One can vary the rules of the chaos game
a little bit, and come up with some
additional interesting shapes and
have a little bit of fun.
So here I am, back at the program
that we used before, and let's start
just going back to what we did previously
so 10,000 points with some small dots.
There are some things that we can change
down below. One is, we can change the
movement ratios. Right now, this says
red point: 0.5 That means
that you move halfway to the red point.
This 0.5 means you move halfway to the green point, halfway to the blue point.
I think that if they are set a little bit
differently, one gets some fun shapes
so let's try this: suppose you move
54 percent to the red, green, and blue points
instead of 50 percent.
then, things shrink up a little bit
and we have a Sierpinski Triangle of sorts
that has kind of broken apart into
smaller Sierpinski Triangles... it's not
fully connected anymore.
We could do something else.
We could go over here and change this
into 0.47...
and then we get some overlap
it's not quite as clean a Sierpinski Triangle as before
The triangles overlap themselves a little bit.
Let's get a few more points...
so the shape is a little bit different.
Let's go back to 10,000
Maybe we can exaggerate this a little more
Let's go to 0.44 and see what that looks like
At some point it starts to look like
a fractal plaid pattern to me.
We can make this colorful if we click on
"Color code" ... That, then is just going
to make the top ones red, the left ones
blue, the right ones green.
That can be a fun thing to play around with.
Let's go back to 0.5 and let me show you
one more thing.
We can also change the relative
probabilities. In the version of the game
that we played, you were equally likely
to go in the red direction when I called "a"
in this direction, or in this direction.
We could change that up, so maybe
I'll make red, now, three times as likely
and hit "Enter" so that change goes into effect.
And the tops of the triangles, now
are heavier. Because the tops
correspond to those "a" moves
and so now we see a Sierpinski Triangle...
the shape is the same, it's just not
uniformly filled out anymore.
The dots spend more time in some parts
of the triangle than others. And this is
interesting to me. It gives it
a 3-D effect, or a faded effect.
Let's try it with 10,000 points...
It's a little less dramatic.
So, there's lots you can play around with.
You can change the movement ratios
You can make them different.
I was keeping them all the same.
You can change the relative probabilities
as well... one last really fun thing you can do...
You can change the number of points.
There's no reason why we have to have
3 points. We could have 4 or 5 or 6 or 7.
And then we would just, with equal probability
move halfway or whatever way to each point
and iterate that.
Let me show you just one more thing here
You can also move these points around
so let's say I want to move the red point.
Then I would click this on "Move"
and then wherever I click my mouse
that's where the red point's going to go
So that can give you... if you want to
change the shape of the triangle.
Here, I'll move the blue point.
You can do that as well.
So there's lots that you can
experiment with... and I'll suggest
a few explorations that you can do
for the homework for this week.