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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 correspon

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.
- westoncb 7y ago
  That makes sense to me. There was a specific thing I was looking for in my original question which was to understand how related vs. arbitrary the shared term 'dimension' is. I think I found an answer after browsing a few Wikipedia articles—but happy to be corrected if it doesn't sound right: The usual notion of dimension that people think of is topological only; fractal dimension (i.e. Hausdorf dimension) is also concerned with metric properties. These metric properties are what creates the need/possibility for fractional measures. In that case, the scenario I was considering with 'fractional axes/directions' is not implied by the fractal dimension concept: fractal dimension is dealing with new subject matter, not just an extension of the old concept to deal with real numbers.
jerf 7y ago
Trying to explain fractional dimensions is probably not conducive to an HN post. but fortunately, 3blue1brown converted this: https://youtu.be/gB9n2gHsHN4 https://youtu.be/gB9n2gHsHN4 Which also nicely and conclusively shows why self-similarity is just a special case of things that have fractional dimension.

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.

westoncb 7y ago
That makes sense to me. There was a specific thing I was looking for in my original question which was to understand how related vs. arbitrary the shared term 'dimension' is. I think I found an answer after browsing a few Wikipedia articles—but happy to be corrected if it doesn't sound right: The usual notion of dimension that people think of is topological only; fractal dimension (i.e. Hausdorf dimension) is also concerned with metric properties. These metric properties are what creates the need/possibility for fractional measures. In that case, the scenario I was considering with 'fractional axes/directions' is not implied by the fractal dimension concept: fractal dimension is dealing with new subject matter, not just an extension of the old concept to deal with real numbers.

jerf 7y ago

Trying to explain fractional dimensions is probably not conducive to an HN post. but fortunately, 3blue1brown converted this: https://youtu.be/gB9n2gHsHN4 https://youtu.be/gB9n2gHsHN4 Which also nicely and conclusively shows why self-similarity is just a special case of things that have fractional dimension.