Now we can write down our precise definition of dimension.
Suppose we do what we've been doing;
that is, we create a geometric structure from a given D-dimensional object like a line, square, or a cube
by repeatedly dividing the length of its sides by a number M.
So this is just what we were doing before.
For example, when we were bisecting the sides, the number M was two (M=2).
When we were trisecting the sides, the number M was three (M=3), etc.
So this is our recipe, and we saw that each new level
we got M raised to the D power (M^D) copies of the figure at the previous level,
all shrunken by a factor of M.
So let's call the number of copies N.
OK, so we have N is equal to M to the D (N = M^D)...we are just renaming this to N.
Now we can define D by taking the logarithm of both sides,
so there we have log N is equal to D times log M (log N = D log M)
using one of the logarithm rules we saw in the previous video.
And then we can say that D, the dimension, is equal to log N over log M (D = log N/log M).
That is, dimension is defined as the logarithm of the number of copies we get divided by the amount by which we reduced the length of the sides.
OK, let's check to make sure this formula actually agrees with our standard notion of dimension.
To check it for Dimension-1,
let's take the bisecting version where M is equal to two (M=2).
That is, we divide each side into two equal halves,
then M is two and N, the number of copies of the line, is two,
so if we say D, dimension is equal to log two over log two, well, we get one (D = log 2/log 2 = 1).
That works because this gives us the correct dimension, Dimension-1.
So now, let's look at that for the trisecting version.
For trisecting, M equals three (M=3).
We divided each side into three equal parts.
So remember, for one dimension N, the number of copies, was three (N=3).
So we get D equals log three over log three, which equals one (D = log 3/log 3 = 1).
Again, this works because we get the answer that the dimension is one (D=1).
So our formula agrees with what we intuitively would consider to be the answer for the correct dimension.
Let's check for 2-dimensional objects.
For 2-dimensional objects, if we're bisecting, then dimension equals log four divided by log two. That's equal to two (D = log 4/log 2 = 2).
You can check that on your calculator if you like.
For trisecting we get M equals three (M=3) and N equals nine (N=9).
So D is equal to log nine divided by log three, which is also equal to two (D = log 9/log 3 = 2), so that works.
And finally, for Dimension-3...well, I'll let you check that, and I promise you it works.
Now all of this has been building up to what we really want to do,
which is to calculate the dimension of the Koch Curve.
So, if you remember from the Koch Curve,
we draw a segment, then we erase its middle third,
then we replace that middle third by an angle whose sides are the same length as the other two sides.
That is, one-third the original segment in this configuration.
Then we iterate that again and again until we finally come up with something that looks like this.
We can keep going for as long as we want.
So, for the Koch Curve, M equals three (M=3).
That is, remember that we're reducing the side of the segments by a factor of three.
N, the number of copies at each level, is four (N=4).
At each level, we replace the figure at the previous level with four copies arrayed in this configuration.
So according to our definition, the dimension of the Koch Curve is
log four divided by log three, that is, log N divided by log M (D = log 4/log 3 = log N/log M)
which, if you put it on your calculator, you'll find is approximately 1.26 (D=1.26).
But what exactly does this mean?
Well, for one thing, it means that the dimension of this curve approximates 1.26.
That is, it gets closer and closer to 1.26 as we iterate through more and more levels.
But what does the 1.26 mean? What does a dimension that's between one and two (1 < D < 2) actually mean?
Well, it's not easy to understand this intuitively,
but I can say that it's a measure of how the number of copies scales with the decrease in the size of the segment.
Very roughly, it's sort of the density of the self-similarity.
For regular geometric figures, this dimension is an integer,
but for fractals, which have possibly a higher density of self-similarity,
dimension can be fractional.
So, just to give you a feel for this, let me quote a little bit from my own book, "Complexity: A Guided Tour"
where I talk about what fractal dimension means. So, to quote:
"I have seen many attempts at intuitive descriptions of what fractal dimension means.
For example, it has been said that fractal dimension represents the roughness, ruggedness, jaggedness, or complictedness of an object.
An object's degree of fragmentation, and how 'dense' the structure of the object is.
But one description I like a lot is the rather poetic notion that fractal dimension 'quantifies the cascade of detail' of an object.
That is, it qualifies how much detail you can see at all scales
as you dive deeper and deeper into the infinite cascade of self-similarity.
For structures that aren't fractals, such as a smooth, round marble
if you keep looking at the structure with increasing magnification
eventually there's a level with no interesting details.
Fractals, on the other hand, have interesting details at all levels,
and fractal dimension, in some sense, quantifies how interesting that detail is
as a function of how much magnification you have to do at each level to see it."
So that was the description from my book,
and, of course, that description of having interesting detail at all levels applies to perfect mathematical fractals
but it's approximated by what we call fractals in nature.
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