5 ms·
How many dimensions is this?
- swayvil 1y agoWhy do people disable zoom on their page?
- magackame 1y agoWdym? Can zoom just fine on desktop.
- brettermeier 1y agoYou cannot zoom on mobile. That really is lame.
- omnicognate 1y agoI'm on mobile (iOS Safari) and seem to be able to zoom fine.
- robin_reala 1y agoI’d suggest it’s a problem with your browser that it respects the zoom restrictions, regardless of the page’s choices. Browsers are meant to be user agents – agents for the user – not agents for the page.
- cvoss 1y agoFirefox > Settings > Accesibility > Zoom on all websites. Many browsers have similar settings.
- tzury 1y agoone of the most precious resources on the internet about dimensions is here https://www.youtube.com/playlist?list=PL3C690048E1531DC7 https://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 despite "outdated" animation (15 years ago), still a great resource.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.
- xg15 1y ago> 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 https://en.wikipedia.org/wiki/Manifold
- dazzaji 1y agoNot 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.
- HelloNurse 1y agoThe 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).
- jerf 1y agoI 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 1y agoBut 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.
- lordnacho 1y agoIt'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.
- IanCal 1y agoThere's a load of things I think are things many kids could do much earlier in maths, though my personal feeling may be off. My son seems to be taking very quickly to the feeling of maths so I'm trying to share some parts that I only realised drastically later on. Rules you learn in maths are often based on much simpler, truer (I'll get back to this), things. I learned loads of "move this denominator up to this side, like an escalator!" and such and only when I was something like 13 I nearly shouted "EQUALS MEANS THE TWO SIDES ARE THE SAME THING" as I realised why so many of these god damn rules existed. It also explained suddenly why some things are OK to do and some aren't (notably taking roots). The other key thing was finding out that these things are decisions. We can choose whatever rules we want and see what happens, and keep whatever is useful. Lots of the more interesting things I've seen later in life have been "well there's no number you can square and get a negative number. What if there was?" or "What if there was a way to raise to a real number, what would that mean?". There's no magical reason we can't divide by 0 but there's not a neat useful answer really, but you can totally define a setup where you can do that and have fun if you want. Imaginary numbers and Argand diagrams are things my 6 year old can mess about with. He's still developing reasoning so certain things trip him up about seeing the process for solving a thing (fascinating to see where his comprehension is up to and the leaps that happen, I remember the first time seeing him pick up a chess piece, go to move it and before putting it anywhere say "no, if I do that you'll do X") but lots of things are very accessible. Powers are fun because you can make big things quickly, factorials too. Basic solving of equations I was only taught in secondary school yet I was taught "two bananas cost 30p and one banana and one apple costs 40p, how much is one apple" years before and that's just swapping fruit for letters. I was always good at maths, but I think it's far more fun now I've understood more about what you can mess about with and how simple parts are at times.
- wodenokoto 1y agoI'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.
- praptak 1y agoYeah, it is not entirely obvious. You need to prove that if you fix a number x from range [0,1] then the point at "x * length(Hn)" converges to a point in 2D, for every x, as n goes to infinity (Hn = nth Hilbert curve). The limit for x is the mapping from x to a point in 2D. The proof is by observing that the subsequent "jumps" are exponentially smaller.
- isoprophlex 1y agoIf you take the length of the entire curve as a number l irrespective of the iteration depth, you can say you are at position 42% of l. Because at iter n, the segments are replaced by a new shape that does not grow beyond the bounds of the segments at n-1, the iterations are 'contractive', and iterating more deeply only makes the x, y location of 'i am at 42%' more precise.
- arutar 1y agoHere's another way that I like to think about it. First, forget briefly about the Hilbert curve and just think about the unit square [0,1]^2. If you take any point (x,y) in the unit square, we can associate x and y with binary coordinates, say x = 0.a1a2a3... and y = 0.b1b2b3... Then we can just define a new number z with binary representation 0.a1b1a2b2a3b3... And going the other way, given z in [0,1], we can take the 'even' binary coordinates to get x, and the 'odd' binary digits to get y. The problem with this specific mapping is that the function is not continuous. But if you are a bit more careful: 1. The first digit says "left half vs right half" 2. The second digit says "top half vs bottom half" (of the rectangle from 1) 3. The third digit says "left half vs right half" (of the square from 2) etc. and then if two numbers share the first n binary digits (i.e. your points are close on the real line) then the corresponding points in the plane will also be quite close together (they are inside the same square / rectangle with side length like 2^(-n/2) at step n). The "reason" why the dimension is different is precisely because of the "n/2": for every n digits of precision you have in the number z, you only get n/2 digits of precision for each of (x, y). This is a bit imprecise because of issues with non-unique binary representation but (at least for me) it captures the spirit of why this should work!
- Patryk27 1y agoThis was also covered by 3blue1brown, highly recommend: https://www.youtube.com/watch?v=gB9n2gHsHN4 https://www.youtube.com/watch?v=gB9n2gHsHN4.
- sans_souse 1y agoSeveral.
- arutar 1y agoOnce 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'. [1]: https://en.wikipedia.org/wiki/Koch_snowflake https://en.wikipedia.org/wiki/Koch_snowflake [2]: https://en.wikipedia.org/wiki/Weierstrass_function https://en.wikipedia.org/wiki/Weierstrass_function [3]: https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%9... [4]: https://en.wikipedia.org/wiki/Heisenberg_group https://en.wikipedia.org/wiki/Heisenberg_group [5]: https://en.wikipedia.org/wiki/Assouad_dimension https://en.wikipedia.org/wiki/Assouad_dimension
- bronlund 1y agoA line has one dimension, and one only. It couldn't care less about whatever you do to it in higher dimensions. That's their problem.
- bronlund 1y agoWhy the downvotes? This is the correct answer to the first question raised: "...how many dimensions does a line have?". You can track a point to create a line, you can shift the line to make a plane, you can move the plane to create a space, you can change the space to create time, you can observe time to create bliss, you can reflect on bliss to create thought, you can use thought to create an idea and you can use that idea to make a thingy. You can create as many dimensions you like, all perpendicular to each other - one way or another. But I promise you this; even though the line is needed for the thingy, the line is blissfully unaware of this fact.
- nh23423fefe 1y agoYou seem to want to be "correct" instead of engaging with the article. Do you think Minkowski was ignorant?
- bronlund 1y agoI'm not criticising Minkowski, I'm criticising the author of the article. He starts out by talking about one dimensional lines, then suddenly jumps to curves as if they could be mistaken for the same thing. And in my humble opinion, it goes downhill from there. He even manage to talk about one dimensional squiggles which is an oxymoron. In one dimension, nothing is squiggly - that is an effect of higher dimensions. That being said, I don't care much for Einstein's relativity or the derived works either. I think Maxwell was on to something and that quantum mechanics more or less agrees with him.
- pavel_lishin 1y agoI 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.
- desertrider12 1y ago> 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.
- non_aligned 1y agoThe 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.
- desertrider12 1y agoWell, consider my ass bitten ;) Your analogy to infinite series helps.