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.