Mathematica. But almost without exception, it has in the past only been applied to systems and questions that have already been investigated by other mathematical means—and that lie very much within the normal tradition of mathematics. My approach in this book, however, is to use computer experiments as a basic way to explore much more general systems—that have never arisen in traditional mathematics, and that are usually far from being accessible by existing mathematical means.
Fractal Geometry. Until recently, the only kinds of shapes widely discussed in science and mathematics were ones that are regular or smooth. But starting in the late 1970s, the field of fractal geometry emphasized the importance of nested shapes that contain arbitrarily intricate pieces, and argued that such shapes are common in nature. In this book we will encounter a fair number of systems that produce such nested shapes. But we will also find many systems that produce shapes which are much more complex, and have no nested structure.
General Systems Theory. Popular especially in the 1960s, general systems theory was concerned mainly with studying large networks of elements—often idealizing human organizations. But a complete lack of anything like the kinds of methods I use in this book made it almost impossible for any definite conclusions to emerge.
Nanotechnology. Growing rapidly since the early 1990s, the goal of nanotechnology is to implement technological systems on atomic scales. But so far nanotechnology has mostly been concerned with shrinking quite familiar mechanical and other devices. Yet what the discoveries in this book now show is that there are all sorts of systems that have much simpler structures, but that can nevertheless perform very sophisticated tasks. And some of these systems seem in many ways much more suitable for direct implementation on an atomic scale.
Nonlinear Dynamics. Mathematical equations that have the property of linearity are usually fairly easy to solve, and so have been used extensively in pure and applied science. The field of nonlinear dynamics is concerned with analyzing more complicated equations. Its greatest success has been with so-called soliton equations for which