One other thing to mention is that the fractals discussed here are also called monofractals, to contrast with something else you can analyze with FracLac and which you can read about later, called multifractals.I call our world Flatland, not because we call it so,īut to make its nature clearer to you, my happy readers, who are privileged Just what we use for N and ε and exactly how this handy technique works for us in box counting with FracLac is explained in the next section. This is a very handy-dandy technique, indeed. The basic equation for finding a fractal dimension from such data approximating scale and detail is nearly what we already know from the scaling rule: D F = lim ε→0where we find the limit as the slope of the regression line for the data. In FracLac, it is the box counting dimension or D B. They have made ways for us to infer the value of complexity from the ratio of changing detail with changing scale (e.g., magnification or resolution in microscopy) approximated by some measure and assigned a number we figure is close enough to its fractal dimension and that is usually a new type of D F. Just how much would you enjoy counting and measuring to find 32 new parts for every 1/8 scaling in a quadric fractal, for instance? Out of kindness and respect for our tolerance of tedium, therefore, our friends, the fractal analysts, have developed methods to assess the D F indirectly. But it is not always easy to calculate a D F this way because the relationship between scale and detail is not always readily observable. The "number of pieces" referred to in the above examples is equivalent to the detail in a pattern, and, for the examples given so far, we needed only to count and measure fairly simple or at least tractable-as-long-as-you-already-know-something-about-them features to find the relationship between scale and detail. Sometimes You Need a Little Help From Your Friends For the 32-segment quadric fractal you surely remember from an earlier page, the pattern scales into 32 new pieces each 1/8 the size of the previous.For the Koch fractal shown earlier, however, D F = log 4/log 3 = 1.26.For anything scaling like the simple line mentioned above, the number of new parts is equal to the scale -1, and D F = log X/log X = 1.00.Now that you have all that scaling rule and log stuff down, you can calculate some fractal dimensions. I stuck an A in there along with N and ε, didn't I? We'll talk later. I calculate a D F for any case like this by solving the general equation for the scaling rule : N= Aε -D F for its variable, D F, using logs, which shows that the D F is the ratio of the log of the number of new parts N, to the log of scale, ε: D F = log N/log ε. Logs and Limits How is a Fractal Dimension Calculated?įor the Koch pattern, you have to get out your math assistance device of choice and calculate the D F. The definitely untrivial point here is that in contrast to the line and square considered above, the scaling rule, D F, for this pattern, even if we could perceive it's infinite nature, is not so obvious-the numbers are 4=(1/3) -D F and this we cannot solve by simple substitution into the scaling rule. This goes on forever and the infinite result, alas, we mere mortals cannot see, but is, nonetheless, the Koch fractal pattern. Basically, what happens is that the starting piece is scaled down to 1/3 the length it was, then that piece is laid down four times to make a new one that is the length of the original but has more pieces. You can see how it is formed in the animation. The Koch fractal line illustrated at the left, for example, scales into 4 new pieces each 1/3 the length of the original. This may seem trivial-that the dimension (or complexity) of a line is 1 and of a filled square 2-but the decidedly untrivial part is that this sort of scaling, the scaling we know and love, is not necessarily the only kind of scaling possible. Maybe you have already expanded this idea and figured out that all this is to say, too, that when scaling a filled square by 1/2, there will always be 4 new pieces, each 1/4 the area of the original, and D would be equal to 2 (e.g., 4=(1/2) -2). But it is kind of interesting to know that it gives us a use for the N in the scaling relationship, and we can figure out that D F = 1.00 in this situation just by substituting into the equation, because 3=(1/3) -1. I won't even draw a diagram because it will surely just bore you. A line, when scaled by, as an example, 1/3, can be seen to be made up of 3 pieces, each 1/3 the length of the original. One shape for which scaling is easy to grasp is a simple line. First consider something you know, patterns such as the familiar Euclidean shapes of elementary geometry. To understand how all this fits together in the calculations for scaling rules and fractal dimensions, let's look at things differently than everyday life usually asks us to.
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