a value of 0.05. The fractal is now really starting to resemble the shape of a spine. Now we can start fine-tuning some of the parameters until we have a shape we're satisfied with. I ended up raising both invCx and invCy from 1 to 1.1 and lowering KleinR from 2 to 1.986 to get a beautiful saxophone-like shape. All that is left to do now is colouring and lighting this beauty. Here are the parameters of the final spine image: Mandelbulb3Dv18{ g....kV....o0.../....2....ESoFS9fzH6.nHqkI3Ic14EkLXbUM1JPzXN10MhnWC..hBJElqHFr.E ................................Quh4SkVdPz1........A./..................y.2...wD ...Uz6.....0..../MU//......c....8.....E3.....cKCI3h1x8sD/.......B.GT0dkpXm1....U z.....kD12../..........wz.................................U0.....y1...sD...../.. .z1...sD6Io.yb8j/yXGl3ZgtDX2zSTPy3uSfUqjKp.9L7VR0xvkNhnWXwYUz2NL5cUv8upjXo7mlypi bx1ghjBE1AyLzAg7CJBMG3sDU.....21..............sD.6....sD..G..................... .............oAnAt1...sD....z.hgwtdbHjgPu/f9DA........................../....k1. .....4iSoz1.......kz.wzzz1.U..6.P....U5...EB....m....c3....F....6/...I1.....SF52 ...U.qFG9yzb2zzzRYoWzz7lz16.pc..zXCc..kvrEtMc7xD6ocyFE0ujz1..........2.28.kFrA0. .Ub96aAIVz9.1se7Umvxz0........../EU0.wzzz1...........s/...................E.2c.. zzzz.............0...................2./8.kzzzD............8.................... /EU0.wzzz1...................................2CcN/UvNPcveeWCNq0.yRiibHJJUk1f..XR SvBmx3CcN/UvNPcvQsLsUa3.ibhVi1bTV1OK.sSq4uCly3CcN/UvNPcvMwLsUa3.ibhVinqTV1OK.sSq 4uCkz3CcN/UvNPcv..EsUa3feeWCNqGQIJ36wk8EwyLsUa3f................................ E....2..F2E.....I....w....UGjBrGipmGgJKOiZYFH/...............................6U. 0.........E9mqtvbOwzzcNaNaNaNauDzzzzzzzzTzHaNaNaNaNuz.........zD........kz1..... ...8./........zD........E.YaNaNaNaNwzcNaNaNaN4zD...................wz........... ........Uz1.............} {Titel: KleinIFS Spine Tutorial} Turning these spines into starfish shapes is just a matter of manipulating the KleinI parameter. A value of 0.98 really seems to be a sweet spot for a nice starfish shape. But of course there's a lot of other values to be tried out. Use the slider to change the values and see what other interesting shapes you can come up with. The starfish shape may not immediately be apparent after changing the KleinI value to 0.98 from our last image. You will see something like this: The reason for this is that our fractal object has suddenly become a lot bigger, and we need to zoom out quite a bit to get it into full view. I also changed the aspect ratio to 4:3 again. All we want to do now is raise the KleinIters parameter to extend the arms and provide more detail to the empty spheres in the fractal. We now end up with a beautiful starfish-like shape. If you like a more 'spherical' look to your fractal, raise the BoxSzY parameter again. A value of 1 has this result: And there you go, an intricate Kleinian starfish! Have fun experimenting! Starfish parameters: Mandelbulb3Dv18{ g.....h....50.../....2....EGgf2LFdML.TSwpkHO0s6EuE.CkN/b2/gDVmtSVHi7.b9mWfgR3C1k ................................OaNaNaNady1........A./..................y.2...wD ...Uz6.....0..../MU//......01...8.....E3.....IzP28XSPtuD/.......1URj0dkpXm1..... z.EnAnQD12../..........wz.................................U0.....y1...sD...../.. .z1...kD8txVBs6hly1KDgNBbiWEzukpbXgPUUtjlbGWW4OPmxfFZnoc2PYgzEWwt/pxwtsjduSSOM6g LyXGr7wBjFxXz6Qg7mLR83vDU.....YN..............sD.6....sD..G..................... .............oAnAt1...sD....z.hgwtdbHjgPu/f9DA........................../....k1. .....4iSoz1.......kz.wzzz1.U..6.P....U5...EB....m....c3....F....6/...I1.....SF52 ...U.qFG9yzb2zzzRYoWzz7lz16.pc..zXCc..kvrEtMc7xD6ocyFE0ujz1..........2.28.kFrA0. .Ub96aAIVz9.1se7Umvxz0........../EU0.wzzz1...........s/...................E.2c.. zzzz.............0...................2./8.kzzzD............8.................... /EU0.wzzz1...................................2CcN/UvNPcveeWCNq0.yRiibHJJUk1f..XR SvBmx3CcN/UvNPcvQsLsUa3.ibhVi1bTV1OK.sSq4uCly3CcN/UvNPcvMwLsUa3.ibhVinqTV1OK.sSq 4uCkz3CcN/UvNPcv..EsUa3feeWCNqGQIJ36wk8EwyLsUa3f................................ E....2..F2E.....I....w....UGjBrGipmGgJKOiZYFH/...............................6U. 0.........E9mqtvbOwzzkpX0LD8QxyD........Uz1........wz.........zD........kz1..... ...B./........zD........E.YaNaNaNaNwzcNaNaNaN4zD...................wz........... ........Uz1.............} {Titel: KleinIFS Starfish Tutorial} .
Load more

Recommended publications
  • 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]
  • 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]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [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]
  • 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.
    [Show full text]
  • Fractal Modeling and Fractal Dimension Description of Urban Morphology
    Fractal Modeling and Fractal Dimension Description of Urban Morphology Yanguang Chen (Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, P. R. China. E-mail: [email protected]) Abstract: The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a measure of scale dependence, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization based on scaling. Fractal geometry is one powerful tool for scaling analysis of cities. Fractal parameters can be defined by entropy and correlation functions. However, how to understand city fractals is still a pending question. By means of logic deduction and ideas from fractal theory, this paper is devoted to discussing fractals and fractal dimensions of urban landscape. The main points of this work are as follows. First, urban form can be treated as pre-fractals rather than real fractals, and fractal properties of cities are only valid within certain scaling ranges. Second, the topological dimension of city fractals based on urban area is 0, thus the minimum fractal dimension value of fractal cities is equal to or greater than 0. Third, fractal dimension of urban form is used to substitute urban area, and it is better to define city fractals in a 2-dimensional embedding space, thus the maximum fractal dimension value of urban form is 2. A conclusion can be reached that urban form can be explored as fractals within certain ranges of scales and fractal geometry can be applied to the spatial analysis of the scale-free aspects of urban morphology.
    [Show full text]
  • Axe-Fx & Axe-Fx ULTRA
    Axe-Fx & Axe-Fx ULTRA Pre-Amp / Effects Processor User’s Manual For Firmware 9.xx and up Table of Contents Table of Contents .............................................................................. 1 Foreword............................................................................................ 3 About the Ultra Model................................................................................ 4 Introduction........................................................................................ 5 What is the Axe-Fx?.................................................................................. 5 Concept..................................................................................................... 8 Getting Set Up ................................................................................. 11 Rear Panel .............................................................................................. 11 Front Panel.............................................................................................. 12 Example Connections ............................................................................. 14 I/O Configuration ..................................................................................... 17 Editing.............................................................................................. 22 Placing Effects......................................................................................... 22 Routing...................................................................................................
    [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]
  • Creating Symmetric Fractals Larry Riddle Ractals Such As the Sierpinski Triangle, the Horizontal Translation by 1
    Symmetric Fractals Seeking Sangaku Ramanujan, Hardy, and Ono Published by the Mathematical Association of America : : November 2016 Figure 1. Clockwise from far left, the Sierpinski triangle, the Koch curve, and the Heighway dragon. Creating Symmetric Fractals Larry Riddle ractals such as the Sierpinski triangle, the horizontal translation by 1. The dragon is the unique Koch curve, and the Heighway dragon, set A, called the attractor of the IFS, that satis- shown in figure 1, are constructed using fi es In fi gure 1, h1(A) is red and simple rules, yet they exhibit beautifully F h2(A) is blue. Both subsets are copies of the Heighway intricate and complex patterns. dragon scaled by a factor of r. All three fractals possess self-similarity—that is, the fractal is composed of smaller copies of itself. But only Bringing in Abstract Algebra the fi rst two fractals have symmetries. In this article In Symmetry in Chaos: A Search for Pattern we show how to use group theory, which is often used in Mathematics, Art and Nature (1992, Oxford to describe the symmetries of objects, to create sym- University Press), Michael Field and Martin metric fractals. Golubitsky showed how to generate a symmetric fractal from a single affi ne transformation and a Iterated Function Systems cyclic group Zn or a dihedral group Dn—groups that Fractals such as those in fi gure 1 can be constructed students encounter in an abstract algebra course. As using sets of functions called iterated function systems we shall see, their method applies equally well when (IFS). The functions in an IFS have a scaling factor applied to an iterated function system.
    [Show full text]