So we've seen that mathematical models of

reaction-diffusion systems are capable of

forming stable spatial patterns.

In varying these parameters a and b, the

different diffusion rates of u and v or

chemical species a and b, gives different

sorts of stripes and dots, sometimes fixed

patterns, sometimes periodic oscillation.

So they're capable of doing a wide range

of things. And this is surprising, I

think, because these are models based on

diffusion, based on a process through

which particles should spread out and

reach some sort of uniformity.

So, this is a pretty amazing mathematical

result and it is, people think it's a

mechanism behind, certainly not all

patterns, of course, but many patterns

that are observed in the natural world,

particularly in biological systems. But

what's maybe even more amazing than this

mathematical result is that one can

actually do experiments with reaction-

diffusion systems and see the same

sorts of general patterns forming.

There are lots of experiments one can do

but I want to describe one just 
very, very briefly.

It's a reaction-diffusion type of system

known as a Belousov–Zhabotinsky reaction.

If one carries out this chemical reaction

just in a beaker, one sees colors that

oscillate. It's a couple of

different chemicals that interact with

each other, not too dissimilar from how

the a and b interacted with each other

in the system we just studied.

So, if we take this chemical system and

we put it not in a well mixed beaker,

but on a surface, usually it's in some

sort of a Petri dish, and one starts with

a more or less uniform distribution of

the two reactants, one can see patterns

emerging just like we did
on the computer screen.

So what I want to do is show you a quick

video of one of those experiments, just

to give you a flavor of what these things

look like, and that this is a real

phenomenon, not just a 
mathematical phenomenon.

And then, I'll show you this video, talk

about it a little bit, and then in the

next subunit I'll draw a few conclusions.

Alright, so let's check out the 
Belousov–Zhabotinsky reaction.

So there are quite a few videos of

Belousov–Zhabotinsky pattern forming

reactions on YouTube and elsewhere,

in the ones I'd like to show you today

are posted by Stephen Morris, he's a

professor of geophysics at University

of Toronto. And I think these videos are

particularly nice and his entire YouTube

channel has all sorts of interesting 
pattern forming systems.

And we're lucky enough that we'll get to 
talk to him next week.

So in Unit 10 I'll have some recorded

conversations, interviews, with some

researchers and teachers in dynamical

systems, and Stephen Morris 
will be one of them.

Alright, the Belousov–Zhabotinsky
reaction.

This is an example of reaction-diffusion.

What you'll see is two different chemicals

are poured into this Petri dish and,

presumably they are liquid and poured

right in and should be uniform but because

of this instability where they are a

little bit of an increase just at random

in one of these activators,

you'll see this reaction that leads to an

expanding wave front. You'll see patterns

appear seemingly out of nowhere.

So, let's run this. There he's pouring in

one liquid. There's another. 
It is well mixed.

But patterns form spontaneously.

And they grow outward. Looks a little

different from the Turing patterns we saw

in the program, but it's the same 
general phenomenon.

Let's look at that once more.

And I should mention that links to these

videos are in the additional resources

section of this unit.

So liquids are poured in, mixed together,

and out of nowhere, it's as if they

sort of, or not quite separate out,

but these reactions form these spatial

structures that persist and grow in time.

Alright, let me show you 
one more of these.

So this is a similar sort of thing.

Let me go back to the beginning.

This one has a soundtrack, I'll mute it.

Don't be alarmed if you watch it on your

own and suddenly the soundtrack jumps out

at you. So this is a Belousov–Zhabotinsky

reaction. Same thing. The video is sped

up a little bit so one can see what

happens. And you'll notice that a hand

will come in with a paperclip and it'll

break some of the circles that are

expanding outward and that has the effect

of turning these circles into spirals.

So let's watch that. So 8x normal speed.

So there they are. They're growing

outwards. And there, breaking some

of those circles and you can see spirals

forming in the middle, a little bit to the

right. And those spirals expand and then

one will end up with these interesting

spiral shapes that sort of merge
into each other.

So I definitely recommend that you watch

these videos if you want on your own.

And there are lots of others on YouTube

or elsewhere if you search for "Belousov–
Zhabotinsky reaction"

or "reaction-diffusion pattern" or
"Turing pattern"

there are lots of nice 
demonstrations of these.

But the main point of this is that this

isn't just a mathematical phenomenon,

it's a real, physical phenomenon as well.

One can get these different chemicals to

interact this way. And this is believed to

be, not this exact equation, but this

general phenomena may be behind some

biological development, how structures

form out of initially homogeneous

tissues or some sort of substrate.

So, again, dynamical systems, now we're

looking at spatial dynamical systems, so

a different mathematical entity but again,

simple, deterministic, local dynamical

systems can produce a certain type of, it

can produce chaotic behavior, it can

produce a certain form of the butterfly

effect if one ran this experiment again

exactly where these spirals would be 
would vary from run to run.

But also, in addition to producing

chaotic behavior we see patterns,

structures forming out of nowhere.

Nobody is designing these spirals.

Nobody from above is making these spirals.

They are literally from the ground up.

A local rule with no global knowledge of

what is going on is able to produce these

large scale, and fairly
intricate structures.