Wolfram 1

The Foundations for a
New Kind of Science
An Outline of Basic Ideas

Three centuries ago science was transformed by the dramatic new idea that rules based on
mathematical equations could be used to describe the natural world. My purpose in this book is to
initiate another such transformation, and to introduce a new kind of science that is based on the much
more general types of rules that can be embodied in simple computer programs.
It has taken me the better part of twenty years to build the intellectual structure that is needed, but I
have been amazed by its results. For what I have found is that with the new kind of science I have
developed it suddenly becomes possible to make progress on a remarkable range of fundamental
issues that have never successfully been addressed by any of the existing sciences before.
If theoretical science is to be possible at all, then at some level the systems it studies must follow
definite rules. Yet in the past throughout the exact sciences it has usually been assumed that these
rules must be ones based on traditional mathematics. But the crucial realization that led me to develop
the new kind of science in this book is that there is in fact no reason to think that systems like those
we see in nature should follow only such traditional mathematical rules.
Earlier in history it might have been difficult to imagine what more general types of rules could be
like. But today we are surrounde

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by computers whose programs in effect implement a huge variety of rules. The programs we use in
practice are mostly based on extremely complicated rules specifically designed to perform particular
tasks. But a program can in principle follow essentially any definite set of rules. And at the core of the
new kind of science that I describe in this book are discoveries I have made about programs with some
of the very simplest rules that are possible.
One might have thought--as at first I certainly did--that if the rules for a program were simple then
this would mean that its behavior must also be correspondingly simple. For our everyday experience
in building things tends to give us the intuition that creating complexity is somehow difficult, and
requires rules or plans that are themselves complex. But the pivotal discovery that I made some
eighteen years ago is that in the world of programs such intuition is not even close to correct.
I did what is in a sense one of the most elementary imaginable computer experiments: I took a
sequence of simple programs and then systematically ran them to see how they behaved. And what I
found--to my great surprise--was that despite the simplicity of their rules, the behavior of the
programs was often far from simple. Indeed, even some of the very simplest programs that I looked at
had behavior that was as complex as anything I had ever seen.
It took me more than a decade to come to terms with this result, and to realize just how
fundamental and far-reaching its consequences are. In retrospect there is no reason the result could not
have been found centuries ago, but increasingly I have come to view it as one of the more important

single discoveries in the whole history of theoretical science. For in addition to opening up vast new
domains of exploration, it implies a radical rethinking of how processes in nature and elsewhere work.
Perhaps immediately most dramatic is that it yields a resolution to what has long been considered
the single greatest mystery of the natural world: what secret it is that allows nature seemingly so
effortlessly to produce so much that appears to us so complex.
It could have been, after all, that in the natural world we would mostly see forms like squares and
circles that we consider simple. But in fact one of the most striking features of the natural world is that

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across a vast range of physical, biological and other systems we are continually confronted with what
seems to be immense complexity. And indeed throughout most of history it has been taken almost for
granted that such complexity--being so vastly greater than in the works of humans--could only be the
work of a supernatural being.
But my discovery that many very simple programs produce great complexity immediately suggests
a rather different explanation. For all it takes is that systems in nature operate like typical programs
and then it follows that their behavior will often be complex. And the reason that such complexity is
not usually seen in human artifacts is just that in building these we tend in effect to use programs that
are specially chosen to give only behavior simple enough for us to be able to see that it will achieve
the purposes we want.
One might have thought that with all their successes over the past few centuries the existing
sciences would long ago have managed to address the issue of complexity. But in fact they have not.
And indeed for the most part they have specifically defined their scope in order to avoid direct contact
with it. For while their basic idea of describing behavior in terms of mathematical equations works
well in cases like planetary motion where the behavior is fairly simple, it almost inevitably fails
whenever the behavior is more complex. And more or less the same is true of descriptions based on
ideas like natural selection in biology. But by thinking in terms of programs the new kind of science
that I develop in this book is for the first time able to make meaningful statements about even
immensely complex behavior.
In the existing sciences much of the emphasis over the past century or so has been on breaking
systems down to find their underlying parts, then trying to analyze these parts in as much detail as
possible. And particularly in physics this approach has been sufficiently successful that the basic
components of everyday systems are by now completely known. But just how these components act
together to produce even some of the most obvious features of the overall behavior we see has in the
past remained an almost complete mystery. Within the framework of the new kind of science that I
develop in this book, however, it is finally possible to address such a question.

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From the tradition of the existing sciences one might expect that its answer would depend on all
sorts of details, and be quite different for different types of physical, biological and other systems. But
in the world of simple programs I have discovered that the same basic forms of behavior occur over

and over again almost independent of underlying details. And what this suggests is that there are quite
universal principles that determine overall behavior and that can be expected to apply not only to
simple programs but also to systems throughout the natural world and elsewhere.
In the existing sciences whenever a phenomenon is encountered that seems complex it is taken
almost for granted that the phenomenon must be the result of some underlying mechanism that is itself
complex. But my discovery that simple programs can produce great complexity makes it clear that this
is not in fact correct. And indeed in the later parts of this book I will show that even remarkably
simple programs seem to capture the essential mechanisms responsible for all sorts of important
phenomena that in the past have always seemed far too complex to allow any simple explanation.
It is not uncommon in the history of science that new ways of thinking are what finally allow
longstanding issues to be addressed. But I have been amazed at just how many issues central to the
foundations of the existing sciences I have been able to address by using the idea of thinking in terms
of simple programs. For more than a century, for example, there has been confusion about how
thermodynamic behavior arises in physics. Yet from my discoveries about simple programs I have
developed a quite straightforward explanation. And in biology, my discoveries provide for the first
time an explicit way to understand just how it is that so many organisms exhibit such great
complexity. Indeed, I even have increasing evidence that thinking in terms of simple programs will
make it possible to construct a single truly fundamental theory of physics, from which space, time,
quantum mechanics and all the other known features of our universe will emerge.
When mathematics was introduced into science it provided for the first time an abstract framework
in which scientific conclusions could be drawn without direct reference to physical reality. Yet despite

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all its development over the past few thousand years mathematics itself has continued to concentrate
only on rather specific types of abstract systems--most often ones somehow derived from arithmetic or
geometry. But the new kind of science that I describe in this book introduces what are in a sense much
more general abstract systems, based on rules of essentially any type whatsoever.
One might have thought that such systems would be too diverse for meaningful general statements
to be made about them. But the crucial idea that has allowed me to build a unified framework for the
new kind of science that I describe in this book is that just as the rules for any system can be viewed
as corresponding to a program, so also its behavior can be viewed as corresponding to a computation.
Traditional intuition might suggest that to do more sophisticated computations would always
require more sophisticated underlying rules. But what launched the whole computer revolution is the
remarkable fact that universal systems with fixed underlying rules can be built that can in effect
perform any possible computation.
The threshold for such universality has however generally been assumed to be high, and to be
reached only by elaborate and special systems like typical electronic computers. But one of the
surprising discoveries in this book is that in fact there are systems whose rules are simple enough to
describe in just one sentence that are nevertheless universal. And this immediately suggests that the

phenomenon of universality is vastly more common and important--in both abstract systems and
nature--than has ever been imagined before.
But on the basis of many discoveries I have been led to a still more sweeping conclusion,
summarized in what I call the Principle of Computational Equivalence: that whenever one sees
behavior that is not obviously simple--in essentially any system--it can be thought of as corresponding
to a computation of equivalent sophistication. And this one very basic principle has a quite
unprecedented array of implications for science and scientific thinking.
For a start, it immediately gives a fundamental explanation for why simple programs can show
behavior that seems to us complex. For like other processes our own processes of perception and
analysis can be

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thought of as computations. But though we might have imagined that such computations would
always be vastly more sophisticated than those performed by simple programs, the Principle of
Computational Equivalence implies that they are not. And it is this equivalence between us as
observers and the systems that we observe that makes the behavior of such systems seem to us
complex.
One can always in principle find out how a particular system will behave just by running an
experiment and watching what happens. But the great historical successes of theoretical science have
typically revolved around finding mathematical formulas that instead directly allow one to predict the
outcome. Yet in effect this relies on being able to shortcut the computational work that the system
itself performs.
And the Principle of Computational Equivalence now implies that this will normally be possible
only for rather special systems with simple behavior. For other systems will tend to perform
computations that are just as sophisticated as those we can do, even with all our mathematics and
computers. And this means that such systems are computationally irreducible--so that in effect the
only way to find their behavior is to trace each of their steps, spending about as much computational
effort as the systems themselves.
So this implies that there is in a sense a fundamental limitation to theoretical science. But it also
shows that there is something irreducible that can be achieved by the passage of time. And it leads to
an explanation of how we as humans--even though we may follow definite underlying rules--can still
in a meaningful way show free will.
One feature of many of the most important advances in science throughout history is that they
show new ways in which we as humans are not special. And at some level the Principle of
Computational Equivalence does this as well. For it implies that when it comes to computation--or
intelligence--we are in the end no more sophisticated than all sorts of simple programs, and all sorts of
systems in nature.
But from the Principle of Computational Equivalence there also emerges a new kind of unity: for
across a vast range of systems, from

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simple programs to brains to our whole universe, the principle implies that there is a basic equivalence
that makes the same fundamental phenomena occur, and allows the same basic scientific ideas and
methods to be used. And it is this that is ultimately responsible for the great power of the new kind of
science that I describe in this book.

Relations to Other Areas

Mathematics. It is usually assumed that mathematics concerns itself with the study of arbitrarily
general abstract systems. But this book shows that there are actually a vast range of abstract systems
based on simple programs that traditional mathematics has never considered. And because these
systems are in many ways simpler in construction than most traditional systems in mathematics it is
possible with appropriate methods in effect to go further in investigating them.
Some of what one finds are then just unprecedentedly clear examples of phenomena already
known in modern mathematics. But one also finds some dramatic new phenomena. Most immediately
obvious is a very high level of complexity in the behavior of many systems whose underlying rules are
much simpler than those of most systems in standard mathematics textbooks.
And one of the consequences of this complexity is that it leads to fundamental limitations on the
idea of proof that has been central to traditional mathematics. Already in the 1930s Gödel's Theorem
gave some indications of such limitations. But in the past they have always seemed irrelevant to most
of mathematics as it is actually practiced.
Yet what the discoveries in this book show is that this is largely just a reflection of how small the
scope is of what is now considered mathematics. And indeed the core of this book can be viewed as
introducing a major generalization of mathematics--with new ideas and methods, and vast new areas
to be explored.
The framework I develop in this book also shows that by viewing the process of doing
mathematics in fundamentally computational terms it becomes possible to address important issues
about the foundations even of existing mathematics.

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Physics. The traditional mathematical approach to science has historically had its great success in
physics--and by now it has become almost universally assumed that any serious physical theory must
be based on mathematical equations. Yet with this approach there are still many common physical
phenomena about which physics has had remarkably little to say. But with the approach of thinking in
terms of simple programs that I develop in this book it finally seems possible to make some dramatic
progress. And indeed in the course of the book we will see that some extremely simple programs seem
able to capture the essential mechanisms for a great many physical phenomena that have previously
seemed completely mysterious.
Existing methods in theoretical physics tend to revolve around ideas of continuous numbers and
calculus--or sometimes probability. Yet most of the systems in this book involve just simple discrete
elements with definite rules. And in many ways it is the greater simplicity of this underlying structure
that ultimately makes it possible to identify so many fundamentally new phenomena.

Ordinary models for physical systems are idealizations that capture some features and ignore
others. And in the past what was most common was to capture certain simple numerical
relationships--that could for example be represented by smooth curves. But with the new kinds of
models based on simple programs that I explore in this book it becomes possible to capture all sorts of
much more complex features that can only really be seen in explicit images of behavior.
In the future of physics the greatest triumph would undoubtedly be to find a truly fundamental
theory for our whole universe. Yet despite occasional optimism, traditional approaches do not make
this seem close at hand. But with the methods and intuition that I develop in this book there is I
believe finally a serious possibility that such a theory can actually be found.
Biology. Vast amounts are now known about the details of biological organisms, but very little in the
way of general theory has ever emerged. Classical areas of biology tend to treat evolution by natural
selection as

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a foundation--leading to the notion that general observations about living systems should normally be
analyzed on the basis of evolutionary history rather than abstract theories. And part of the reason for
this is that traditional mathematical models have never seemed to come even close to capturing the
kind of complexity we see in biology. But the discoveries in this book show that simple programs can
produce a high level of complexity. And in fact it turns out that such programs can reproduce many
features of biological organisms--and for example seem to capture some of the essential mechanisms
through which genetic programs manage to generate the actual biological forms we see. So this means
that it becomes possible to make a wide range of new models for biological systems--and potentially
to see how to emulate the essence of their operation, say for medical purposes. And insofar as there
are general principles for simple programs, these principles should also apply to biological
organisms--making it possible to imagine constructing new kinds of general abstract theories in
biology.
Social Sciences. From economics to psychology there has been a widespread if controversial
assumption--no doubt from the success of the physical sciences--that solid theories must always be
formulated in terms of numbers, equations and traditional mathematics. But I suspect that one will
often have a much better chance of capturing fundamental mechanisms for phenomena in the social
sciences by using instead the new kind of science that I develop in this book based on simple
programs. No doubt there will quite quickly be all sorts of claims about applications of my ideas to the
social sciences. And indeed the new intuition that emerges from this book may well almost
immediately explain phenomena that have in the past seemed quite mysterious. But the very results of
the book show that there will inevitably be fundamental limits to the application of scientific methods.
There will be new questions formulated, but it will take time before it becomes clear when general
theories are possible, and when one must instead inevitably rely on the details of judgement for
specific cases.

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Computer Science. Throughout its brief history computer science has focused almost exclusively on
studying specific computational systems set up to perform particular tasks. But one of the core ideas
of this book is to consider the more general scientific question of what arbitrary computational
systems do. And much of what I have found is vastly different from what one might expect on the
basis of existing computer science. For the systems traditionally studied in computer science tend to
be fairly complicated in their construction--yet yield fairly simple behavior that recognizably fulfills
some particular purpose. But in this book what I show is that even systems with extremely simple
construction can yield behavior of immense complexity. And by thinking about this in computational
terms one develops a new intuition about the very nature of computation.
One consequence is a dramatic broadening of the domain to which computational ideas can be
applied--in particular to include all sorts of fundamental questions about nature and about
mathematics. Another consequence is a new perspective on existing questions in computer science--
particularly ones related to what ultimate resources are needed to perform general types of
computational tasks.
Philosophy. At any period in history there are issues about the universe and our role in it that seem
accessible only to the general arguments of philosophy. But often progress in science eventually
provides a more definite context. And I believe that the new kind of science in this book will do this
for a variety of issues that have been considered fundamental even since antiquity. Among them are
questions about ultimate limits to knowledge, free will, the uniqueness of the human condition and the
inevitability of mathematics. Much has been said over the course of philosophical history about each
of these. Yet inevitably it has been informed only by current intuition about how things are supposed
to work. But my discoveries in this book lead to radically new intuition. And with this intuition it
turns out that one can for the first time begin to see resolutions to many longstanding issues--typically
along rather different lines from those expected on the basis of traditional general arguments in
philosophy

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Art. It seems so easy for nature to produce forms of great beauty. Yet in the past art has mostly just
had to be content to imitate such forms. But now, with the discovery that simple programs can capture
the essential mechanisms for all sorts of complex behavior in nature, one can imagine just sampling
such programs to explore generalizations of the forms we see in nature. Traditional scientific
intuition--and early computer art--might lead one to assume that simple programs would always
produce pictures too simple and rigid to be of artistic interest. But looking through this book it
becomes clear that even a program that may have extremely simple rules will often be able to generate
pictures that have striking aesthetic qualities--sometimes reminiscent of nature, but often unlike
anything ever seen before.
Technology. Despite all its success, there is still much that goes on in nature that seems more complex
and sophisticated than anything technology has ever been able to produce. But what the discoveries in
this book now show is that by using the types of rules embodied in simple programs one can capture
many of the essential mechanisms of nature. And from this it becomes possible to imagine a whole
new kind of technology that in effect achieves the same sophistication as nature. Experience with

traditional engineering has led to the general assumption that to perform a sophisticated task requires
constructing a system whose basic rules are somehow correspondingly complicated. But the
discoveries in this book show that this is not the case, and that in fact extremely simple underlying
rules--that might for example potentially be implemented directly at the level of atoms--are often all
that is needed. My main focus in this book is on matters of basic science. But I have little doubt that
within a matter of a few decades what I have done will have led to some dramatic changes in the
foundations of technology--and in our basic ability to take what the universe provides and apply it for
our own human purposes.

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Some Past Initiatives

My goals in this book are sufficiently broad and fundamental that there have inevitably been
previous attempts to achieve at least some of them. But without the ideas and methods of this book
there have been basic issues that have eventually ended up presenting almost insuperable barriers to
every major approach that has been tried.
Artificial Intelligence. When electronic computers were first invented, it was widely believed that it
would not be long before they would be capable of human-like thinking. And in the 1960s the field of
artificial intelligence grew up with the goal of understanding processes of human thinking and
implementing them on computers. But doing this turned out to be much more difficult than expected,
and after some spin-offs, little fundamental progress was made. At some level, however, the basic
problem has always been to understand how the seemingly simple components in a brain can lead to
all the complexities of thinking. But now finally with the framework developed in this book one
potentially has a meaningful foundation for doing this. And indeed building on both theoretical and
practical ideas in the book I suspect that dramatic progress will eventually be possible in creating
technological systems that are capable of human-like thinking.
Artificial Life. Ever since machines have existed, people have wondered to what extent they might be
able to imitate living systems. Most active from the mid-1980s to the mid-1990s, the field of artificial
life concerned itself mainly with showing that computer programs could be made to emulate various
features of biological systems. But normally it was assumed that the necessary programs would have
to be quite complex. What the discoveries in this book show, however, is that in fact very simple
programs can be sufficient. And such programs make the fundamental mechanisms for behavior
clearer--and probably come much closer to what is actually happening in real biological systems.
Catastrophe Theory. Traditional mathematical models are normally based on quantities that vary
continuously. Yet in nature discrete changes are often seen. Popular in the 1970s, catastrophe theory
was

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concerned with showing that even in traditional mathematical models, certain simple discrete changes
could still occur. In this book I do not start from any assumption of continuity--and the types of
behavior I study tend to be vastly more complex than those in catastrophe theory.
Chaos Theory. The field of chaos theory is based on the observation that certain mathematical
systems behave in a way that depends arbitrarily sensitively on the details of their initial conditions.
First noticed at the end of the 1800s, this came into prominence after computer simulations in the
1960s and 1970s. Its main significance is that it implies that if any detail of the initial conditions is
uncertain, then it will eventually become impossible to predict the behavior of the system. But despite
some claims to the contrary in popular accounts, this fact alone does not imply that the behavior will
necessarily be complex. Indeed, all that it shows is that if there is complexity in the details of the
initial conditions, then this complexity will eventually appear in the large-scale behavior of the
system. But if the initial conditions are simple, then there is no reason for the behavior not to be
correspondingly simple. What I show in this book, however, is that even when their initial conditions
are very simple there are many systems that still produce highly complex behavior. And I argue that it
is this phenomenon that is for example responsible for most of the obvious complexity we see in
nature.
Complexity Theory. My discoveries in the early 1980s led me to the idea that complexity could be
studied as a fundamental independent phenomenon. And gradually this became quite popular. But
most of the scientific work that was done ended up being based only on my earliest discoveries, and
being very much within the framework of one or another of the existing sciences--with the result that
it managed to make very little progress on any general and fundamental issues. One feature of the new
kind of science that I describe in this book is that it finally makes possible the development of a basic
understanding of the general phenomenon of complexity, and its origins.

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Computational Complexity Theory. Developed mostly in the 1970s, computational complexity
theory attempts to characterize how difficult certain computational tasks are to perform. Its concrete
results have tended to be based on fairly specific programs with complicated structure yet rather
simple behavior. The new kind of science in this book, however, explores much more general classes
of programs--and in doing so begins to shed new light on various longstanding questions in
computational complexity theory.
Cybernetics. In the 1940s it was thought that it might be possible to understand biological systems on
the basis of analogies with electrical machines. But since essentially the only methods of analysis
available were ones from traditional mathematics, very little of the complex behavior of typical
biological systems was successfully captured.
Dynamical Systems Theory. A branch of mathematics that began roughly a century ago, the field of
dynamical systems theory has been concerned with studying systems that evolve in time according to
certain kinds of mathematical equations--and in using traditional geometrical and other mathematical
methods to characterize the possible forms of behavior that such systems can produce. But what I
argue in this book is that in fact the behavior of many systems is fundamentally too complex to be
usefully captured in any such way.

Evolution Theory. The Darwinian theory of evolution by natural selection is often assumed to
explain the complexity we see in biological systems--and in fact in recent years the theory has also
increasingly been applied outside of biology. But it has never been at all clear just why this theory
should imply that complexity is generated. And indeed I will argue in this book that in many respects
it tends to oppose complexity. But the discoveries in the book suggest a new and quite different
mechanism that I believe is in fact responsible for most of the examples of great complexity that we
see in biology.
Experimental Mathematics. The idea of exploring mathematical systems by looking at data from
calculations has a long history, and has gradually become more widespread with the advent of
computers and

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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

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careful manipulation leads to a property similar to linearity. But the kinds of systems that I discuss in
this book typically show much more complex behavior, and have no such simplifying properties.
Scientific Computing. The field of scientific computing has usually been concerned with taking
traditional mathematical models--most often for various kinds of fluids and solids--and trying to
implement them on computers using numerical approximation schemes. Typically it has been difficult
to disentangle anything but fairly simple phenomena from effects associated with the approximations
used. The kinds of models that I introduce in this book involve no approximations when implemented
on computers, and thus readily allow one to recognize much more complex phenomena.
Self-Organization. In nature it is quite common to see systems that start disordered and featureless,
but then spontaneously organize themselves to produce definite structures. The loosely knit field of
self-organization has been concerned with understanding this phenomenon. But for the most part it has
used traditional mathematical methods, and as a result has only been able to investigate the formation
of fairly simple structures. With the ideas in this book, however, it becomes possible to understand
how vastly more complex structures can be formed.
Statistical Mechanics. Since its development about a century ago, the branch of physics known as
statistical mechanics has mostly concerned itself with understanding the average behavior of systems
that consist of large numbers of gas molecules or other components. In any specific instance, such
systems often behave in a complex way. But by looking at averages over many instances, statistical
mechanics has usually managed to avoid such complexity. To make contact with real situations,
however, it has often had to use the so-called Second Law of Thermodynamics, or Principle of
Entropy Increase. But for more than a century there have been nagging difficulties in understanding
the basis for this principle. With the ideas in this book, however, I believe that there is now a
framework in which these can finally be resolved.

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The Personal Story of the Science in This Book

I can trace the beginning of my serious interest in the kinds of scientific issues discussed in this
book rather accurately to the summer of 1972, when I was twelve years old. I had bought a copy of the
physics textbook on the right, and had become very curious about the process of randomization
illustrated on its cover. But being far from convinced by the mathematical explanation given in the
book, I decided to try to simulate the process for myself on a computer.
The computer to which I had access at that time was by modern standards a very primitive one.
And as a result, I had no choice but to study a very simplified version of the process in the book. I
suspected from the start that the system I constructed might be too simple to show any of the
phenomena I wanted. And after much programming effort I managed to convince myself that these
suspicions were correct.
Yet as it turns out, what I looked at was a particular case of one of the main kinds of systems--
cellular automata--that I consider in this book. And had it not been for a largely technical point that
arose from my desire to make my simulations as physically realistic as possible, it is quite possible

that by 1974 I would already have discovered some of the principal phenomena that I now describe in
this book.
As it was, however, I decided at that time to devote my energies to what then seemed to be the
most fundamental area of science: theoretical particle physics. And over the next several years I did
indeed manage to make significant progress in a few areas of particle physics and cosmology. But
after a while I began to suspect that many of the most important and fundamental questions that I was
encountering were quite independent of the abstruse details of these fields.
And in fact I realized that there were many related questions even about common everyday
phenomena that were still completely unanswered. What for example is the fundamental origin of the
complicated patterns that one sees in turbulent fluids? How are the intricate patterns of snowflakes
produced? What is the basic mechanism that allows plants and animals to grow in such complex
ways?

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To my surprise, very little seemed to have been done on these kinds of questions. At first I thought
it might be possible to make progress just by applying some of the sophisticated mathematical
techniques that I had used in theoretical physics. But it soon became clear that for the phenomena I
was studying, traditional mathematical results would be very difficult, if not impossible, to find.
So what could I do? It so happened that as an outgrowth of my work in physics I had in 1981 just
finished developing a large software system that was in some respects a forerunner to parts of
Mathematica. And at least at an intellectual level the most difficult part of the project had been
designing the symbolic language on which the system was based. But in the development of this
language I had seen rather clearly how just a few primitive operations that I had come up with could
end up successfully covering a vast range of sophisticated computational tasks.
So I thought that perhaps I could do something similar in natural science: that there might be some
appropriate primitives that I could find that would successfully capture a vast range of natural
phenomena. My ideas were not so clearly formed at the time, but I believe I implicitly imagined that
the way this would work is that such primitives could be used to build up computer programs that
would simulate the various natural systems in which I was interested.
There were in many cases well-established mathematical models for the individual components of
such systems. But two practical issues stood in the way of using these as a basis for simulations. First,
the models were usually quite complicated, so that with realistic computer resources it was very
difficult to include enough components for interesting phenomena to occur. And second, even if one
did see such phenomena, it was almost impossible to tell whether in fact they were genuine
consequences of the underlying models or were just the result of approximations made in
implementing the models on a computer.
But what I realized was that at least for many of the phenomena I wanted to study, it was not
crucial to use the most accurate possible models for individual components. For among other things
there was evidence from nature that in many cases the details of the components did not matter much--
so that for example the same complex patterns

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of flow occur in both air and water. And with this in mind, what I decided was that rather than starting
from detailed realistic models, I would instead start from models that were somehow as simple as
possible--and were easy to set up as programs on a computer.
At the outset, I did not know how this would work, and how complicated the programs I would
need would have to be. And indeed when I looked at various simple programs they always seemed to
yield behavior vastly simpler than any of the systems I wanted to study.
But in the summer of 1981 I did what I considered to be a fairly straightforward computer
experiment to see how all programs of a particular type behaved. I had not really expected too much
from this experiment. But in fact its results were so surprising and dramatic that as I gradually came to
understand them, they forced me to change my whole view of science, and in the end to develop the
whole intellectual structure of the new kind of science that I now describe in this book.
The picture on the right shows a reproduction of typical output from my original experiment. The
graphics are primitive, but the elaborate patterns they contain were like nothing I had ever seen before.
At first I did not believe that they could possibly be correct. But after a while I became convinced that
they were--and I realized that I had seen a sign of a quite remarkable and unexpected phenomenon:
that even from very simple programs behavior of great complexity could emerge.
But how could something as fundamental as this never have been noticed before? I searched the
scientific literature, talked to many people, and found out that systems similar to the ones I was
studying had been named "cellular automata" some thirty years earlier. But despite a few close
approaches, nobody had ever actually tried anything quite like the type of experiment I had.
Yet I still suspected that the basic phenomenon I had seen must somehow be an obvious
consequence of some known scientific principle. But while I did find that ideas from areas like chaos
theory and fractal geometry helped in explaining some specific features, nothing even close to the
phenomenon as a whole seemed to have ever been studied before.
My early discoveries about the behavior of cellular automata stimulated a fair amount of activity in
the scientific community. And

Captions on this page:
A reproduction of the computer printout that first gave me a hint of some of the central phenomena in this book.

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by the mid-1980s, many applications had been found in physics, biology, computer science,
mathematics and elsewhere. And indeed some of the phenomena I had discovered were starting to be
used as the basis for a new area of research that I called complex systems theory.
Throughout all this, however, I had continued to investigate more basic questions, and by around
1985 I was beginning to realize that what I had seen before was just a hint of something still much
more dramatic and fundamental. But to understand what I was discovering was difficult, and required
a major shift in intuition.
Yet I could see that there were some remarkable intellectual opportunities ahead. And my first idea
was to try to organize the academic community to take advantage of them. So I started a research

center and a journal, published a list of problems to attack, and worked hard to communicate the
importance of the direction I was defining.
But despite growing excitement--particularly about some of the potential applications--there
seemed to be very little success in breaking away from traditional methods and intuition. And after a
while I realized that if there was going to be any dramatic progress made, I was the one who was
going to have to make it. So I resolved to set up the best tools and infrastructure I could, and then just
myself pursue as efficiently as possible the research that I thought should be done.
In the early 1980s my single greatest impediment had been the practical difficulty of doing
computer experiments using the various rather low-level tools that were available. But by 1986 I had
realized that with a number of new ideas I had it would be possible to build a single coherent system
for doing all kinds of technical computing. And since nothing like this seemed likely to exist
otherwise, I decided to build it.
The result was Mathematica.
For five years the process of building Mathematica and the company around it absorbed me. But in
1991--now no longer an academic, but instead the CEO of a successful company--I was able to return
to studying the kinds of questions addressed in this book.
And equipped with Mathematica I began to try all sorts of new experiments. The results were
spectacular--and within the space of a few months I had already made more new discoveries about
what

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simple programs do than in all the previous ten years put together. My earlier work had shown me the
beginnings of some unexpected and very remarkable phenomena. But now from my new experiments
I began to see the full force and generality of these phenomena.
As my methodology and intuition improved, the pace of my discoveries increased still more, and
within just a couple of years I had managed to take my explorations of the world of simple programs
to the point where the sheer volume of factual information I had accumulated would be the envy of
many long-established fields of science.
Quite early in the process I had begun to formulate several rather general principles. And the
further I went, the more these principles were confirmed, and the more I realized just how strong and
general they were.
When I first started at the beginning of the 1980s, my goal was mostly just to understand the
phenomenon of complexity. But by the mid-1990s I had built up a whole intellectual structure that
was capable of much more, and that in fact provided the foundations for what could only be
considered a fundamentally new kind of science.
It was for me a most exciting time. For everywhere I turned there were huge untouched new areas
that I was able to explore for the first time. Each had its own particular features. But with the overall
framework I had developed I was gradually able to answer essentially all of what seemed to be the
most obvious questions that I had raised.
At first I was mostly concerned with new questions that had never been particularly central to any
existing areas of science. But gradually I realized that the new kind of science I was building should
also provide a fundamentally new way to address basic issues in existing areas.

So around 1994 I began systematically investigating each of the various major traditional areas of
science. I had long been interested in fundamental questions in many of these areas. But usually I had
tended to believe most of the conventional wisdom about them. Yet when I began to study them in the
context of my new kind of science I kept on seeing signs that large parts of this conventional wisdom
could not be correct.
The typical issue was that there was some core problem that traditional methods or intuition had
never successfully been able to address--and which the field had somehow grown to avoid. Yet over

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and over again I was excited to find that with my new kind of science I could suddenly begin to make
great progress--even on problems that in some cases had remained unanswered for centuries.
Given the whole framework I had built, many of the things I discovered seemed in the end
disarmingly simple. But to get to them often involved a remarkable amount of scientific work. For it
was not enough just to be able to take a few specific technical steps. Rather, in each field, it was
necessary to develop a sufficiently broad and deep understanding to be able to identify the truly
essential features--that could then be rethought on the basis of my new kind of science.
Doing this certainly required experience in all sorts of different areas of science. But perhaps most
crucial for me was that the process was a bit like what I have ended up doing countless times in
designing Mathematica: start from elaborate technical ideas, then gradually see how to capture their
essential features in something amazingly simple. And the fact that I had managed to make this work
so many times in Mathematica was part of what gave me the confidence to try doing something
similar in all sorts of areas of science.
Often it seemed in retrospect almost bizarre that the conclusions I ended up reaching had never
been reached before. But studying the history of each field I could in many cases see how it had been
led astray by the lack of some crucial piece of methodology or intuition that had now emerged in the
new kind of science I had developed.
When I made my first discoveries about cellular automata in the early 1980s I suspected that I had
seen the beginning of something important. But I had no idea just how important it would all
ultimately turn out to be. And indeed over the past twenty years I have made more discoveries than I
ever thought possible. And the new kind of science that I have spent so much effort building has
seemed an ever more central and critical direction for future intellectual development.

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The Crucial Experiment
How Do Simple Programs Behave?

New directions in science have typically been initiated by certain central observations or
experiments. And for the kind of science that I describe in this book these concerned the behavior of
simple programs.

In our everyday experience with computers, the programs that we encounter are normally set up to
perform very definite tasks. But the key idea that I had nearly twenty years ago--and that eventually
led to the whole new kind of science in this book--was to ask what happens if one instead just looks at
simple arbitrarily chosen programs, created without any specific task in mind. How do such programs
typically behave?
The mathematical methods that have in the past dominated theoretical science do not help much
with such a question. But with a computer it is straightforward to start doing experiments to
investigate it. For all one need do is just set up a sequence of possible simple programs, and then run
them and see how they behave.
Any program can at some level be thought of as consisting of a set of rules that specify what it
should do at each step. There are many possible ways to set up these rules--and indeed we will study
quite a few of them in the course of this book. But for now, I will consider a particular class of
examples called cellular automata, that were the very first kinds of simple programs that I investigated
in the early 1980s.

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An important feature of cellular automata is that their behavior can readily be presented in a visual
way. And so the picture below shows what one cellular automaton does over the course of ten steps.
The cellular automaton consists of a line of cells, each colored either black or white. At every step
there is then a definite rule that determines the color of a given cell from the color of that cell and its
immediate left and right neighbors on the step before.
For the particular cellular automaton shown here the rule specifies--as in the picture below--that a
cell should be black in all cases where it or either of its neighbors were black on the step before.
And the picture at the top of the page shows that starting with a single black cell in the center this
rule then leads to a simple growing pattern uniformly filled with black. But modifying the rule just
slightly one can immediately get a different pattern.
As a first example, the picture at the top of the facing page shows what happens with a rule that
makes a cell white whenever both of its neighbors were white on the step before--even if the cell itself
was black before. And rather than producing a pattern that is uniformly filled with black, this rule now
instead gives a pattern that repeatedly alternates between black and white like a checkerboard.

A visual representation of the behavior of a cellular automaton, with each row of cells corresponding to one step.
At the first step the cell in the center is black and all other cells are white. Then on each successive step, a
particular cell is made black whenever it or either of its neighbors were black on the step before. As the picture
shows, this leads to a simple expanding pattern uniformly filled with black.

A representation of the rule for the cellular automaton shown above. The top row in each box gives one of the
possible combinations of colors for a cell and its immediate neighbors. The bottom row then specifies what color
the center cell should be on the next step in each of these cases. In the numbering scheme described in Chapter
3, this is cellular automaton rule 254.

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This pattern is however again fairly simple. And we might assume that at least with the type of
cellular automata that we are considering, any rule we might choose would always give a pattern that
is quite simple. But now we are in for our first surprise.
The picture below shows the pattern produced by a cellular automaton of the same type as before,
but with a slightly different rule.

A cellular automaton with a slightly different rule. The rule makes a particular cell black if either of its neighbors
was black on the step before, and makes the cell white if both its neighbors were white. Starting from a single
black cell, this rule leads to a checkerboard pattern. In the numbering scheme of Chapter 3, this is cellular
automaton rule 250.

A cellular automaton that produces an intricate nested pattern. The rule in this case is that a cell should be black
whenever one or the other, but not both, of its neighbors were black on the step before. Even though the rule is
very simple, the picture shows that the overall pattern obtained over the course of 50 steps starting from a single
black cell is not so simple. The particular rule used here can be described by the formula ai'= Mod[ai-1 + ai+1, 2].
In the numbering scheme of Chapter 3, it is cellular automaton rule 90.

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This time the rule specifies that a cell should be black when either its left neighbor or its right
neighbor--but not both--were black on the step before. And again this rule is undeniably quite simple.
But now the picture shows that the pattern it produces is not so simple.
And if one runs the cellular automaton for more steps, as in the picture below, then a rather
intricate pattern emerges. But one can now see that this pattern has very definite regularity. For even
though it is intricate, one can see that it actually consists of many nested triangular pieces that all have
exactly the same form. And as the picture shows, each of these pieces is essentially just a smaller copy
of the whole pattern, with still smaller copies nested in a very regular way inside it.
So of the three cellular automata that we have seen so far, all ultimately yield patterns that are
highly regular: the first a simple uniform pattern, the second a repetitive pattern, and the third an
intricate but still nested pattern. And we might assume that at least for

A larger version of the pattern from the previous page, now shown without a grid explicitly indicating each cell. The
picture shows five hundred steps of cellular automaton evolution. The pattern obtained is intricate, but has a
definite nested structure. Indeed, as the picture illustrates, each triangular section is essentially just a smaller copy
of the whole pattern, with still smaller copies nested inside it. Patterns with nested structure of this kind are often
called "fractal" or "self-similar".

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cellular automata with rules as simple as the ones we have been using these three forms of behavior
would be all that we could ever get.
But the remarkable fact is that this turns out to be wrong.
And the picture below shows an example of this. The rule used--that I call rule 30--is of exactly
the same kind as before, and can be described as follows. First, look at each cell and its right-hand
neighbor. If both of these were white on the previous step, then take the new color of the cell to be
whatever the previous color of its left-hand neighbor was. Otherwise, take the new color to be the
opposite of that.
The picture shows what happens when one starts with just one black cell and then applies this rule
over and over again. And what one sees is something quite startling--and probably the single most
surprising scientific discovery I have ever made. Rather than getting a simple regular pattern as we
might expect, the cellular automaton instead produces a pattern that seems extremely irregular and
complex.

A cellular automaton with a simple rule that generates a pattern which seems in many respects random. The rule
used is of the same type as in the previous examples, and the cellular automaton is again started from a single
black cell. But now the pattern that is obtained is highly complex, and shows almost no overall regularity. This
picture is our first example of the fundamental phenomenon that even with simple underlying rules and simple
initial conditions, it is possible to produce behavior of great comDiscHyphenplexity. In the numbering scheme of
Chapter 3, the cellular automaton shown here

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But where does this complexity come from? We certainly did not put it into the system in any
direct way when we set it up. For we just used a simple cellular automaton rule, and just started from a
simple initial condition containing a single black cell.
Yet the picture shows that despite this, there is great complexity in the behavior that emerges. And
indeed what we have seen here is a first example of an extremely general and fundamental
phenomenon that is at the very core of the new kind of science that I develop in this book. Over and
over again we will see the same kind of thing: that even though the underlying rules for a system are
simple, and even though the system is started from simple initial conditions, the behavior that the
system shows can nevertheless be highly complex. And I will argue that it is this basic phenomenon
that is ultimately responsible for most of the complexity that we see in nature.
The next two pages [29, 30] show progressively more steps in the evolution of the rule 30 cellular
automaton from the previous page. One might have thought that after maybe a thousand steps the
behavior would eventually resolve into something simple. But the pictures on the next two pages [29,
30] show that nothing of the sort happens.
Some regularities can nevertheless be seen. On the left-hand side, for example, there are obvious
diagonal bands. And dotted throughout there are various white triangles and other small structures.
Yet given the simplicity of the underlying rule, one would expect vastly more regularities. And
perhaps one might imagine that our failure to see any in the pictures on the next two pages [29, 30] is
just a reflection of some kind of inadequacy in the human visual system.

But it turns out that even the most sophisticated mathematical and statistical methods of analysis
seem to do no better. For example, one can look at the sequence of colors directly below the initial
black cell. And in the first million steps in this sequence, for example, it never repeats, and indeed
none of the tests I have ever done on it show any meaningful deviation at all from perfect randomness.
In a sense, however, there is a certain simplicity to such perfect randomness. For even though it
may be impossible to predict what

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Five hundred steps in the evolution of the rule 30 cellular automaton from page 27. The pattern produced
continues to expand on both left and right, but only the part that fits across the page is shown here. The
asymmetry between the left and right-hand sides is a direct consequence of asymmetry that exists in the particular
underlying cellular automaton rule used.

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(gambar)

Fifteen hundred steps of rule 30 evolution. Some regularities are evident, particularly on the left. But even after all
these steps there are no signs of overall regularity--and indeed even continuing for a million steps many aspects of
the pattern obtained seem perfectly random according to standard mathematical and statistical tests. The picture
here shows a total of just under two million individual cells.

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color will occur at any specific step, one still knows for example that black and white will on average
always occur equally often.
But it turns out that there are cellular automata whose behavior is in effect still more complex--and
in which even such averages become very difficult to predict. The pictures on the next several pages
[32, 33, 34, 35, 36, 37, 38] give a rather dramatic example. The basic form of the rule is just the same
as before. But now the specific rule used--that I call rule 110--takes the new color of a cell to be black
in every case except when the previous colors of the cell and its two neighbors were all the same, or
when the left neighbor was black and the cell and its right neighbor were both white.
The pattern obtained with this rule shows a remarkable mixture of regularity and irregularity. More
or less throughout, there is a very regular background texture that consists of an array of small white
triangles repeating every 7 steps. And beginning near the left-hand edge, there are diagonal stripes that
occur at intervals of exactly 80 steps.
But on the right-hand side, the pattern is much less regular. Indeed, for the first few hundred steps
there is a region that seems essentially random. But by the bottom of the first page all that remains of
this region is three copies of a rather simple repetitive structure.
Yet at the top of the next page the arrival of a diagonal stripe from the left sets off more
complicated behavior again. And as the system progresses, a variety of definite localized structures
are produced.

Some of these structures remain stationary, like those at the bottom of the first page while others
move steadily to the right or left at various speeds. And on their own, each of these structures works in
a fairly simple way. But as the pictures illustrate, their various interactions can have very complicated
effects.
And as a result it becomes almost impossible to predict--even approximately--what the cellular
automaton will do.
Will all the structures that are produced eventually annihilate each other, leaving only a very
regular pattern? Or will more and more structures appear until the whole pattern becomes quite
random?
The only sure way to answer these questions, it seems, is just to run the cellular automaton for as
many steps as are needed, and to

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(gambar)

A cellular automaton whose behavior seems neither highly regular nor completely random. The picture is obtained
by applying the simple rule shown for a total of 150 steps, starting with a single black cell. Note that the particular
rule used here yields a pattern that expands on the left but not on the right. In the scheme defined in Chapter 3, the
rule is number 110.

More steps in the pattern shown above. Each successive page shows a total of 700 steps. The pattern continues
to expand on the left forever, but only the part that fits across each page is shown. For a long time it is not clear
how the right-hand part of the pattern will eventually look. But after 2780 steps, a fairly simple repetitive structure
emerges. Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for
individual cells a total of about 12 million times.

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(gambar)

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(Gambar)

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(gambar)

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(gambar)

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(gambar)

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(gambar)

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watch what happens. And as it turns out, in the particular case shown here, the outcome is finally clear
after about 2780 steps: one structure survives, and that structure interacts with the periodic stripes
coming from the left to produce behavior that repeats every 240 steps.
However certain one might be that simple programs could never do more than produce simple
behavior, the pictures on the past few pages [32, 33, 34, 35, 36, 37, 38] should forever disabuse one of
that notion. And indeed, what is perhaps most bizarre about the pictures is just how little trace they
ultimately show of the simplicity of the underlying cellular automaton rule that was used to produce
them.
One might think, for example, that the fact that all the cells in a cellular automaton follow exactly
the same rule would mean that in pictures like the last few pages [32, 33, 34, 35, 36, 37, 38] all cells
would somehow obviously be doing the same thing. But instead, they seem to be doing quite different
things. Some of them, for example, are part of the regular background, while others are part of one or
another localized structure. And what makes this possible is that even though individual cells follow
the same rule, different configurations of cells with different sequences of colors can together produce
all sorts of different kinds of behavior.
Looking just at the original cellular automaton rule one would have no realistic way to foresee all
of this. But by doing the appropriate computer experiments one can easily find out what actually
happens--and in effect begin the process of exploring a whole new world of remarkable phenomena
associated with simple programs.

The Need for a New Intuition

The pictures in the previous section plainly show that it takes only very simple rules to produce
highly complex behavior. Yet at first this may seem almost impossible to believe. For it goes against
some of our most basic intuition about the way things normally work.

A single picture of the behavior from the previous five pages [33, 34, 35, 36, 37]. A total of 3200 steps are shown.
Note that this is more than twice as many as in the picture on page 30.

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For our everyday experience has led us to expect that an object that looks complicated must have
been constructed in a complicated way. And so, for example, if we see a complicated mechanical

device, we normally assume that the plans from which the device was built must also somehow be
correspondingly complicated.
But the results at the end of the previous section show that at least sometimes such an assumption
can be completely wrong. For the patterns we saw are in effect built according to very simple plans--
that just tell us to start with a single black cell, and then repeatedly to apply a simple cellular
automaton rule. Yet what emerges from these plans shows an immense level of complexity.
So what is it that makes our normal intuition fail? The most important point seems to be that it is
mostly derived from experience with building things and doing engineering--where it so happens that
one avoids encountering systems like the ones in the previous section.
For normally we start from whatever behavior we want to get, then try to design a system that will
produce it. Yet to do this reliably, we have to restrict ourselves to systems whose behavior we can
readily understand and predict--for unless we can foresee how a system will behave, we cannot be
sure that the system will do what we want.
But unlike engineering, nature operates under no such constraint. So there is nothing to stop
systems like those at the end of the previous section from showing up. And in fact one of the
important conclusions of this book is that such systems are actually very common in nature.
But because the only situations in which we are routinely aware both of underlying rules and
overall behavior are ones in which we are building things or doing engineering, we never normally get
any intuition about systems like the ones at the end of the previous section.
So is there then any aspect of everyday experience that should give us a hint about the phenomena
that occur in these systems? Probably the closest is thinking about features of practical computing.
For we know that computers can perform many complex tasks. Yet at the level of basic hardware a
typical computer is capable of executing just a few tens of kinds of simple logical, arithmetic and
other instructions. And to some extent the fact that by executing large numbers of such

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instructions one can get all sorts of complex behavior is similar to the phenomenon we have seen in
cellular automata.
But there is an important difference. For while the individual machine instructions executed by a
computer may be quite simple, the sequence of such instructions defined by a program may be long
and complicated. And indeed--much as in other areas of engineering--the typical experience in
developing software is that to make a computer do something complicated requires setting up a
program that is itself somehow correspondingly complicated.
In a system like a cellular automaton the underlying rules can be thought of as rough analogs of the
machine instructions for a computer, while the initial conditions can be thought of as rough analogs of
the program. Yet what we saw in the previous section is that in cellular automata not only can the
underlying rules be simple, but the initial conditions can also be simple--consisting say of just a single
black cell--and still the behavior that is produced can be highly complex.
So while practical computing gives a hint of part of what we saw in the previous section, the whole
phenomenon is something much larger and stronger. And in a sense the most puzzling aspect of it is
that it seems to involve getting something from nothing.

For the cellular automata we set up are by any measure simple to describe. Yet when we ran them
we ended with patterns so complex that they seemed to defy any simple description at all.
And one might hope that it would be possible to call on some existing kind of intuition to
understand such a fundamental phenomenon. But in fact there seems to be no branch of everyday
experience that provides what is needed. And so we have no choice but to try to develop a whole new
kind of intuition.
And the only reasonable way to do this is to expose ourselves to a large number of examples. We
have seen so far only a few examples, all in cellular automata. But in the next few chapters [3, 4, 5, 6]
we will see many more examples, both in cellular automata and in all sorts of other systems. And by
absorbing these examples, one is in the end able to develop an intuition that makes the basic
phenomena that I have discovered seem somehow almost obvious and inevitable

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Why These Discoveries Were Not Made Before

The main result of this chapter--that programs based on simple rules can produce behavior of great
complexity--seems so fundamental that one might assume it must have been discovered long ago. But
it was not, and it is useful to understand some of the reasons why it was not.
In the history of science it is fairly common that new technologies are ultimately what make new
areas of basic science develop. And thus, for example, telescope technology was what led to modern
astronomy, and microscope technology to modern biology. And now, in much the same way, it is
computer technology that has led to the new kind of science that I describe in this book.
Indeed, this chapter and several of those that follow can in a sense be viewed as an account of
some of the very simplest experiments that can be done using computers. But why is it that such
simple experiments were never done before?
One reason is just that they were not in the mainstream of any existing field of science or
mathematics. But a more important reason is that standard intuition in traditional science gave no
reason to think that their results would be interesting.
And indeed, if it had been known that they were worthwhile, many of the experiments could
actually have been done even long before computers existed. For while it may be somewhat tedious, it
is certainly possible to work out the behavior of something like a cellular automaton by hand. And in
fact, to do so requires absolutely no sophisticated ideas from mathematics or elsewhere: all it takes is
an understanding of how to apply simple rules repeatedly.
And looking at the historical examples of ornamental art on the facing page there seems little
reason to think that the behavior of many cellular automata could not have been worked out many
centuries or even millennia ago. And perhaps one day some Babylonian artifact created using the rule
30 cellular automaton from page 27 will be unearthed. But I very much doubt it. For I tend to think
that if pictures like the one on page 27 had ever in fact been seen in ancient times then science would
have been led down a very different path from the one it actually took.

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[No text on this page]

Historical examples of ornamental art. Repetitive patterns are common and some nested patterns are seen, but
the more complicated kinds of patterns discussed in this chapter do not ever appear to have been used. Note that
the second-to-last picture is not an abstract design, but is instead text written in a highly stylized form of Arabic
script.

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Even early in antiquity attempts were presumably made to see whether simple abstract rules could
reproduce the behavior of natural systems. But so far as one can tell the only types of rules that were
tried were ones associated with standard geometry and arithmetic. And using these kinds of rules, only
rather simple behavior could be obtained--adequate to explain some of the regularities observed in
astronomy, but unable to capture much of what is seen elsewhere in nature.
And perhaps because of this, it typically came to be assumed that a great many aspects of the
natural world are simply beyond human understanding. But finally the successes based on calculus in
the late 1600s began to overthrow this belief. For with calculus there was finally real success in taking
abstract rules created by human thought and using them to reproduce all sorts of phenomena in the
natural world.
But the particular rules that were found to work were fairly sophisticated ones based on particular
kinds of mathematical equations. And from seeing the sophistication of these rules there began to
develop an implicit belief that in almost no important cases would simpler rules be useful in
reproducing the behavior of natural systems.
During the 1700s and 1800s there was ever-increasing success in using rules based on
mathematical equations to analyze physical phenomena. And after the spectacular results achieved in
physics in the early 1900s with mathematical equations there emerged an almost universal belief that
absolutely every aspect of the natural world would in the end be explained by using such equations.
Needless to say, there were many phenomena that did not readily yield to this approach, but it was
generally assumed that if only the necessary calculations could be done, then an explanation in terms
of mathematical equations would eventually be found.
Beginning in the 1940s, the development of electronic computers greatly broadened the range of
calculations that could be done. But disappointingly enough, most of the actual calculations that were
tried yielded no fundamentally new insights. And as a result many people came to believe--and in
some cases still believe today--that computers could never make a real contribution to issues of basic
science.

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But the crucial point that was missed is that computers are not just limited to working out
consequences of mathematical equations. And indeed, what we have seen in this chapter is that there

are fundamental discoveries that can be made if one just studies directly the behavior of even some of
the very simplest computer programs.
In retrospect it is perhaps ironic that the idea of using simple programs as models for natural
systems did not surface in the early days of computing. For systems like cellular automata would have
been immensely easier to handle on early computers than mathematical equations were. But the issue
was that computer time was an expensive commodity, and so it was not thought worth taking the risk
of trying anything but well-established mathematical models.
By the end of the 1970s, however, the situation had changed, and large amounts of computer time
were becoming readily available. And this is what allowed me in 1981 to begin my experiments on
cellular automata.
There is, as I mentioned above, nothing in principle that requires one to use a computer to study
cellular automata. But as a practical matter, it is difficult to imagine that anyone in modern times
would have the patience to generate many pictures of cellular automata by hand. For it takes roughly
an hour to make the picture on page 27 by hand, and it would take a few weeks to make the picture on
page 29.
Yet even with early mainframe computers, the data for these pictures could have been generated in
a matter of a few seconds and a few minutes respectively. But the point is that one would be very
unlikely to discover the kinds of fundamental phenomena discussed in this chapter just by looking at
one or two pictures. And indeed for me to do it certainly took carrying out quite large-scale computer
experiments on a considerable number of different cellular automata.
If one already has a clear idea about the basic features of a particular phenomenon, then one can
often get more details by doing fairly specific experiments. But in my experience the only way to find
phenomena that one does not already know exist is to do very systematic and general experiments, and
then to look at the results with as few preconceptions as possible. And while it takes only rather basic

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computer technology to make single pictures of cellular automata, it requires considerably more to do
large-scale systematic experiments.
Indeed, many of my discoveries about cellular automata came as direct consequences of using
progressively better computer technology.
As one example, I discovered the classification scheme for cellular automata with random initial
conditions described at the beginning of Chapter 6 when I first looked at large numbers of different
cellular automata together on high-resolution graphics displays. Similarly, I discovered the
randomness of rule 30 (page 27) when I was in the process of setting up large simulations for an early
parallel-processing computer. And in more recent years, I have discovered a vast range of new
phenomena as a result of easily being able to set up large numbers of computer experiments in
Mathematica.
Undoubtedly, therefore, one of the main reasons that the discoveries I describe in this chapter were
not made before the 1980s is just that computer technology did not yet exist powerful enough to do
the kinds of exploratory experiments that were needed.
But beyond the practicalities of carrying out such experiments, it was also necessary to have the
idea that the experiments might be worth doing in the first place. And here again computer technology

played a crucial role. For it was from practical experience in using computers that I developed much
of the necessary intuition.
As a simple example, one might have imagined that systems like cellular automata, being made up
of discrete cells, would never be able to reproduce realistic natural shapes. But knowing about
computer displays it is clear that this is not the case. For a computer display, like a cellular automaton,
consists of a regular array of discrete cells or pixels. Yet practical experience shows that such displays
can produce quite realistic images, even with fairly small numbers of pixels.
And as a more significant example, one might have imagined that the simple structure of cellular
automaton programs would make it straightforward to foresee their behavior. But from experience in
practical computing one knows that it is usually very difficult to foresee what even a simple program
will do. Indeed, that is exactly why bugs in programs are so common. For if one could just look at a
program

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and immediately know what it would do, then it would be an easy matter to check that the program
did not contain any bugs.
Notions like the difficulty of finding bugs have no obvious connection to traditional ideas in
science. And perhaps as a result of this, even after computers had been in use for several decades,
essentially none of this type of intuition from practical computing had found its way into basic
science. But in 1981 it so happened that I had for some years been deeply involved in both practical
computing and basic science, and I was therefore in an almost unique position to apply ideas derived
from practical computing to basic science.
Yet despite this, my discoveries about cellular automata still involved a substantial element of
luck. For as I mentioned on page 19, my very first experiments on cellular automata showed only very
simple behavior, and it was only because doing further experiments was technically very easy for me
that I persisted.
And even after I had seen the first signs of complexity in cellular automata, it was several more
years before I discovered the full range of examples given in this chapter, and realized just how easily
complexity could be generated in systems like cellular automata.
Part of the reason that this took so long is that it involved experiments with progressively more
sophisticated computer technology. But the more important reason is that it required the development
of new intuition. And at almost every stage, intuition from traditional science took me in the wrong
direction. But I found that intuition from practical computing did better. And even though it was
sometimes misleading, it was in the end fairly important in putting me on the right track.
Thus there are two quite different reasons why it would have been difficult for the results in this
chapter to be discovered much before computer technology reached the point it did in the 1980s. First,
the necessary computer experiments could not be done with sufficient ease that they were likely to be
tried. And second, the kinds of intuition about computation that were needed could not readily have
been developed without extensive exposure to practical computing.

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But now that the results of this chapter are known, one can go back and see quite a number of
times in the past when they came at least somewhat close to being discovered.
It turns out that two-dimensional versions of cellular automata were already considered in the early
1950s as possible idealized models for biological systems. But until my work in the 1980s the actual
investigations of cellular automata that were done consisted mainly in constructions of rather
complicated sets of rules that could be shown to lead to specific kinds of fairly simple behavior.
The question of whether complex behavior could occur in cellular automata was occasionally
raised, but on the basis of intuition from engineering it was generally assumed that to get any
substantial complexity, one would have to have very complicated underlying rules. And as a result,
the idea of studying cellular automata with simple rules never surfaced, with the result that nothing
like the experiments described in this chapter were ever done.
In other areas, however, systems that are effectively based on simple rules were quite often
studied, and in fact complex behavior was sometimes seen. But without a framework to understand its
significance, such behavior tended either to be ignored entirely or to be treated as some kind of
curiosity of no particular fundamental significance.
Indeed, even very early in the history of traditional mathematics there were already signs of the
basic phenomenon of complexity. One example known for well over two thousand years concerns the
distribution of prime numbers (see page 132). The rules for generating primes are simple, yet their
distribution seems in many respects random. But almost without exception mathematical work on
primes has concentrated not on this randomness, but rather on proving the presence of various
regularities in the distribution.
Another early sign of the phenomenon of complexity could have been seen in the digit sequence of
a number like π 3.141592653… (see page 136). By the 1700s more than a hundred digits of π had
been computed, and they appeared quite random. But this fact was treated essentially as a curiosity,
and the idea never appears to have arisen that

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there might be a general phenomenon whereby simple rules like those for computing π could produce
complex results.
In the early 1900s various explicit examples were constructed in several areas of mathematics in
which simple rules were repeatedly applied to numbers, sequences or geometrical patterns. And
sometimes nested or fractal behavior was seen. And in a few cases substantially more complex
behavior was also seen. But the very complexity of this behavior was usually taken to show that it
could not be relevant for real mathematical work--and could only be of recreational interest.
When electronic computers began to be used in the 1940s, there were many more opportunities for
the phenomenon of complexity to be seen. And indeed, looking back, significant complexity probably
did occur in many scientific calculations. But these calculations were almost always based on
traditional mathematical models, and since previous analyses of these models had not revealed
complexity, it tended to be assumed that any complexity in the computer calculations was just a
spurious consequence of the approximations used in them.
One class of systems where some types of complexity were noticed in the 1950s are so-called
iterated maps. But as I will discuss on page 149, the traditional mathematics that was used to analyze

such systems ended up concentrating only on certain specific features, and completely missed the
main phenomenon discovered in this chapter.
It is often useful in practical computing to produce sequences of numbers that seem random. And
starting in the 1940s, several simple procedures for generating such sequences were invented. But
perhaps because these procedures always seemed quite ad hoc, no general conclusions about
randomness and complexity were drawn from them.
Along similar lines, systems not unlike the cellular automata discussed in this chapter were studied
in the late 1950s for generating random sequences to be used in cryptography. Almost all the results
that were obtained are still military secrets, but I do not believe that any phenomena like the ones
described in this chapter were discovered.
And in general, within the context of mainstream science, the standard intuition that had been
developed made it very difficult for anyone to imagine that it would be worth studying the behavior of
the

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very simple kinds of computer programs discussed in this chapter. But outside of mainstream science,
some work along such lines was done. And for example in the 1960s early computer enthusiasts tried
running various simple programs, and found that in certain cases these programs could succeed in
producing nested patterns.
Then in the early 1970s, considerable recreational computing interest developed in a specific two-
dimensional cellular automaton known as the Game of Life, whose behavior is in some respects
similar to the rule 110 cellular automaton discussed in this chapter. Great effort was spent trying to
find structures that would be sufficiently simple and predictable that they could be used as idealized
components for engineering. And although complex behavior was seen it was generally treated as a
nuisance, to be avoided whenever possible.
In a sense it is surprising that so much could be done on the Game of Life without the much
simpler one-dimensional cellular automata in this chapter ever being investigated. And no doubt the
lack of a connection to basic science was at least in part responsible.
But whatever the reasons, the fact remains that, despite many hints over the course of several
centuries, the basic phenomenon that I have described in this chapter was never discovered before.
It is not uncommon in the history of science that once a general new phenomenon has been
identified, one can see that there was already evidence of it much earlier. But the point is that without
the framework that comes from knowing the general phenomenon, it is almost inevitable that such
evidence will have been ignored.
It is also one of the ironies of progress in science that results which at one time were so unexpected
that they were missed despite many hints eventually come to seem almost obvious. And having lived
with the results of this chapter for nearly two decades, it is now difficult for me to imagine that things
could possibly work in any other way. But the history that I have outlined in this section--like the
history of many other scientific discoveries--provides a sobering reminder of just how easy it is to
miss what will later seem obvious.

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The World of Simple Programs
The Search for General Features

At the beginning of the last chapter we asked the basic question of what simple programs typically
do. And as a first step towards answering this question we looked at several specific examples of a
class of programs known as cellular automata.
The basic types of behavior that we found are illustrated in the pictures on the next page. In the
first of these there is pure repetition, and a very simple pattern is formed. In the second, there are
many intricate details, but at an overall level there is still a very regular nested structure that emerges.
In the third picture, however, one no longer sees such regularity, and instead there is behavior that
seems in many respects random. And finally in the fourth picture there is what appears to be still more
complex behavior--with elaborate localized structures being generated that interact in complex ways.
At the outset there was no indication that simple programs could ever produce behavior so diverse
and often complex. But having now seen these examples, the question becomes how typical they are.
Is it only cellular automata with very specific underlying rules that produce such behavior? Or is it in
fact common in all sorts of simple programs?
My purpose in this chapter is to answer this question by looking at a wide range of different kinds
of programs. And in a sense my

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approach is to work like a naturalist--exploring and studying the various forms that exist in the world
of simple programs.
I start by considering more general cellular automata, and then I go on to consider a whole
sequence of other kinds of programs--with underlying structures further and further away from the
array of black and white cells in the cellular automata of the previous chapter.
And what I discover is that whatever kind of underlying rules one uses, the behavior that emerges
turns out to be remarkably similar to the basic examples that we have already seen in cellular
automata.
Throughout the world of simple programs, it seems, there is great universality in the types of
overall behavior that can be produced. And in a sense it is ultimately this that makes it possible for me
to construct the coherent new kind of science that I describe in this book--and to use it to elucidate a
large number of phenomena, independent of the particular details of the systems in which they occur.

Four basic examples from the previous chapter of behavior produced by cellular automata with simple underlying
rules. In each case, the most obvious features that are seen are different. Note that all the pictures are shown on
the same scale; the last picture appears coarser because the structures it contains are larger.

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More Cellular Automata

The pictures below show the rules used in the four cellular automata on the facing page. The
overall structure of these rules is the same in each case; what differs is the specific choice of new
colors for each possible combination of previous colors for a cell and its two neighbors.
There turn out to be a total of 256 possible sets of choices that can be made. And following my
original work on cellular automata these choices can be numbered from 0 to 255, as in the picture
below.
But so how do cellular automata with all these different rules behave? The next page shows a few
examples in detail, while the following two pages [55, 56] show what happens in all 256 possible
cases.
At first, the diversity of what one sees is a little overwhelming. But on closer investigation,
definite themes begin to emerge.
In the very simplest cases, all the cells in the cellular automaton end up just having the same color
after one step. Thus, for example, in

The rules used for the four examples of cellular automata on the facing page. In each case, these specify the new
color of a cell for each possible combination of colors of that cell and its immediate neighbors on the previous step.
The rules are numbered according to the scheme described below.

The sequence of 256 possible cellular automaton rules of the kind shown above. As indicated, the rules can
conveniently be numbered from 0 to 255. The number assigned is such that when written in base 2, it gives a
sequence of 0's and 1's that correspond to the sequence of new colors chosen for each of the eight possible cases
covered by the rule.

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Evolution of cellular automata with a sequence of different possible rules, starting in all cases from a single black
cell

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rules 0 and 128 all the cells become white, while in rule 255 all of them become black. There are also
rules such as 7 and 127 in which all cells alternate between black and white on successive steps.
But among the rules shown on the last few pages [54, 55, 56] the single most common kind of
behavior is one in which a pattern consisting of a single cell or a small group of cells persists.
Sometimes this pattern remains stationary, as in rules 4 and 123. But in other cases, such as rules 2
and 103, it moves to the left or right.
It turns out that the basic structure of the cellular automata discussed here implies that the
maximum speed of any such motion must be one cell per step. And in many rules, this maximum
speed is achieved--although in rules such as 3 and 103 the average speed is instead only half a cell per
step.
In about two-thirds of all the cellular automata shown on the last few pages [54, 55, 56], the
patterns produced remain of a fixed size. But in about one-third of cases, the patterns instead grow
forever. Of such growing patterns, the simplest kind are purely repetitive ones, such as those seen in
rules 50 and 109. But while repetitive patterns are by a small margin the most common kind, about
14% of all the cellular automata shown yield more complicated kinds of patterns.
The most common of these are nested patterns, like those on the next page. And it turns out that
although 24 rules in all yield such nested patterns, there are only three fundamentally different forms
that occur. The simplest and by far the most common is the one exemplified by rules 22 and 60. But as
the pictures on the next page show, other nested forms are also possible. (In the case of rule 225, the
width of the overall pattern does not grow at a fixed rate, but instead is on average proportional to the
square root of the number of steps.)

The behavior of all 256 possible cellular automata with rules involving two colors and nearest neighbors. In each
case, thirty steps of evolution are shown, starting from a single black cell. Note that some of the rules are related
just by interchange of left and right or black and white (e.g. rules 2 and 16 or rules 126 and 129). There are 88
fundamentally inequivalent such elementary rules

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Repetition and nesting are widespread themes in many cellular automata. But as we saw in the
previous chapter, it is also possible for cellular automata to produce patterns that seem in many
respects random. And out of the 256 rules discussed here, it turns out that 10 yield such apparent
randomness. There are three basic forms, as illustrated on the facing page.


Examples of cellular automata that produce nested or fractal patterns. Rule 22--like rule 90 from page 26--gives a
pattern with fractal dimension Log[2,3] 1.58; rule 150 gives one with fractal dimension Log[2, 1+Sqrt[5]]
1.69. The width of the pattern obtained from rule 225 increases like the square root of the number of steps.

Examples of cellular automata that produce patterns with many apparently random features. Three hundred steps
of evolution are shown, starting in each case from a single black cell.

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Beyond randomness, the last example in the previous chapter was rule 110: a cellular automaton
whose behavior becomes partitioned into a complex mixture of regular and irregular parts. This
particular cellular automaton is essentially unique among the 256 rules considered here: of the four
cases in which such behavior is seen, all are equivalent if one just interchanges the roles of left and
right or black and white.
So what about more complicated cellular automaton rules?
The 256 "elementary" rules that we have discussed so far are by most measures the simplest
possible--and were the first ones I studied. But one can for example also look at rules that involve
three colors, rather than two, so that cells can not only be black and white, but also gray. The total
number of possible rules of this kind turns out to be immense--7,625,597,484,987 in all--but by
considering only so-called "totalistic" ones, the number becomes much more manageable.
The idea of a totalistic rule is to take the new color of each cell to depend only on the average
color of neighboring cells, and not on their individual colors. The picture below shows one example of
how this works. And with three possible colors for each cell, there are 2187 possible totalistic rules,
each of which can conveniently be identified by a code number as illustrated in the picture. The facing
page shows a representative sequence of such rules.
We might have expected that by allowing three colors rather than two we would immediately get
noticeably more complicated behavior.

Example of a totalistic cellular automaton with three possible colors for each cell. The rule is set up so that the new
color of every cell is determined by the average of the previous colors of the cell and its immediate neighbors. With
0 representing white, 1 gray and 2 black, the rightmost element of the rule gives the result for average color 0,
while the element immediately to its left gives the result for average color 1/3--and so on. Interpreting the sequence
of new colors as a sequence of base 3 digits, one can assign a code number to each totalistic rule

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A sequence of totalistic cellular automata with three possible colors for each cell. Although their basic rules are
more complicated, the cellular automata shown here do not seem to have fundamentally more complicated
behavior than the two-color cellular automata shown on previous pages. Note that in the sequence of rules shown
here, those that change the white background are not included. The symmetry of all the patterns is a consequence
of the basic structure of totalistic rules.

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But in fact the behavior we see on the previous page is not unlike what we already saw in many
elementary cellular automata a few pages back [55, 56]. Having more complicated underlying rules
has not, it seems, led to much greater complexity in overall behavior.
And indeed, this is a first indication of an important general phenomenon: that at least beyond a
certain point, adding complexity to the underlying rules for a system does not ultimately lead to more
complex overall behavior. And so for example, in the case of cellular automata, it seems that all the
essential ingredients needed to produce even the most complex behavior already exist in elementary
rules.
Using more complicated rules may be convenient if one wants, say, to reproduce the details of
particular natural systems, but it does not add fundamentally new features. Indeed, looking at the
pictures on the previous page one sees exactly the same basic themes as in elementary cellular
automata. There are some patterns that attain a definite size, then repeat forever, as shown below,
others that continue to grow, but have a repetitive form, as at the top of the facing page, and still
others that produce nested or fractal patterns, as at the bottom of the page.

Examples of three-color totalistic rules that yield patterns which attain a certain size, then repeat forever. The
maximum repetition period is found to be 78 steps, and is achieved by the rule with code number 1329. In the
pictures shown here and on the following pages, the initial condition used contains a single gray cell.

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Examples of three-color totalistic rules that yield patterns which grow forever but have a fundamentally repetitive
structure.

Examples of three-color totalistic rules which yield nested patterns. In most cases, these patterns have an overall
form that is similar to what was found with two-color rules. But code 420, for example, yields a pattern with a
slightly different structure.

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