> Yet, this common-sense definition is unsatisfying if we consider that a lower-dimensional object might end up straddling a higher-dimensional space. If a line segment is rotated or bent, does that make it 2D? Or is that object forever one-dimensional, somehow retaining the memory of its original orientation and curvature?

Isn't that exactly what topological manifolds are for?

https://en.wikipedia.org/wiki/Manifold

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codethief 4 days ago | parent | next [-]

For generic fractal curves the topology induced on the curve from the surrounding space will typically prevent that curve from being a topological manifold, since locally it won't look like R^n. It will merely be a topological space.

If, however, the (image of) the curve is a topological manifold (with the induced topology), then the (integer) dimension of the manifold will agree with the Minkowski dimension that's explained in the article (and also with the more commonly used Hausdorff dimension[0]).

[0]: https://en.m.wikipedia.org/wiki/Hausdorff_dimension

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HelloNurse 4 days ago | parent | prev | next [-]

The article seems to miss the difference between just using low-dimensional manifolds (e.g. any line) and escalating to a higher-dimensional space to distinguish different ones (e.g. various lines in a plane).

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jerf 4 days ago | parent [-]

I don't think it was "missed" so much as "the author chose not to discuss that". Just as it isn't productive to say that the Minkowski dimension, the main point of the article, "misses" that concept... it just isn't part of the definition. There are plenty of other mathematical ways to approach the situation but if you're always too busy talking about something else to talk about any particular concept you'll never get anywhere. Every article can't teach everything about any particular subject.

	▲	HelloNurse 3 days ago \| parent [-]
		But choosing not to discuss the obvious answer to a rhetorical question is a typical device of useless engagement bait posts on social media, hidden and made worse by the somewhat difficult subject. Ignorance is a much more charitable assumption than malicious waste of time.

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dazzaji 4 days ago | parent | prev [-]

Not quite - as I understand it box-counting measures global space-filling, manifolds handle local coordinate structure. Consider that the Earth is locally flat but globally spherical, and a Möbius strip vs cylinder are locally identical but globally different. Related problems, but the tools reveal different aspects of geometry. So I think whether “this is exactly what topological manifolds are for” depends what you’re trying to understand.