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