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TIMESTAMPS
  • 00:25
    Benoit B Mandelbrot is the father of fractal geometry. -- Math Joke: What does the B in Benoit B Mandelbrot stand for? Benoit B Mandelbrot&65279&59;
  • 01:13
    Fractals are a rebellion against calculus. Fractals are a way to model roughness without curves.
  • 06:00
    In actuality, fractal dimensionality is generally what we mean by the term dimensional.
  • 09:14
    This specialized case is called self-similarity dimension.
  • 13:02
    Applying the line touching box scaling method to a non self-similar shape, such as the coast line of Britain works as well.
  • 15:10
    Fractals are shapes whose dimension is not an integer.
  • 15:18
    This is a quantitative way to say fractals are shares that are rough, and stay rough even if you zoom in.
  • 18:00
    The coast of Britain has a dimension of about 1.21 across a wide variety of zoom levels.
  • 19:21
    Fractals can be used to separate man made from computer generated. If something can be defined by fractal dimensionality, it is generally from nature. Otherwise, it is man made.

Fractal Dimension Mandelbrot Explained

What fractal dimension is, and how this is the core concept defining what fractals themselves are. One technical note: It's possible to have fractals with an integer dimension. The example to have in mind is some *very* rough curve, which just so happens to achieve roughness level exactly 2. Slightly rough might be around 1.1-dimension; quite rough could be 1.5; but a very rough curve could get up to 2.0 (or more). A classic example of this is the boundary of the Mandelbrot set. The Sierpinski pyramid also has dimension 2 (try computing it!). The proper definition of a fractal, at least as Mandelbrot wrote it, is a shape whose "Hausdorff dimension" is greater than its "topological dimension". Hausdorff dimension is similar to the box-counting one I showed in this video, in some sense counting using balls instead of boxes, and it coincides with box-counting dimension in many cases. But it's more general, at the cost of being a bit harder to describe. Topological dimension is something that's always an integer, wherein (loosely speaking) curve-ish things are 1-dimensional, surface-ish things are two-dimensional, etc. For example, a Koch Curve has topological dimension 1, and Hausdorff dimension 1.262. A rough surfaces might have topological dimension 2, but fractal dimension 2.3. And if a curve with topological dimension 1 has a Hausdorff dimension that *happens* to be exactly 2, or 3, or 4, etc., it would be considered a fractal, even though it's fractal dimension is an integer. Pick a random fractal from a hat, though, and it will almost certainly have a non-integer dimension.






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