Fractals, Visualization and J is a text that uses fractals as a motivational goal for the study of visualization. The language J is introduced as needed for the topics at hand. Included are chapters: Introduction to J and Graphics, Plots, Verbs and First Fractals, Time Series and Fractals, Iterated function systems and Raster Fractals, Color, Contours and Animations, Complex Dynamics, Cellular Automata.
Author(s): Clifford Reiter
Edition: 4
Year: 2016
Language: English
Pages: 147
Tags: APL, J, Fractals
Preface ................................................................................................................................................ v
Getting Ready: Addon Files ...................................................................................................... vi
Chapter 1 Introduction to J and Graphics ........................................................................................ 1
1.1 Some Arithmetic with J ........................................................................................................ 1
1.2 Lists, Arrays and Trigonometric Functions ......................................................................... 3
1.3 Experiment: Plotting Polygons ............................................................................................ 4
1.4 Constructing Arrays .............................................................................................................. 5
1.5 Experiment: Creating a Raster Image .................................................................................. 7
1.6 Object versus Raster Graphics ............................................................................................. 8
1.7 Defining Functions ............................................................................................................... 8
1.8 On Language ....................................................................................................................... 12
1.9 Errors and Getting Help ..................................................................................................... 12
1.10 More Nouns and Array Computations ............................................................................. 15
1.11 Exercises ............................................................................................................................ 18
Chapter 2 Plots, Verbs and First Fractals ....................................................................................... 23
2.1 Function Composition and Plots ........................................................................................ 23
2.2 Experiment: Plotting Time Series, Functions and Curves ................................................ 25
2.3 More Function Composition .............................................................................................. 26
2.4 Experiment: The Koch Snowflake ..................................................................................... 30
2.5 Transformations of the Plane and Homogeneous Coordinates ....................................... 32
2.6 Experiment: Transformations and Animations ................................................................ 34
2.7 Gerunds and Multiplots ..................................................................................................... 36
2.8 Experiment: Collages of Transformations ........................................................................ 38
2.9 Simple Verbs ...................................................................................................................... 39
2.10 Exercises ........................................................................................................................... 41
Chapter 3 Time Series and Fractals ................................................................................................ 45
3.1 Statistics and Least Squares Fit ......................................................................................... 45
3.2 Experiment: Plot Driver ..................................................................................................... 47
3.3 Random Walks ................................................................................................................... 47
3.4 Experiment: Observing Trends ......................................................................................... 50
3.5 R/S Analysis, the Hurst Exponent, and Sunspots............................................................. 52
3.6 Autocorrelation Functions ................................................................................................. 55
3.7 Experiment: Random Midpoint Displacement ................................................................. 56
3.8 Experiment: Forecasting via Best Analogs........................................................................ 58
3.9 Exercises ............................................................................................................................. 61
Chapter 4 Iterated Function Systems and Raster Fractals ............................................................ 65
4.1 Agenda and the 3x+1 Function .......................................................................................... 65
4.2 Experiment: Probabilistic Iterated Function Systems...................................................... 66
4.3 Remarks on Iterated Function Systems ............................................................................ 69
4.4 Weighted Selection of Random Transformations .............................................................. 71
4.5 Experiment: The Chaos Game ........................................................................................... 72
4.6 Fractal Dimension .............................................................................................................. 75
4.7 Fractal Dimension via Raster Box Counting ..................................................................... 76
4.8 Exercises ............................................................................................................................. 78
Chapter 5 Color, Contours and Animations .................................................................................... 83
5.1 The RGB Color Model ......................................................................................................... 83
5.2 Adverbs and Conjunctions ................................................................................................. 85
5.3 Experiment: Color Contour Plots ...................................................................................... 86
5.4 Animations ......................................................................................................................... 89
5.5 Plasma Clouds .................................................................................................................... 90
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5.6 Experiment: Palettes and Inner Product Fractals ............................................................ 92
5.7 Inverse Iterated Function Systems .................................................................................... 96
5.8 Exercises ............................................................................................................................. 98
Chapter 6 Complex Dynamics ........................................................................................................101
6.1 Experiment: Julia Sets ......................................................................................................101
6.2 Experiment: Julia sets for Elliptic Curves ....................................................................... 103
6.3 The Mandelbrot Set .......................................................................................................... 104
6.4 The 3x+1 Function in the Complex Plane ....................................................................... 106
6.5 Newton's Method in the Complex Plane ......................................................................... 108
6.6 Exercises ............................................................................................................................ 112
Chapter 7 Cellular Automata .......................................................................................................... 113
7.1 One Dimensional Automata .............................................................................................. 113
7.2 Fuzzy Logic and Fuzzy Automata ..................................................................................... 117
7.3 Experiment: The Game of Life .......................................................................................... 121
7.4 Majority Rule and Spot Formation .................................................................................. 124
7.5 Cyclic Cellular Automata .................................................................................................. 126
7.6 Experiment: The Hodgepodge Rule ................................................................................ 128
7.7 Hexagonal Lattice and the Packard-Wolfram Snowflake ................................................ 131
7.8 A Snowflake Model Using Intermediate Values .............................................................. 133
7.9 Exercises ........................................................................................................................... 134
Bibliography and References ......................................................................................................... 137
Index............................................................................................................................................... 139
