We’ll begin our exploration into methods for
generated fractals by looking again a Koch curve.
We’ll see that a few see simple modifications
to the process generates the Koch curve yields fractals
that are much more interesting and
much more realistic in many ways.
Throughout this unit,
I’ll make use of several online programs
that can be founded at different several websites.
And links to all of the programs
that I referred to in this videos.
As well as a few others can be found in a subunit
towards to end of this one called computer programs.
So all the links that you need should be there.
Alright,
Let’s go start with a Koch curve.
So here is the Koch curve
We call that generated what one does is
replace a line segment with a bend line segment
at every step.
This is the basic operation.
We take a line segment
and we turn it into that.
And we repeat this process.
So this is the basic process.
This is for the generating step for this fractal.
So we might start with a line segment like this.
So n equals 0, that would be in black.
And then the next step would be replaced
this line segment with a bend line segment.
So the blue curve, that’s n equals 1.
Now I have 1,2,3,4 line segments.
In the Koch curve we should all be the same length.
This is I was not a perfect drawing.
Step n equals 2,
I replace each blue segments by a bend segment.
So that becomes,
So that’s n equals 2.
And then I could repeat that one more time.
There are, let’s see, 1,2,3,4
16 of this segments
Each of those 16 line segments gets
replaced with a bend segment.
Alright,
So there is a step n equals 3.
And one could do a few more steps but
one would quickly run out of patience.
So this is not the most beatiful drawing
I am clearly not an artist.
It does give the idea though
But if you want an experiment with this
go to larger n’s and see what happens
if we tried slightly different rules over here.
We’re gonna need a computer
to make some pictures for us.
So what I will do next
I’ll show you a website
that will let you make Koch curves like this
and lots of other cool things.
So this is a program that will let you draw
Koch curves and variations thereof.
So let me show you how it works and than
we’ll start experimenting little bit.
And then you can experiment some as well.
So first,
Let’s go down here,
and I wanna make sure that
“Snap to grid” is choosen.
And what that’ll do as I move red points around,
it’ll make them go to points on the grid.
There is a very faint grade in grey
might be hard to see on the screen
But I can make the points sort of sticks there.
So I am gonna move these points around.
So we have something like this
So what I have done is I’ve created that shape.
that sort of a generator that bend line shape
That’s at the core of the Koch curve construction.
And then I can change the iterations
go from n equals 0 to 1, 2 like before.
So I usually start with n equals 1.
So I can see what that first step is
I go down the iterations
and we start with a straight line.
The first step is to take the straight line
replace it with a bend line.
Then we have 4 lines.
We’ve replaced each of those with a bend line.
And then we’ve replaced each of those
with a bend line and so on.
And we have a nice Koch curve that
the program generates for us.
Certainly much nicer than
when I just drew by hand.
Let’s go down may be a few iterations.
If you wanna to see the original step.
You could click on “view also” the 0 iteration.
Then it makes this yellow line down here.
I don’t find it very helpful but sometimes
that’s nice to see what the first iteration was.
Because that’s the basic geometric transformation
that gets repeated again and again and again.
So I like kinda keeping eye on sometimes
So I can see so reminds me that
I am taking straight line and
turning it into bend line this thing in blue.
And then we can iterate that
again
and again
So now I have a tool
that will makes Koch curves for us.
And I’ll show you in the next video
that it can do lots more as well.