Let’s do another example.
We’ll consider a different fractal and calculate its self-similarity dimension.
The fractal we’ll consider now is a famous fractal
known as Sierpinski Triangle. You have likely seen it before.
Here are the steps in its construction
when starts with a solid triangle and then removed from that triangle,
the middle triangle like this
when one left this three triangles, 1,2,3,
>From each of those remove the middle triangle,
now nine triangles from each of those
remove the middle triangle and so on.
And as this repeats, one gets a fractal shape.
So here is this carried out a few generations farther
and it is a fractal because
we see these triangles within triangles
within triangles repeating in many different scales itself similar.
Ok, so now let’s calculate the self-similarity dimension.
So, number of small copies,
I guess I look at this version of it and I see one, two, three small copies.
One, two, three. So the number of small copies is three.
The magnification factor is two.
You can see that they need to take this
and stretch it by a factor two this way
and to this way to have the small copy be as big as large one.
I guess you could also see that
here this triangle this side is half of that,
the base is half of that.
So we need to magnify or stretch this triangle by two
in order to have it be as big as the full one.
So the magnification factor is two and the dimension is D
We need to solve for D
So again we’ll use logarithms to do that and let’s see we get.
So the equation was three equals two to the D.
We’ll take the log of both sides.
We use the exponent property of log’s.
And then solve for the D by dividing both sides by log2.
So there is an answer.
The dimension of the Sierpinski triangle is log3 over log2.
And we can get an approximate decimal value
for that from the calculator.
So three log divided by two log equals
and I get that D is about 1.585
So again we see a non-integer dimension.
The dimension of the Sierpienski triangle is between one and two.
So this object has a dimension about 1.585
The next couple of quizzes will give you a chance to practice
this idea of calculating the self-similarity dimension.
So I’ll give you a few other fractals and
you’ll calculate the self-similarity dimension.
I should mention that in my experience teaching this
it can sometimes take a little while
to see this magnification factor in small copies.
It is so geometric and visual that
I find it little bit hard to explain with words
but if you do a couple of examples
eventually you’ll start to see it.
So try the quizzes out.
If you don’t get them right away, don't worry.
I’ll go over them in the videos after the quiz.