9/6/2023 0 Comments Fractal dimension![]() 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.Molecular complexity has been a topic of research for more than sixty years. 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. ![]()
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