So let's vary the rule that generates

the Koch curve a little bit and

see what happens.

So here's the program again.

Go down and select "Snap" to grid

and I'm going to move the points around.

So let's see here.

So, then I'm going to get something

that's going to make a Koch curve.

There is it. Okay.

So, there's that shape.

And, let's see...I can move this up

a little bit and get a pointier Koch curve.

I can move this up even more,

then I'll get a really pointy curve.

It starts to overflow itself

or to override itself.

So then what one can do is

put a little bit of

asymmetry in here instead of having this be

of having this be

centered, I can move this point over there

and then we get an interesting shape

like that. Even though this is

still a deterministic rule--here, let me put

that in--

it's starting to look much more

realistic. More like a real

coastline or a jagged rock or torn piece

of paper or something. So a little bit

of irregularity in the rule, something

that's not quite symmetric can

make things pretty interesting.

I would encourage you to play around

with other possibilities. And

you can do even more if you have more

points to play around with.

So, let me just give you a feel for

what one can do. Put a bump like this

and then we can do a bigger

bump down there. Let's see

what this looks like.

Again, something pretty jagged

and rough. So, I would suggest

if you're at all inclined to

play around with this program

or a similar one on the web.

And just kind of get a feel for

how a simple rule with

a little bit of regularity in it--

So a simple rule, again, is

that we're replacing every line segment

with sort of a wiggle line segment

that's bent like that. If we iterate that

rule, we get this funny

sort of shape.

You can even change the position

of these points on the fly

and I found that a little hard

to know what's going on when I do that,

but it's still fun.

And the main point here

is that this very simple iterative

process can lead to a pretty wide array

of different looking fractals.