So, I think the phenomenon of universality in physical systems is incredibly amazing. We see period doubling in physical systems and that period doubling is described by a delta, and that delta has a value that is the same as for these one-dimensional systems that we started studying with - the logistic equation, and so on. If you're not already convinced that this is amazing, let me try to convince you by going over the line of reasoning that led us to this point. So, I started a long time ago in this course talking about the logistic equation. And when I presented it I said don't take this equation very seriously - it is certainly not a realistic model of how rabbits might grow, or how almost anything might grow. But it's more like a characature or sketch or cartoon - it gives us some rough picture of how a population might grow where there is some limit to the growth. And then we found that there is surprising richness to this equation, amazing richness when we form the bifurcation diagram and zoomed in again and again and again and saw these fractal patterns and complicated behavior. But we might have thought...it would be very reasonable to think "well we're just playing with a mathematical game - it's just an equation that somebody told a story about rabbits to motivate - but it's just a little equation and it's really neat that it does all this stuff - but what might it say about the real world?" And we talked about that sum, and I appealed to the argument that this shows that even the simplest possible... or just about the simplest possible equation ...iterated function...can show the butterfly effect and sensitive dependence and that's an important realization and that certainly is. But still the logistic equation seemed just like a little toy model - a fun thing to play with. Then, in this chapter, we started looking at ...or in this unit...we started looking at bifurcation diagrams for different equations - the sine equation, the cubic equation, we could have done lots of others - and we saw that the bifurcation diagram was kind of similar. Then, I presented the notion of this quantity delta, looking at the ratios of the lengths of period two and then period four and then period eight regions. And we saw that the same number appeared - 4.669201 - so that all of these different functions turn out to behave the same...not exactly the same, but they undergo this transition to chaos from periodic to chaotic behavior in the same way. We're still in the realm of math. It's an amazing mathematical fact that we're still in the realm of math. We might wonder what this has to say about the real world. So now, experimentalists, scientists, and physicists have gone out and actually measured period-doubling in a whole bunch of different physical phenomena and they've calculated this quantity delta - this ratio of period one to period two, period two to period four, and so on. And they find numbers consistent with that predicted by this theory - 4.669. So, we started with a ridiculously simple, deliberately unrealistic equation, and nevertheless, that equation contains some information in the form of a number that one can go out and measure in real physical systems. So this surely tells us something important and profound - that nature and mathematics are somehow constrained in the ways that a transition from periodic to aperiodic behavior - from periodic-ordered behavior to chaotic behavior - can occur. That when it undergoes period-doubling, almost always what it's going to do in the large-end limit, it's going to do this transition in the same way. So all of these different phenomena - period-doubling in rabbit equations on the computer and in sine equations on the computer , and in electronic circuits, and in convection rolls, and more - all have some similar feature about them. There is some way that this transition is constrained and that certain featyres can only occur in one way. So this also says that some of the details of our mathematical models apparently don't matter because we see the same behavior across a very wide range of systems. That's what it means to say that this quantity delta is universal. So, of course this leads to a bunch of questions - how can this possibly be? And there are at least two parts of that question I think. The first is a mathematical one - how can it be that all these different one-dimensional functions have the same delta? That they undergo period-doubling in essentially the same way - this 4.669 number. The other question is what in the world would a dripping faucet have to do with a one-dimensional equation? Or fluid mechanics where we have convection rolls wiggling around in a little box? These are surely high-dimensional systems - you would model them with more than just a one-dimensional equation, and they're also continuous systems - they're not discrete like iterated functions. The second question - what one-dimensional iterated functions might have to do with higher-dimensional real physical systems - I'll discuss in the next several units. In the next sub-unit , the next set of lectures here, I'll try to give some feel for how it can be mathematically that there are these similarities among all of these different one-dimensional functions.