Wikipedia:Reference desk/Archives/Mathematics/2021 September 1
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September 1
[edit]Snubification in Conway notation
[edit]How can snubification (or alternation) be written in Conway notation? Existent human being (talk) 09:37, 1 September 2021 (UTC)
Difference between Hausdorff dimension and Fractal dimension
[edit]I first need to state that I'm a bit of a math layperson who enjoys recreational math a bit, and also watches a lot of math YouTube videos, such as 3blue1brown, Standup Maths, Mathologer, Numberphile, etc. I had recently rewatched Grant Sanderson's video on fractals, and that led me to try to do a little bit of Wikipedia research on fractals, where I came across two articles Hausdorff dimension and Fractal dimension which appear to be discussing basically the exact same concept; in fact one used to be a redirect to the other (way back in Wikipedia's infancy, from like 2003-2006 or so) when they were split and have developed separately since then. Can someone give a lay-person's explanation for the difference between those concepts? Is one a subset of the other? Are they two terms for the same thing? How are they related? Thanks! --Jayron32 18:14, 1 September 2021 (UTC)
- OK. Early on in the Fractal Dimension page we see the promising-looking phrase "There are several formal mathematical definitions of fractal dimension", which contains a link. The link is actually to a position within the same page labelled "specific_dimensions" - which is terrible Wikipedia practice I think - and that label doesn't seem to exist any more, so it confusingly goes to the section "Fractal Surface Structures". But I think the intended information is still on the page, in section 6, "Examples". There for instance "Box-counting dimension" is actually a redirect to Minkowski–Bouligand dimension. This contains a formal definition of fractal dimension which is almost, but not quite, exactly the same as Hausdorff dimension, which apparently succeeded it as another formal definition. In other words I think fractal dimension is the concept and Hausdorff (and any other alternative that may exist) is an implementaton, or (more formally) a formal definition. Card Zero (talk) 19:08, 1 September 2021 (UTC)
- So if I get this straight, there's a general concept called "Fractal dimension", and there are different formulations of how to specifically calculate/express/etc. the concept, of which the "Hausdorff dimension" is one. Did I get that correct? --Jayron32 19:44, 1 September 2021 (UTC)
- You've correctly understood my somewhat shaky interpretation, yes. Card Zero (talk) 19:58, 1 September 2021 (UTC)
- So if I get this straight, there's a general concept called "Fractal dimension", and there are different formulations of how to specifically calculate/express/etc. the concept, of which the "Hausdorff dimension" is one. Did I get that correct? --Jayron32 19:44, 1 September 2021 (UTC)
- I hesitate to use the term "implementation", because the difference is already in the specification. Two extremes in the world of fractal planar curves are the straight lines – not visually exciting, as well as being "godless and immoral" – and the plane-filling curves such as the Hilbert curve – the limit of which also does not yield an interesting image. The interesting fractal curves are somewhere between these extremes. A straight line has dimension 1, and the image of a plane-filling curve has dimension 2. Inasmuch as "fractal dimension" is a concept, it is the idea of finding a generalization of the concept of "dimension" that agrees with the usual integer-valued concept for these extremes but also assigns a numeric value to the in-between cases. Different generalizations have been proposed, which do not only differ in how they are defined, but can also yield different results when applied – or fail to assign a definite value. For the well-known mathematically defined fractal curves, based on a recursive construction, the two generalizations known as Hausdorff dimension and Minkowski–Bouligand dimension are both defined and always agree in the assigned value. For even more exotic planar images, this is not necessarily the case. Divergence between these two generalizations is discussed in the section Minkowski–Bouligand dimension § Relations to the Hausdorff dimension. --Lambiam 20:27, 1 September 2021 (UTC)
- Thank you, that was very clear! --Jayron32 10:50, 2 September 2021 (UTC)
- I hesitate to use the term "implementation", because the difference is already in the specification. Two extremes in the world of fractal planar curves are the straight lines – not visually exciting, as well as being "godless and immoral" – and the plane-filling curves such as the Hilbert curve – the limit of which also does not yield an interesting image. The interesting fractal curves are somewhere between these extremes. A straight line has dimension 1, and the image of a plane-filling curve has dimension 2. Inasmuch as "fractal dimension" is a concept, it is the idea of finding a generalization of the concept of "dimension" that agrees with the usual integer-valued concept for these extremes but also assigns a numeric value to the in-between cases. Different generalizations have been proposed, which do not only differ in how they are defined, but can also yield different results when applied – or fail to assign a definite value. For the well-known mathematically defined fractal curves, based on a recursive construction, the two generalizations known as Hausdorff dimension and Minkowski–Bouligand dimension are both defined and always agree in the assigned value. For even more exotic planar images, this is not necessarily the case. Divergence between these two generalizations is discussed in the section Minkowski–Bouligand dimension § Relations to the Hausdorff dimension. --Lambiam 20:27, 1 September 2021 (UTC)