In the last subunit, I introduced the self-similarity dimension
and we calculated the self-similarity dimension for three different fractals.
The fractals snowflake, the Sierpienski triangle and the Koch curve.
In this subunit, I’ll make a number of other remarks about fractals
including thinking about what self-similarity means and
what it means to say something is scale-free.
And also I’ll try to address a question.
It’s a quite likely on your mind which is
what’s up with non-integer dimensions,
how can it make sense
for an object to have dimension that’s not 1 or 2
but it’s something in between. How can we even think about that?
So to begin addressing that
get some new ways to think about this
we’re gonna start by asking the question
how long is the Koch curve.
So the question before I say is how long is the Koch curve.
Reminder you calculated its dimension in the previous section.
Here are the steps and its construction.
So what I’m gonna do is come up with an expression
for the length of zero step, the initial step.
And then after 1 step, 2 step, 3 step and so on.
And we’ll see what happens.
So I’m gonna say, let’s call this initial length 1.
Just for a convenience say
Let’s say this is a meter or something.
So I’m gonna fill out this table here,
so it is step 0 initially
how long is each line segment
but there was only one segment
and it has length 1.
So the total length is 1.
There is one line segment of length 1, the total length is 1.
Ok, what about at the first step,
now there are 4 line segments, 1, 2, 3, 4.
And each line segment is one third the length of the original line segment.
So step 1, length of each line segment is a third,
the number of line segments is 4,
and if we have four things each of which have a length of a third,
the total length is four third.
Right, what about the step 2?
Well, the process repeats.
If I just look at this section here,
I go from one line segment, 1,2,3,4,.. four line segments.
But that same story happens here, here and here.
So there are now 16, …1,2,3,4,5,6,7,8 and so on line segments.
Right, so step 2, there are 16 line segments,
by the way 16 is 4 squared.
I have 4 segments here
and then 1,2,3,4, shape like that 4 times 4 is 16.
Ok, so now this length is a third.
This length then into the other lengths is a third of third.
So that’ s nine.
So that’s 1/9 which is a third of third
which is also a third squared.
So if I had 16 things
each of which have a length of a ninth,
and the total lengths,
length is 16 over 9
which happens to be 4 third squared.
Right, probably see the pattern that’s taking shape here.
Let’s do step 3 quickly,
so if I just focus on this little piece here,
I go from this length to this length,
And I now have 4 times as many segments as I did before
and every step the number of line segments
goes up by a factor 4, it’s multiplied by 4.
So now I am gonna have, that’s step 3,
4 times 4 times 4, which is 4 cubed.
At each step, the length of the line segment was down by a third.
This is one. This is third. This is a third of third.
This a third of a third of a third.
That’s a third cubed.
So let’s say that third cubed.
So if I have 4 cubed things
each of which has a length of a third cubed,
then the total length
is going to be 4 third cubed.
Notice by the way that the length is getting longer
and that’s pretty easy to see.
This is length 1, now I added the detour.
It’s clearly this path with the bend in it is longer than this.
Every time I add a bend in a path, I am adding length.
So the length is getting larger and larger.
We can generalize this, we see the pattern now.
At the nth step, each line segments has been cut in third n times,
and every step the number of line segments is multiplied by 4.
So that the total length is 4 third to the n.
So now the question is what happens
as n gets larger and larger and larger.
And you can see what’s happening here.
This number is getting larger and larger and larger.
4 third multiplied by itself again and again and again.
It’s always getting larger
you are adding 33 percent every time.
So as n gets larger and larger,
the total length also gets larger and larger.
Say that it goes to infinity.
So here is a picture of the Koch curve.
This, I did in computer, ought to I think 16 steps.
The resolution wouldn’t let me go any farther. We wouldn’t see it.
And so as we continue the iteration process
at every step adding a bend and every line
then the length of this curve becomes infinite.
So this curve here it’s gonna have infinite length.
So if we have a curve it's infinitely long
but it’s clearly contained a finite area.
I can draw a circle around it.
The area in here is definitely finite.
This is just some little oval on my paper.
It’s about size of my hand.
Nevertheless I can fit one of these wiggly Koch curves in here.
That actually has infinite length.
So one of things about fractals is that
they combine quality is of finite and infinite in interesting way.
And in fact, that was some of the original motivation
for these constructions in the early 1900.
Mathematicians were looking at set theory
and trying to think about different properties of inifinities.
So in any event the Koch curve has an infinite length,
although clearly you can fit that infinite length in a finite area.
So now let’s think about the dimension of the Koch curve.
Recall that we found previously that is dimension log4 over log3
which is approximately 1.262
so it’s in between in 1 and 2 dimensions.
So one way of think about this is that
objects with this sort of dimensions between 1 and 2
have some features of qualities of 1 dimensional
and some therein two-dimensional.
So it’s one-dimensional in that it’s made up out of lines.
Remember in our construction we started with a line
and put a wiggle in it, put a bump in it, put bumps in it,
put bumps on the bums and so on, so on and so on.
And we end it up with a shape.
So it’s one-dimensional in a sense
or it has one dimensional qualities
because it started as a line.
However it has so many bumps in it.
And the shape becomes so jagged
so many teeth teeth on teeth on teeth
and bumps and bumps and bumps
and that it starts a bump up,
sort of push up into the second dimension.
It’s not fully two-dimensional, like an area.
It doesn’t really take up space.
But the line becomes so bumpy that
it has some sort of two-dimensional features.
That’s not a precise argument by any means
but just qualitatively it’s a way
to think about this non-integer dimension.
They combine so something that’s between dimensions one and two,
is sort of one-dimensional, sort of two-dimensional.
So another thing to think about with a Koch curve is
as we said in a finite area,
it has an infinite length.
So that would be really beneficial
if this was may be a part of along or something
some organ of structure that was trying to exchange
that wanted to have a large surface area.
May be this is a shape
that’s trying to dissipate heat,
so it could be something physical,
it doesn’t have to be biological.
So this large surface area would be
i.e. this large long perimeter of this large length
would be really beneficial.
Because that will be a large length
to cross which the heat could dissipate,
or oxygen could be exchanged or something.
So that’s another interesting feature
of these sorts of fractals
so said before they combined finite and infinite
and they could have some real structural advantages
depending on the particular application.