In this subunit we’ll learn
how to calculate the dimension of certain mathematical fractals.
To start we consider really simple shapes
Line segments, squares and cubes.
And by considering these shapes
we’ll see how to generalize our intuitive idea of dimension.
So it can apply in some interesting and important ways to fractals.
So our starting point for thinking about dimension
is to think about how many small copies
of a shape fit inside a big copy.
And I am gonna make a table here and will fill out
And we are gonna do three simple examples
To start by thinking about line segment,
so suppose I have a line segment like this.
And then suppose I have a line segment
that’s three times as long 1,2,3
And I asked how many of the small lines segments
fit inside this line segment
that’s strecthed by 3 and the answer is 3.
1,2,3 little line segments
inside this big line segment.
So for the shape of a line,
magnification factor of 3,
the number of small copies in a big copy is 3.
And so what I mean by magnification factor here
is that I need to take a small copy
and magnified 3 times,
streched it 3 times in this direction.
In order to have it be the same size
as the large shape, this large line segment.
Ok, so much for line segment.
Now, let’s think about a square.
So here is a small square.
imagine this is the square
with a side of 1 for instance.
And then what if I had a square
that has a side of 3.
I can ask how many this little squares fit in here.
And if draw it like this.
We can see that 1,2,3,4,5,6,7,8,9
9 little squares fit inside the big square.
This hand drawing isn’t perfectly ideal
at this size is,
this square is the same size as that.
So there are 3 times 3
equals 9 small squares in this big square.
And the big square is magnified by a factor of 3.
Right it’s stretched by 3 in this direction,
3 in that direction.
So going back to my table.
The shape of a square
Magnification factor of 3
Number of small copies in the big copy is 9.
Let’s think about one more shape.
Cube
So here is a little cube
And cube has a side of 1.
And now suppose we have a cube
that’s 3 times as large.
So it’s a little bit challenging for me to draw.
Ok, so how many of these small cubes
fit inside this cube?
Let me draw on some lines,
that will may be make a little clearer.
Ok, not a perfect drawing
but I hope it illustrates the point.
So let’s think about how many small cubes
are in this big cube.
Well, in this top layer there 1,2,3,4,5,6,7,8,9
And then there is another layer
or slice here another 9.
And another 9 here on this bottom slice.
So 9 plus 9 plus 9 gives me 27
So there are 27 small cubes in this larger cube.
And the magnification factor again is 3
to go from this small cube or
this small cube here to the big one.
I need to magnify the entire shape by 3.
Meaning that it 3 times as long,
3 times as deep and 3 times as high.
So I can complete this table now.
So the last shape was a cube.
Magnification factor is 3, and
the number of small copies in the big copy is 27.
So let’s see
what we can learn from this table.
The question is what geometric property
or features of these shapes determine
what goes over here.
Determines how many small copies
go in the big copy.
Because the magnification factor
for all of these is the same.
So the answer of this question
is that it’s the dimension.
Lines, squares and cubes
have a different dimension and
so the numbers that appear over here are different.
So let me write down the relationship
between this and this using the dimension.
So here is an equation that relates
magnification factor to the number of small copies
in a big copy of the shape.
So the equation is that
the number of small copies equals
a magnification factor raised to the D power.
And D in this equation
is the self similarity dimension.
And I’ll just call the dimension for short now.
Since there is no other dimension
around to be thinking about.
So let’s think about this.
What is it? what is it telling us?
So let’s go to, let’s start with a square first.
So square has a magnification factor of 3
and the number of small copies
in the big copy is 9.
And so then what D would that be ?
Let’s go back here.
So number of small copies is 9
And the magnification factor is 3.
So we wonder ok, what’s the dimension?
Well, D would have to be 2.
Because 9 is 3 squared.
So we’ll say that the square has a dimension of 2.
And that makes sense.
That’s consistent with our intuitive idea of dimension.
A square has extended in 1 and 2 directions.
So we think of a square is 2 dimensional.
We can do similar things with a cube,
to the cube first.
So the magnification factor
So the number of small copies we decided was 27.
And that’s gonna be the magnification factor
to the dimension D.
And so we see than this case.
So D is 3. So the cube is 3 dimensional.
And that’s consistent with our notion of dimension.
1,2,3 directions for a cube.
So we would say it’s 3 dimensional.
And we can do the same thing with a line.
This equation is little bit boring.
Number of small copies in the line segment is 3.
And the magnification factor is 3.
So what’s the exponent?
Well 3 to the 1.
Remember that a number raised to the first power
is just that number.
Right, so, x to the 1 is just x for instance.
So this says that dimension of a line is 1.
So going back to this picture.
This is the key equation for this unit.
Magnification factor to the D is a number
of small copies and D is a dimension.
This is may be an unusual way
of thinking about dimension.
But it does tell us that
Lines are 1 dimensional.
Squares are 2 dimensional.
And cubes are 3 dimensional like we would expect.
So in the next video we’ll look at applying
this idea of dimension to a fractal.
But before we do that I’ll suggest doing
a quick quiz just to practice this formula
to make sure you see how it works.