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.