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There is a common misconception, repeated here, that fractals are defined by their self-similarity. This is not true. Fractals are mathematical objects which ha

by uoaei 7y ago

There is a common misconception, repeated here, that fractals are defined by their self-similarity. This is not true. Fractals are mathematical objects which have non-integral dimensionalities: that is, they exist in a space somewhere between (n)D space and (n+1)D space for some integer n.[1] The exact Hausdorff dimension, the number quantifying their dimensionality, is not in general the same for each construction. For the Koch curve, the Hausdorff dimension is ln(4)/ln(3) = log_3(4) approximately 1.2619.[2] This is because for each iteration of the curve's construction, the perimeter of the curve grows by a factor of 4 while the area enclosed by the curve only grows by a factor of 3. That makes it fill space faster than a growing 1D line, but slower than a growing 2D surface. [1] https://www.youtube.com/watch?v=gB9n2gHsHN4 https://www.youtube.com/watch?v=gB9n2gHsHN4 [2] https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_...

oarabbus_ 7y ago
>There is a common misconception, repeated here, that fractals are defined by their self-similarity. This is not a misconception. Self-similarity at all length scales is indeed a defining factor of fractals. From that fact it _does_ follow that _most_ fractals have a non-integer fractal dimension. However it is absolutely not true that fractals have non-integral dimensionalities; for example the most famous fractal of them all, the Mandelbrot set, has an integer Hausdorff dimension of 2. [0] https://arxiv.org/abs/math/9201282 https://arxiv.org/abs/math/9201282
- JadeNB 7y ago
  There'll doubtless be more argument about this, but I think that, before disagreeing with you, people should note that 'fractal' is not a disembodied term, but the creation of one person who has the right to define it—but who unfortunately can't make up his mind. Per Wikipedia (https://en.wikipedia.org/wiki/Fractal https://en.wikipedia.org/wiki/Fractal): > Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." More formally, in 1982 Mandelbrot stated that "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way." Still later, Mandelbrot settled on this use of the language: "… to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants". (Note that the mention of dimension only requires one notion of dimension to exceed another, not that either dimension be non-integral!)
westoncb 7y ago
To what extent is the concept of 'dimension' here the same as in everyday experience? (or, e.g. Linear Algebra, where the number of vectors in a basis correspond to dimension. I see that concept as roughly similar to the 'everyday' concept.) From this comment I would guess this is an alternate meaning of 'dimension' which captures one key property from the usual meaning (i.e. ratio of surface to interior space depends on dimension), but leaves out the rest. Like is there a sense in which we can rotate onto a fractional axis or something like that?
- dreamcompiler 7y ago
  It's a generalization of the everyday concept in an analogous way to how the reals generalize the integers. Obviously fractals exist in ordinary integer-dimensional space, but the Hausdorff (or fractal) dimension provides additional descriptive mathematical insight. https://en.m.wikipedia.org/wiki/Hausdorff_dimension https://en.m.wikipedia.org/wiki/Hausdorff_dimension
  - westoncb 7y ago
    I think that's a misleading characterization. It's actually a particularization not a generalization: the ordinary notion only deals with topological properties, fractal/Hausdorff dimension deals with metric properties as well. That's the main distinction: the old notion doesn't care about roughness or changes in measurement at varied scales since it has nothing to do with metric properties. The generalization from integers to reals in the actual values is not indicative of the relationship between the two concepts.
- andrepd 7y ago
  First it's useful to understand why e.g. a Koch curve has fractional dimension. The answer informally is that it has "too much area" to be just a line, but it's "too much a line" to be a 2d figure. Just notice that its length (1D measure) is infinite (just like a square has "infinite length"), so its dimension must be >1D, but also its area) 2D measure is zero (just like a line has 0 area), so it must also be <2D. Then if you want to make this intuition precise, you need to look into the box-counting dimension, and then the Hausdorf dimension.
77544cec 7y ago
This is how I understood the notion of Hausdorff dimension when reading Mandelbrot's book. Pick a stick to measure the length of the curve. As you break it into smaller sticks, you can fit them in tighter niches within the curve, and your measure of the curve's length becomes greater and greater. In the case of the Koch curve, breaking your stick in 3 means you'll be able to measure a length 4 times greater.

by uoaei 7y ago

oarabbus_ 7y ago

>There is a common misconception, repeated here, that fractals are defined by their self-similarity. This is not a misconception. Self-similarity at all length scales is indeed a defining factor of fractals. From that fact it _does_ follow that _most_ fractals have a non-integer fractal dimension. However it is absolutely not true that fractals have non-integral dimensionalities; for example the most famous fractal of them all, the Mandelbrot set, has an integer Hausdorff dimension of 2. [0] https://arxiv.org/abs/math/9201282 https://arxiv.org/abs/math/9201282

JadeNB 7y ago
There'll doubtless be more argument about this, but I think that, before disagreeing with you, people should note that 'fractal' is not a disembodied term, but the creation of one person who has the right to define it—but who unfortunately can't make up his mind. Per Wikipedia (https://en.wikipedia.org/wiki/Fractal https://en.wikipedia.org/wiki/Fractal): > Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." More formally, in 1982 Mandelbrot stated that "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way." Still later, Mandelbrot settled on this use of the language: "… to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants". (Note that the mention of dimension only requires one notion of dimension to exceed another, not that either dimension be non-integral!)

westoncb 7y ago

To what extent is the concept of 'dimension' here the same as in everyday experience? (or, e.g. Linear Algebra, where the number of vectors in a basis correspond to dimension. I see that concept as roughly similar to the 'everyday' concept.) From this comment I would guess this is an alternate meaning of 'dimension' which captures one key property from the usual meaning (i.e. ratio of surface to interior space depends on dimension), but leaves out the rest. Like is there a sense in which we can rotate onto a fractional axis or something like that?

dreamcompiler 7y ago
It's a generalization of the everyday concept in an analogous way to how the reals generalize the integers. Obviously fractals exist in ordinary integer-dimensional space, but the Hausdorff (or fractal) dimension provides additional descriptive mathematical insight. https://en.m.wikipedia.org/wiki/Hausdorff_dimension https://en.m.wikipedia.org/wiki/Hausdorff_dimension
- westoncb 7y ago
  I think that's a misleading characterization. It's actually a particularization not a generalization: the ordinary notion only deals with topological properties, fractal/Hausdorff dimension deals with metric properties as well. That's the main distinction: the old notion doesn't care about roughness or changes in measurement at varied scales since it has nothing to do with metric properties. The generalization from integers to reals in the actual values is not indicative of the relationship between the two concepts.
andrepd 7y ago
First it's useful to understand why e.g. a Koch curve has fractional dimension. The answer informally is that it has "too much area" to be just a line, but it's "too much a line" to be a 2d figure. Just notice that its length (1D measure) is infinite (just like a square has "infinite length"), so its dimension must be >1D, but also its area) 2D measure is zero (just like a line has 0 area), so it must also be <2D. Then if you want to make this intuition precise, you need to look into the box-counting dimension, and then the Hausdorf dimension.

77544cec 7y ago

This is how I understood the notion of Hausdorff dimension when reading Mandelbrot's book. Pick a stick to measure the length of the curve. As you break it into smaller sticks, you can fit them in tighter niches within the curve, and your measure of the curve's length becomes greater and greater. In the case of the Koch curve, breaking your stick in 3 means you'll be able to measure a length 4 times greater.