1. The whole is greater than the sum of the parts
By this I mean that the topics of computation, fractals, chaos,
complex systems, and adaptation are far more interesting considered
together than by themselves. Each of the topics is related to the
others in a nontrivial way; moreover, each can be seen as the result
of a few simple principles such as recursion, parallelism, and
nonlinearity.


Let's examine this idea further. Starting from a computational
framework, fractals are special ``programs'' that build selfsimilar
structures.
Computation 
structural 

selfsimilarity 

Fractals 
The program in this case often involves a form of recursion that
specifies how one part of a fractal
looks like another portion
of it
(or of the whole fractal). Other times, as in the case of the
Mandelbrot set
and in
Julia sets,
structural selfsimilarity appears
almost as a sideeffect because the compact rules that describe such
objects do not explicitly contain specifications for how things should
look.
In either case, when one allows the rules of a fractal to be
expressed ad infinitum strange mathematical properties can be
found in the structural forms of the fractal. In the most extreme
cases, fractals such as the Mandelbrot set can be seen as mathematical
statements that are similar in complexity to incomputable problems
found in computer science.
Chaotic systems are similar to fractals but contain functional
selfsimilarity that occurs at different scales.
Fractals 
functional 

selfsimilarity 

Chaos 
More specifically, while we could visually confirm that a portion
of a fractal is similar to itself, it is also possible to see how a
chaotic system is functionally selfsimilar. Often times, this
functional selfsimilarity shows itself as a characteristic behavior
that occurs on multiple time and/or spatial scales. This is why a
strange attractor
(which is a characterization of a chaotic system's
behavior) will usually have a fractal shape to it.
Chaos, which depends on nonlinearity, is usually studied in systems
that can be described with only a few equations. However, another type
of chaos is found in large collections that operate in parallel.
Examples of this latter type include ecosystems, economies, and
chemical soups. We refer to such collections as complex systems
because even though a single agent in the collection can be understood
(e.g., a predator, stock trader, or reactant) the composite behavior
of the system is far more complicated.
Chaos 
multiplicity 

& parallelism 

Complex Systems 
Hence, by adding multiplicity and parallelism to collections of
simple nonlinear agents, it is possible to produce behavior that is
not only chaotic, but computationally profound as well.
One of the more interesting facts concerning complex systems is that
sophisticated behavior can be found even when the interactions among
the system's agents are constrained to be local. Sometimes the
details of how the interaction occur can be subtly modified, as in the
case when a synapse that connects two neurons is allowed to strengthen
or weaken.
Complex Systems 
feedback 



Adaptation 
Since agents in a complex system are all indirectly coupled to one
another, it is possible for a sophisticated form of feedback to occur
that allows the agents (and the system as a whole) to adapt in a
seemingly intelligent manner. While an obvious form of adaptation
through feedback is found in the brain, more subtle forms can be found
in market economies, immune systems, and through evolution.
Since adaptation is ultimately a form of modeling (where events
external to the adapted system are mirrored internally), adaptive
systems that alter their environment are indirectly altering
themselves.
Adaptation 
self 

reference 

Computation 
This means that some adaptive systems can be selfreferential (or
recursive), which is a requirement for the ability to perform complex
computations. Adaptive systems that contain such selfreference (such
as recurrent neural networks) are capable of universal computation.
Having come fullcircle in seeing how all of these topics relate in a
sequential manner, it is now worthwhile to see what some of the
crossconnections are between the five topics. The table below
contains a list of subjects that relate to two or more of the book
topics.

Computation 
Fractals 
Chaos 
Complex Systems 
Adaptation 
Computation 
 
recursion 
incomputability 
cellular automata 
coadaptation 
Fractals 
recursion 
 
strange attractors 
growth models 
hierarchical models 
Chaos 
incomputability 
strange attractors 
 
phase transitions 
emergence 
Complex Systems 
cellular automata 
growth models 
phase transitions 
 
selforganization 
Adaptation 
coadaptation 
hierarchical models 
emergence 
selforganization 
 
As can be seen, there are many subjects that are clearly related to
more than one book part. By viewing all of these topics in a
multidisciplinary manner, one can appreciate how the topics of
computation, fractals, chaos, complex system, and adaptation are far
more interesting when considered together.