Surrounding the Koch coastline withīoxes is a way to determine its dimension ![]() There are, however, experimental techniques. A narrow and quick search of the popular literature reveals nothing on the ease or impossibility of this task. If there are analytic techniques for calculating the fractal dimension of an arbitrary Julia set they are well hidden. 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. Julia and Mandelbrot sets, fractals produced by the iterated mapping of continuous complex functions, are another matter. Fractals produced by simple iterative scaling procedures like the Koch coastline are very easy to handle analytically. Whether this is convenient or simple is another matter. It should be possible to use analytic methods like those described above on all sorts of fractal objects. Even though its Hausdorff-Besicovitch dimension is a whole number (D = 2) its topological dimension (D T = 1) is strictly less than this. Thus, the Peano space-filling curve is also a fractal as we would expect it to be. A better definition is that a fractal is any entity whose Hausdorff-Besicovitch dimension strictly exceeds its topological dimension (D > D T). Actually fractals can have whole number dimensions so this is a bit of a misnomer. Its dimension is not a whole number but a fraction. ![]() ![]() The Koch coastline is somewhere between a line and a plane. 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. This agrees with the topological dimension of the space. In the limit we find a dimension of log 2 n 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. Is this really a dimension? Apply the procedure to the unit line segment. Thus the Koch coastline has a Hausdorff-Besicovitch dimension which is the limit of the sequence log 1 Where N(h) is the number of disks of size h needed to cover the object. The Hausdorff-Besicovitch dimension of an object in a metric space is given by the formula D = If we apply this procedure to any entity in any metric space we can define a quantity that is the equivalent of a dimension. In general, it takes 4 n disks of radius (⅓) n to cover the Koch coastline. 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. How many disks does it take to cover the Koch coastline? Well, it depends on their size of course. In euclidean three-space disks would be balls while in a two-space with a Manhattan metric they would be squares. The term disk is used because such regions are disk-shaped in the coordinate plane with the usual metric but any shape is possible. 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. (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. How far is the corner of 33rd and 1st from 69th and 5th? Answer: 36 blocks and 4 avenues or 40 units. One of the more famous, non-euclidean metrics is the Manhattan metric (or taxicab metric). Such distance establishing relationships are called metrics and a space that has a metric associated with it is called a metric space. In relativity, the "distance" between any two events in space-time is given by the proper time On the Euclidean coordinate plane the distance between any two points is given by the Pythagorean theorem On a line segment like the Koch coastline, we arbitrarily chose the length of one side of the first iterate as a unit length. To be a bit more precise, every space that feels "real" has associated with it a sense of distance between any two points. Such an activity hints at the existence of a quantifiable characteristic. Using a focus that reveals details 1/9 the first focus gives us a coastline 16 times longer and so on. Double the resolution by the same factor. ![]() The curve is now four times longer or 4 units. Sharpen the focus a bit so that you can resolve details that are ⅓ as big as those seen with the first approximation. Take the Koch coastline and examine it through a badly focused lens. Whatever doesn't kill you only makes you stronger. Luckily, mathematics was fortified by the study of the monsters and not destroyed by them. Here were creations so twisted and distorted that they did not fit into the box of contemporary mathematics. There really was a reason to fear pathological entities like the Koch coastline and Peano's monster curve. San Marco dragon rendered with Julia's Dream. Data calculated using Fractal Dimension Calculator.
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