PS1: Why isn't the Hilbert curve constructed more simply as an "accordion" curve that just zig-zags up and down the whole height of the unit square?
Well, we could conceptually do that, but if we represent the length of the curve as a normalized interval (e.g., 0-1) and then pick a point on the curve this way (say, pos = 0.6), the 2D position of that point would in most cases abruptly change every time we modify length by inserting another zig and zag. This means the resulting mapping has no defined limit at infinity: it oscillates instead of converging on something stable.
In contrast, on the Hilbert curve, that 2D position converges: the points get pushed around a bit in the beginning, but the motion decreases as we iterate. The construction method preserves locality by modifying the length of the curve in a more uniform way.
PS2: There is an interesting critique of the Hilbert curve. Because of the iterative scale-and-copy construction method, it would seem that the curve can only ever cross through x / y coordinates that can be expressed as finite fractions. Yet, the x-y plane is not just that; in standard Euclidean geometry, the axes are reals and include irrational values such as √2 or π. So, don't we end up with some gaps in the mapping?
One way to tackle this is a throwback to the following article: https://lcamtuf.substack.com/p/09999-1. There is asymmetry baked into common math: we allow processes to continue forever (to infinity), but we define reals as an Archimedean group that doesn't contain infinite or infinitesimal ("1/∞"-style) numbers. It's an ad-hoc axiom, but the alternatives are worse. If we get rid of process ("potential") infinity, that kneecaps a lot of math. If we allow infinitesimals and infinite values, we soon discover that number-like ("actual") infinities have a bunch of distinct flavors that interact with each other in funky ways (and come with some provably-unprovable properties in tow).
Anyway, for a finite number of iterations, the earlier observation is correct: the curve only covers rational coordinates. If you iterate longer, you can get arbitrarily close to irrational coords, but you never hit the bullseye.
But when we're talking about the end state "at infinity", we're no longer merely "arbitrarily" close to the target; we're infinitely close (again, loosely, "1/∞"). And if there are no infinitesimals in reals, that's necessarily synonymous with "equal to".
(Adding infinitesimals doesn't necessarily change the outcome because, as hinted earlier, infinities come in flavors. More about that in the aforementioned article.)
PS3, for the very angry person on Hackernews talking about "useless engagement bait" and "ignorance [being] a much more charitable assumption than malicious waste of time"... the article doesn't talk about the problem from the perspective of topological manifolds for two reasons:
1) We'd need a long segue into topology, non-metric spaces, and set theory to explain the topological approach (Lebesgue covering dimensions).
2) The outcome of that wouldn't be terribly useful for the main "gotcha" I wanted to talk about, which are space-filling curves and the problem of measuring the dimensionality of fractal shapes.
I was recently asked to explain the difference between the schoolroom arithmetic and genuine mathematical thinking. This dimension puzzle acts as a vivid illustration: it shows what it means to fiddle with mathematical objects, and why a bit of algebra is a useful tool for this process, helping to structure the reasoning.
PS1: Why isn't the Hilbert curve constructed more simply as an "accordion" curve that just zig-zags up and down the whole height of the unit square?
Well, we could conceptually do that, but if we represent the length of the curve as a normalized interval (e.g., 0-1) and then pick a point on the curve this way (say, pos = 0.6), the 2D position of that point would in most cases abruptly change every time we modify length by inserting another zig and zag. This means the resulting mapping has no defined limit at infinity: it oscillates instead of converging on something stable.
In contrast, on the Hilbert curve, that 2D position converges: the points get pushed around a bit in the beginning, but the motion decreases as we iterate. The construction method preserves locality by modifying the length of the curve in a more uniform way.
PS2: There is an interesting critique of the Hilbert curve. Because of the iterative scale-and-copy construction method, it would seem that the curve can only ever cross through x / y coordinates that can be expressed as finite fractions. Yet, the x-y plane is not just that; in standard Euclidean geometry, the axes are reals and include irrational values such as √2 or π. So, don't we end up with some gaps in the mapping?
One way to tackle this is a throwback to the following article: https://lcamtuf.substack.com/p/09999-1. There is asymmetry baked into common math: we allow processes to continue forever (to infinity), but we define reals as an Archimedean group that doesn't contain infinite or infinitesimal ("1/∞"-style) numbers. It's an ad-hoc axiom, but the alternatives are worse. If we get rid of process ("potential") infinity, that kneecaps a lot of math. If we allow infinitesimals and infinite values, we soon discover that number-like ("actual") infinities have a bunch of distinct flavors that interact with each other in funky ways (and come with some provably-unprovable properties in tow).
Anyway, for a finite number of iterations, the earlier observation is correct: the curve only covers rational coordinates. If you iterate longer, you can get arbitrarily close to irrational coords, but you never hit the bullseye.
But when we're talking about the end state "at infinity", we're no longer merely "arbitrarily" close to the target; we're infinitely close (again, loosely, "1/∞"). And if there are no infinitesimals in reals, that's necessarily synonymous with "equal to".
(Adding infinitesimals doesn't necessarily change the outcome because, as hinted earlier, infinities come in flavors. More about that in the aforementioned article.)
PS3, for the very angry person on Hackernews talking about "useless engagement bait" and "ignorance [being] a much more charitable assumption than malicious waste of time"... the article doesn't talk about the problem from the perspective of topological manifolds for two reasons:
1) We'd need a long segue into topology, non-metric spaces, and set theory to explain the topological approach (Lebesgue covering dimensions).
2) The outcome of that wouldn't be terribly useful for the main "gotcha" I wanted to talk about, which are space-filling curves and the problem of measuring the dimensionality of fractal shapes.
^ < What , and how many dimensional is my logo here a 2D projection of ?
I was recently asked to explain the difference between the schoolroom arithmetic and genuine mathematical thinking. This dimension puzzle acts as a vivid illustration: it shows what it means to fiddle with mathematical objects, and why a bit of algebra is a useful tool for this process, helping to structure the reasoning.
Nice!