Graphics Reference
In-Depth Information
CHAPTER 22
Chaos and Fractals
Prerequisites:
Chapter 5 and Chapter 7 (for Theorem 22.3.1) in [AgoM05]
22.1
Introduction
Physics, and science in general, is filled with the study of
dynamical systems
, that is,
processes that evolve with time and where future states are determined by past states.
The evolution of the universe and the motion of the planets around the sun are two
examples of dynamical systems. Describing what happens in such systems, even ones
that are defined in rather simple ways, is often extremely difficult. What is the final
state of the universe? What orbits will the planets eventually assume? Answering such
questions involves a lot of topology but is also a natural context in which to discuss
chaos and fractals. Fractals are the de facto way to model natural phenomena in com-
puter graphics (see Section 5.4). The interconnection between all these topics is what
this chapter is all about. We can only provide a brief outline of some highlights and
the reader who wants to learn more will need to consult the references.
Section 22.2 defines dynamical systems and some basic terminology associated to
them. We also explain when such systems are said to be chaotic. In Section 22.3 we
define what is meant by the dimension of a topological space and discuss the con-
nection of this concept with that of a fractal. Finally, Section 22.4 is on iterated func-
tion systems on which many of the common algorithms for generating fractals are
based.
22.2
Dynamical Systems and Chaos
Mathematically, the study of dynamical systems reduces to
a space
X
(the states),
a map f:
X
Æ
X
(f corresponds to the laws which control a process),
