Elegant Fractals

The following sections are included:

• Dedication
• Preface
• Contents

This chapter will introduce some of the fractal and chaos concepts and terminology so that the book is self-contained and readable even if you are unfamiliar with the subject. It will also describe how the images are produced using the Hénon map as the main example. It should enable you to get a quick start producing your own elegant fractal images.

This chapter will describe efficient methods for finding new examples of elegant attractors among the much more numerous unbounded and inelegant solutions that occur. It will introduce some additional nomenclature and quantitative measures used throughout the book and will show examples of attractors found using those methods for a particular six-dimensional quadratic map. Finally, it will suggest ways to automate the selection of those attractors most likely to appeal to a human.

This chapter will apply the methods in the previous chapters to the search for elegant attractors for a variety of iterated maps of a particular type. The main difference is in the form of the nonlinear map function. It will also describe the role of symmetry in the visual appeal of the resulting images.

Most physical processes in nature are better described by variables that change continuously in time rather than at discrete instants. Such systems are governed by systems of ordinary differential equations rather than by iterated maps. This chapter give examples of continuous-time systems whose attractors have a somewhat different appearance than those in the preceding chapters.

One of the most common and versatile methods for producing fractals makes use of iterated function systems. They are a kind of iterated map, and you can use the methods developed earlier to produce the images and to automatically select ones that are likely to be elegant from the vast majority that are completely dull. This chapter will show some such examples and will suggest ways to extend the method.

Some of the most elegant fractals are produced by plots showing in color the number of iterations required for an orbit starting at a particular point to escape beyond some boundary surrounding an attractor. In fact, the famous Mandelbrot set and its associated Julia sets are the best known such examples. This chapter shows a number of other less familiar examples employing the maps and flows in the previous chapters as well as a special map that is a generalization of the usual Julia sets.

All the previous examples involved systems in which single orbits were followed in time as they visited successive points in space. Here we consider systems with many points in space that evolve simultaneously according to their state and the state of their near neighbors. Such systems are more computationally intensive, but they represent better models of real-world systems such as weather patterns and landscapes, even if the resulting images are generally less elegant than the previous ones.

All the previous examples are from deterministic dynamical systems, although the rules, initial conditions, and colors are sometimes chosen randomly. As a result, the same initial conditions will give exactly the same pattern every time, and the resulting fractal patterns are perfectly self-similar. Here we consider stochastic cases where the rules involve randomness and the resulting fractals are only statistically self-similar, and sometimes not even that.

The final example of fractals will involve tilings of the plane. Often such patterns are tediously repetitious, but we will consider examples that are aperiodic and that have structure down to the smallest visible scale. Some of the examples are deterministic, while others are random and thus represent an extension of the cases in the previous chapter.

This short chapter will not introduce any new fractal types, but rather it will describe the computer program used to produce the images in the book. It will also summarize the state of the art in automating the selection of elegant fractal images. Finally, it will offer suggestions for future work and possible additional books on other forms of elegance.

The following sections are included:

• Bibliography
• Index
• About the Author