> 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?
It's always puzzled me how many of these interesting things are not in a standard math course.
I recall doing a 4th year "advanced mathematical techniques" course where they went over the basics of graph theory and RSA algorithm. Descrete maths. This kind of thing is not the intensive calculus that people get told is math in high school, but a high schooler could do it if you showed them.
I'm not sure I understand how you can use a single number to say where on the Hilbert curve you are. As the curve is infintely, iteratively defined, you either can't say where you are on the curve, or you can say "length x, iterations y", which makes it 2 dimensional.
Once one exists the realm of differentiable manifolds, it is not really reasonable to talk about a single notion of 'dimension'.
Topological dimension is indeed something one can define: e.g. the Koch snowflake [1] or the graph of the Weierstrass function [2] have topological dimension 1. Actually, the first is homeomorphic to the unit circle and the second is homeomorphic to real line. It's great if you are doing topology and you only care about how things look like up to homeomorphism. But if you have metric structure (and you care about it), it is not so useful.
Minkowski dimension is certainly easy to define but it has some problems: sets which are "very small" (like a sequence `1/log(n)`) can have Minkowski dimension 1. The article has a minor technical oversight: the limit certainly does not need to exist. Minkowski defined it as the limit supremum of the sequence (actually, he defined it in terms of the decay rate of the size of the neighbourhood of the set, but this is equivalent). But one could analogously define a "lower" variant by taking the limit infimum instead.
Hausdorff dimension is not discussed in this article, but it is probably the most "robust" notion of dimension one can define. The Hausdorff dimension of any sequence is 0. But even then, lots of sets with Hausdorff dimension 1 can be very small, like the fat Cantor set which has dimension 1 but has length 0 [3]. So this 'dimension' does not necessarily line up with the intuition for "1-dimensional" in esoteric circumstances.
But even Hausdorff / Minkowski dimension does not capture the essence of some matters. For example, one might be interested in when a certain space can be mapped into another space without too much distortion (let's say by a map which respects the metric, like a bi-Lipschitz map). It can easily happen that a set has small (finite) Hausdorff or Minkowski dimension, but it cannot be embedded in a non-distorting way in any finite dimensional Euclidean space. This happens for instance with the real Heisenberg group [4]. If you are interested in this type problem then you want something like Assouad dimension [5].
The moral of the story is: the correct notion of dimension depends critically on what you want to do with your notion of 'dimension'. For sets which are very nice (smooth manifolds) all "reasonable" notions of dimension will coincide with what you expect; but beyond this there is an infinite zoo of ways to define dimension which are all reasonable in various ways, but capture genuinely different notions of 'size'.
I wish this blog post had gone back to the cross-shape above, with mostly 1 degree of freedom, except in the center, and shown us what the logical dimensionality of it is.
> the gaps are reduced to zero and the curve crosses through each and every point within its build envelope
I think that's oversimplifying an important point. If you build a Hilbert curve in a 1x1 square, the vertices of the curve always have rational coordinates. So all points on its line segments must always have at least one rational coordinate.
There's no way it can cross through every point in a square region of R^2.
A better way to say it might be "the gaps are reduced towards zero and the curve will pass arbitrarily close by every point in its envelope". That still explains why its Minkowski dimension must be 2.
The limit of a process at infinity is not necessarily subject to the same constraints as the outcome you're seeing after a finite number of steps. The assumption you're making here is intuitive, but it bites students in the butt every now and then.
Most simply, after "infinitely many" steps, the "arbitrarily close" in your mental model becomes "infinitely close", and in real numbers, "infinitely close" is the same as "equal to", because you don't have infinitesimals.
It's the same reason why we can construct irrational or transcendental numbers as a limit of an infinite series of rational numbers.
13 comments
[ 1.8 ms ] story [ 30.6 ms ] threadhttps://www.youtube.com/playlist?list=PL3C690048E1531DC7
walks you through step by step, builds the intuition and provides some historical context in the background.
despite "outdated" animation (15 years ago), still a great resource.
also available in 7 other languages.
Isn't that exactly what topological manifolds are for?
https://en.wikipedia.org/wiki/Manifold
I recall doing a 4th year "advanced mathematical techniques" course where they went over the basics of graph theory and RSA algorithm. Descrete maths. This kind of thing is not the intensive calculus that people get told is math in high school, but a high schooler could do it if you showed them.
Topological dimension is indeed something one can define: e.g. the Koch snowflake [1] or the graph of the Weierstrass function [2] have topological dimension 1. Actually, the first is homeomorphic to the unit circle and the second is homeomorphic to real line. It's great if you are doing topology and you only care about how things look like up to homeomorphism. But if you have metric structure (and you care about it), it is not so useful.
Minkowski dimension is certainly easy to define but it has some problems: sets which are "very small" (like a sequence `1/log(n)`) can have Minkowski dimension 1. The article has a minor technical oversight: the limit certainly does not need to exist. Minkowski defined it as the limit supremum of the sequence (actually, he defined it in terms of the decay rate of the size of the neighbourhood of the set, but this is equivalent). But one could analogously define a "lower" variant by taking the limit infimum instead.
Hausdorff dimension is not discussed in this article, but it is probably the most "robust" notion of dimension one can define. The Hausdorff dimension of any sequence is 0. But even then, lots of sets with Hausdorff dimension 1 can be very small, like the fat Cantor set which has dimension 1 but has length 0 [3]. So this 'dimension' does not necessarily line up with the intuition for "1-dimensional" in esoteric circumstances.
But even Hausdorff / Minkowski dimension does not capture the essence of some matters. For example, one might be interested in when a certain space can be mapped into another space without too much distortion (let's say by a map which respects the metric, like a bi-Lipschitz map). It can easily happen that a set has small (finite) Hausdorff or Minkowski dimension, but it cannot be embedded in a non-distorting way in any finite dimensional Euclidean space. This happens for instance with the real Heisenberg group [4]. If you are interested in this type problem then you want something like Assouad dimension [5].
The moral of the story is: the correct notion of dimension depends critically on what you want to do with your notion of 'dimension'. For sets which are very nice (smooth manifolds) all "reasonable" notions of dimension will coincide with what you expect; but beyond this there is an infinite zoo of ways to define dimension which are all reasonable in various ways, but capture genuinely different notions of 'size'.
[1]: https://en.wikipedia.org/wiki/Koch_snowflake
[2]: https://en.wikipedia.org/wiki/Weierstrass_function
[3]: https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%9...
[4]: https://en.wikipedia.org/wiki/Heisenberg_group
[5]: https://en.wikipedia.org/wiki/Assouad_dimension
I think that's oversimplifying an important point. If you build a Hilbert curve in a 1x1 square, the vertices of the curve always have rational coordinates. So all points on its line segments must always have at least one rational coordinate. There's no way it can cross through every point in a square region of R^2.
A better way to say it might be "the gaps are reduced towards zero and the curve will pass arbitrarily close by every point in its envelope". That still explains why its Minkowski dimension must be 2.
Most simply, after "infinitely many" steps, the "arbitrarily close" in your mental model becomes "infinitely close", and in real numbers, "infinitely close" is the same as "equal to", because you don't have infinitesimals.
It's the same reason why we can construct irrational or transcendental numbers as a limit of an infinite series of rational numbers.