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I was watching this video by 3blue1brown and he discusses the idea of fractal dimensions. He notes that fractals can have non-integer number of dimensions. For example, the Sierpinski triangle has a dimension of 1.5849.

+Itai Efrat asked the question in the comments section "In physics the idea of dimension is usually expressed as the number of degrees of freedom needed to describe the movement of a particle. Is there a sense in which a particle moving in a fractal has a non integer number of degrees of freedom?"

Unsurprisingly, the combined intelligence of the Youtube comments section (including myself) was unable to come to a conclusion.

So, does a particle moving in a fractal have a non integer number of degrees of freedom?

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    $\begingroup$ I'm not sure that "degrees of freedom" is really the right way to think about fractal notions of dimension. Degrees of freedom has a vector-spacey feel to it---it is, more or less, a description of how many basis vectors you have in a vector space. On the other hand, most notions of fractal dimension deal more with how balls scale; these are metric (and measure theoretic) notions, rather than algebraic notions. $\endgroup$ – Xander Henderson Nov 29 '17 at 1:00
  • $\begingroup$ I am not sure how "particle moving in a fractal" can be implemented using mechanical constraints, but if it is done then it will trivially (by definition) have the number of "degrees of freedom" equal to the fractal's dimension. The problem with the question is that "degrees of freedom" are just independent variables describing the system, i.e. "dimensions", however one extends "dimensions" beyond integers "degrees of freedom" can go along with it. $\endgroup$ – Conifold Nov 29 '17 at 1:00
  • $\begingroup$ @Conifold I think that your "trivial" is my "highly nontrivial." Though perhaps that is begin swept under the rug of "defining a motion on a fractal." Of potential interest: Differential Equations on Fractals. $\endgroup$ – Xander Henderson Nov 29 '17 at 1:17
  • $\begingroup$ @Conifold Well, we see fractals in a 2-dimensional space, so we can change the mechanical degrees of freedom and we can imagine a particle moving in those degrees of freedom although that might not be the "natural" state of things. To better explain what I mean, imagine a 3-dimensional particle as seen by a 2-dimensional being. The 3-dimensional particle would appear to teleport places and change sizes as it moved through the plane of the 2-dimensional being's view. This is obviously very unnatural for a 2-dimensional being, but it's possible given this hypothetical scenario. $\endgroup$ – Byte11 Nov 29 '17 at 1:30
  • $\begingroup$ @Conifold Now applying this notion to fractals, would a two-dimensional movement cross over a space that isn't "within the scope" of the fractal just as a 3-dimensional particle can move beyond the scope of perception of a two-dimensional being? And also, would two different fractals with the same dimension (if that's even possible) define a particle's movement in different ways? $\endgroup$ – Byte11 Nov 29 '17 at 1:32

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