Hey, everybody! This is the very last unit of this course.

I'm going to do a quick review of all the different ideas that we've gone over in this course,

and see how they address the main goals of complex systems research that I laid out in the very first unit.

Then it's time for us to go on our course field trip - course virtual field trip -

in which we're goimg to visit the Santa Fe Institute

and see how things work there and talk to some of the main people who are there.

So pack up your sunglasses and your hats.

We're going to the high desert of Santa Fe, New Mexico - 7,000 feet above sea level -

and up a steep hill to the Santa Fe Institute. I'll see you there.

We started out this course with the question, "What are complex systems?"

Our informal answer was, "They are large networks of simple interacting elements which, following simple rules,

produce emergent collective complex behavior."

And I told you that this course would provide some insight into what this all means.

The purpose of this subunit is to go over what we've covered and to see how much insight we've gained.

So remember that I mentioned four core disciplines of the sciences of complexity:

the discipline of dynamics; information; computation; and evolution and learning.

And the goals of this course were to give you an overview of what these topics are about,

and to give you a sense of how these topics are integrated into the study of complex systems.

And while doing so, to give you a sense of how we can use idealized models to study these topics.

And we've looked at a lot of idealized models in NetLogo.

So if you've gotten this far in the course,

you should be very proud of yourself because we've covered quite a lot.

And hopefully you've learned quite a lot about complex systems.

So let's review what we've done, very quickly.

We looked at dynamics and chaos,

and learned how it can provide a vocabulary for describing how complex systems change over time.

The vocabulary included ideas such as fixed points, periodic attractors, chaos,

sensitive dependence on initial conditions, and other terms.

Dynamics showed us how complex behavior can arise from iteration,

and the iteration of simple rules such as the logistic map.

And we were able to characterize the complexity of behavior

in terms of the particular kinds of dynamics we saw,

whether they be fixed points, cycles, or chaos.

Also the field of dynamics showed a contrast between intrinsic unpredictability, which we saw in chaotic systems,

and universal properties such as the period doubling route to chaos and Feigenbaum's Constant.

Our next topic was fractals.

Fractals showed us how a new kind of geometry can be developed that characterizes real-world patterns

in a more realistic way than Euclidean geometry.

Like dynamics, the study of fractals shows us how complex patterns can arise from the iteration of simple rules.

And we're able to characterize complexity in a different way here in terms of fractal dimension.

Information theory was next.

And we learned how information theory makes an analogy between information and physical entropy;

and also a different way of characterizing complexity; that is, in terms of information content.

So now we've seen several different ways in which complexity can be characterized.

Genetic algorithms was next.

And that showed us how idealized models of evolution and adaptation can be constructed.

And it also demonstrated how complex behavior, or complex shapes, can emerge from the simple rules of evolution

(which themselves can be thought of as iterative).

Cellular automata: again, we saw how cellular automata were idealized models of complex systems.

This was another way in which complex patterns emerged from the iteration of simple rules.

And we learned about the idea of Wolfram classes,

which characterize the complexity of cellular automata behavior in terms of these classes of patterns.

We looked at several models of self-organization in biology: like firefly synchronization; bird flocking and fish schooling;

ant foraging; ant task allocation; and there are many other possible models that we didn't cover.

We saw how we could build idealized models such as these NetLogo models of self-organizing behavior.

And we made an attempt to isolate some of the common principles of these systems in terms of their dynamics,

the information that they process, the computation that they do, and their adaptation.

We looked at models of cooperation;

in particular, the Prisoner's Dilemma model and the El Farol problem model.

This gave us a sense of how idealized models can explain self-organized cooperation in social systems;

and in general, how idealized models can be used to study very complex phenomena.

Then we looked at networks.

Networks gave us a vocabulary for describing the structure and dynamics of networks in the real world,

in terms of concepts such as small-world, scale-free, degree distribution, clustering, path-length, et cetera.

The models that we explored captured some aspects of real-world network structure;

such as, preferential attachment showed us how we can get scale-free structure in a network.

This captured the idea of a power law in the degree distribution of networks.

After networks we covered scaling, in which we looked at some theories of metabolic scaling in biology,

and the very new area of urban scaling.

We saw that, looking at how complex systems scale as the size is increased,

can give clues to the underlying structure and dynamics of these systems, such as fractal distribution networks.

In the beginning of this course, I said that there were two goals of the sciences of complexity.

The first is to provide cross-disciplinary insights into complex systems.

And the second one is to develop a general theory of complex systems.

Well, clearly we've managed to accomplish the first.

We've seen a lot of cross-disciplinary insights that we get from studying different complex systems.

But, in this class at least, we haven't talked about what it might mean to have a general theory of complex systems.

Many people are asking, "Can we develop some kind of general or unified theory of complex systems?"

That is, can we develop a mathematical language that's going to unify the core disciplines of dynamics,

information processing, computation, and evolution in these systems?

Some people have referred to this hypothetical language as a "calculus of complexity".

In my book "Complexity: A Guided Tour", I make an analogy with some history.

In the late 1600s, Isaac Newton, along with Gottfried Leibniz, was developing the calculus.

As James Gleick said in his biography of Isaac Newton,

"He was hampered by the chaos of language - words still vaguely defined and words not quite existing...

Newton believed he could marshal a complete science of motion, if only he could find the appropriate lexicon..."

Well, when I read this, this sounded very much like the state of complex systems today.

Newton had a lot of concepts floating around in his head that had been developed by previous mathematicians;

notions - that he was trying to bring together in a unified whole - such as infinitesimal, derivative, integral, limit.

And he was able finally to unify these different concepts to develop the appropriate lexicon for understanding motion.

And that's the mathematics that we call calculus.

Well, the state of complex systems today is very much like the state of mathematics back before calculus was invented.

We have all these notions that seem separate in many ways;

I've listed a few of them here - a kind of floating word cloud.

And the idea is that we need to unify them in some way to have a mathematics that unifies all these disparate notions.

I'm not sure that that's going to happen; it's a very exciting prospect!

But I'll leave you with the question of whether complex systems is going to find its own Isaac Newton,

either in the near or far future, perhaps.

To conclude, I'll give you a quotation that I like very much, attributed to Oliver Wendell Holmes:

"I do not give a fig for the simplicity on this side of complexity,

but I would give my life for the simplicity on the other side of complexity".

With that, it's time to take our virtual field trip.