The Chaos Hypertextbook
© 19952007 by Glenn Elert
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San Marco dragon rendered with Julia's Dream
There really was a reason to fear pathological entities like the Koch coastline and Peano's monster curve. Here were creations so twisted and distorted that they did not fit into the box of contemporary mathematics. Luckily, mathematics was fortified by the study of the monsters and not destroyed by them. Whatever doesn't kill you only makes you stronger.
Take the Koch coastline and examine it through a badly focused lens. It appears to have a certain length. Let's call it 1 unit. Sharpen the focus a bit so that you can resolve details that are ⅓ as big as those seen with the first approximation. The curve is now four times longer or 4 units. Double the resolution by the same factor. Using a focus that reveals details 1/9 the first focus gives us a coastline 16 times longer and so on. Such an activity hints at the existence of a quantifiable characteristic.
To be a bit more precise, every space that feels "real" has associated with it a sense of distance between any two points. On a line segment like the Koch coastline, we arbitrarily chose the length of one side of the first iterate as a unit length. On the Euclidean coordinate plane the distance between any two points is given by the Pythagorean theorem
s^{2} = x^{2} + y^{2}.
In relativity, the "distance" between any two events in spacetime is given by the proper time
s^{2} = c^{2}t^{2} − x^{2} − y^{2} − z^{2}.
Such distance establishing relationships are called metrics and a space that has a metric associated with it is called a metric space. One of the more famous, noneuclidean metrics is the Manhattan metric (or taxicab metric). How far is the corner of 33rd and 1st from 69th and 5th? Answer: 36 blocks and 4 avenues or 40 units. (We have to bend reality a bit and assume that city blocks in Manhattan are square and not rectangular.) Metrics are also used to create neighborhoods in a space. Pick a point in a metric space. This point plus all others lying less than or equal to a certain distance away comprise a region of the space called a closed disk. The term disk is used because such regions are diskshaped in the coordinate plane with the usual metric but any shape is possible. In euclidean threespace disks would be balls while in a twospace with a Manhattan metric they would be squares.
How many disks does it take to cover the Koch coastline? Well, it depends on their size of course. 1 disk with diameter 1 is sufficient to cover the whole thing, 4 disks with diameter ⅓, 16 disks with diameter 1/9, 64 disks with diameter 1/27, and so on. In general, it takes 4^{n} disks of radius (⅓)^{n} to cover the Koch coastline. If we apply this procedure to any entity in any metric space we can define a quantity that is the equivalent of a dimension. The HausdorffBesicovitch dimension of an object in a metric space is given by the formula
D = 

log N(h)  
log (1/h) 
where N(h) is the number of disks of size h needed to cover the object. Thus the Koch coastline has a HausdorffBesicovitch dimension which is the limit of the sequence
log 1  ,  log 4  ,  log 16  ,  log 64  , …  log 4^{n}  =  n log 4  =  log 4  = 1.261859507 … 
log 1  log 3  log 9  log 27  log 3^{n}  n log 3  log 3 
Is this really a dimension? Apply the procedure to the unit line segment. It takes 1 disk of diameter 1, 2 disks of diameter ½, 4 disks of diameter ¼, and so on to cover the unit line segment. In the limit we find a dimension of
log 2^{n}  =  n log 2  =  log 2  = 1 
log 2^{n}  n log 2  log 2 
This agrees with the topological dimension of the space.
The problem now is, how do we interpret a result like 1.261859507…? This does not agree with the topological dimension of 1 but neither is it 2. The Koch coastline is somewhere between a line and a plane. Its dimension is not a whole number but a fraction. It is a fractal. Actually fractals can have whole number dimensions so this is a bit of a misnomer. A better definition is that a fractal is any entity whose HausdorffBesicovitch dimension strictly exceeds its topological dimension (D > D_{T}). Thus, the Peano spacefilling curve is also a fractal as we would expect it to be. Even though its HausdorffBesicovitch dimension is a whole number (D = 2) its topological dimension (D_{T} = 1) is strictly less than this. The monster has been tamed.
It should be possible to use analytic methods like those described above on all sorts of fractal objects. Whether this is convenient or simple is another matter. Fractals produced by simple iterative scaling procedures like the Koch coastline are very easy to handle analytically. Julia and Mandelbrot sets, fractals produced by the iterated mapping of continuous complex functions, are another matter. There's no obvious fractal structure to the quadratic mapping, no hint that a "monster" curve lurks inside, and no simple way to extract an exact fractal dimension. If there are analytic techniques for calculating the fractal dimension of an arbitrary Julia set they are well hidden. A narrow and quick search of the popular literature reveals nothing on the ease or impossibility of this task. There are, however, experimental techniques.
Surrounding the Koch Coastline with Boxes (a way to determine its dimension) 
Take any plane geometric object of finite extent (fractal or otherwise) and cover it with a single closed disk. Any type of disk will do, so to make life easy we will use a square; the disk of the Manhattan metric in the plane. Record its dimension and call it "h". Repeat the procedure with a smaller box. Record its dimension and the number of boxes "N(h)" required to cover the object. Repeat with ever smaller boxes until you have reached the limit of your resolving power as shown in the figure to the right. Plot the results on a graph with "log N(h)" on the vertical axis and "log (1/h)" on the horizontal axis. The slope of the best fit line of the data will be an approximation of the HausdorffBesicovitch dimension of the object. The following are the results of a few sample experiments using this boxcounting method. I think with a bit of refinement, the deviations could all be brought below 5%.
Although this chapter is ending, this is not the last word on dimension. A study of twelve definitions of dimension appeared in 1981 (see Harrison) and an entire book on Dimension Theory was written in 1941 (Hurewicz & Wallman). The topics of chaos, fractals, and dimension are rich and strange. They are immensely interesting and serious consideration should be given to incorporating them into high school mathematics. I can easily envision these three topics as the central themes of a final year high school math course incorporating the basics, advanced topics, current events in science, and computer applications.
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