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An Old Conjecture Falls, Making Spheres a Lot More Complicated
- xtajv 3y agoI'm terrible - one might even go so far as to say that the telescope conjecture has collapsed.
- e12e 3y agoI enjoy this way more pretending it's the prelude to a Philip K Dick or H P Lovecraft story - than trying to actually grasp the math :)
- alan-crowe 3y agoIf you enjoy the intersection of H P Lovecraft and mathematics you may enjoy https://www.hulver.com/scoop/story/2009/1/15/182727/390 https://www.hulver.com/scoop/story/2009/1/15/182727/390
- smaddox 3y agoLooking beyond the Lovecraftian mysticism, this is actually pretty fascinating in and of itself. I'm not a true mathematician (only a lowly electrical engineer with some training and study in mathematics), but I would not have expected that equality to hold for complex values, let alone quaternions, etc.
- datavirtue 3y agoThe first sentence should have been the ball-is-equal-to-egg explanation with mention of topology. Before that I had no idea what they were talking about. P.s. I have to assume the rules forbid shapes with surfaces of zero thickness. Otherwise I can just smash a ball into an inner-tube. If the shapes have thickness mandated, what is it? Are the thickness of the surfaces a consideration when morphing from one shape to another? Is the surface thickness negative or positive from zero? All of these questions stem from my experience in 3D modeling where these parameters must be defined.
- dullcrisp 3y agoThere is no thickness (or it’s zero if you like). The deformations have to be continuous mathematical functions, so punching a hole isn’t possible. The study is about the properties of (higher dimensional) shapes rather than concrete objects. It’s like asking what’s the thickness of a circle.
- wcarron 3y agoEasy, it's 0.38mm.
- datavirtue 3y agoIf it's zero I can make a doughnut from a ball without tearing.
- dullcrisp 3y agoYou can’t. The informal proof may not be very convincing but it’s that the torus has two circles that remain distinct no matter how you deform the space: the smaller and larger circles in this picture [1]. But on a sphere, every circle can be deformed to any other circle. If the torus were itself the deformation of a sphere, you’d be able to deform it the same way as the sphere to get one circle to the other. Again though, the version of these objects that mathematicians study is formalized such that this is unambiguous. [1] https://en.m.wikipedia.org/wiki/File:Tesseract_torus.png https://en.m.wikipedia.org/wiki/File:Tesseract_torus.png
- teraflop 3y agoNo, you can't. If you'd like an analogy from 3D modeling to see why not: Any polyhedral mesh has an integer called its "Euler characteristic", which is simply calculated by taking the number of vertices, subtracting the number of edges, and adding the number of faces. (V-E+F) Obviously, smoothly deforming a surface by moving vertices around doesn't change its Euler characteristic. A bit less obviously, any sequence of local refinements to "patches" of the mesh can't change its Euler characteristic either. (For example, splitting one face into smaller regions that are still connected to their surroundings in the same way.) Anything that you might reasonably call a "smooth" transformation will keep the Euler characteristic unchanged. You can convince yourself of this by experimentation with whatever 3D modeling software you like. But a spherical mesh has Euler characteristic 2, and a torus mesh has Euler characteristic 0. So no smooth deformation can transform one into the other. The only way to change the Euler characteristic would be to change the mesh topology itself, which would mean there's at least one pair of faces that are connected by an edge in one mesh and not connected in the other, which means the mesh has been "torn" along that edge. With a lot of math, you can extend this argument to arbitrary continuous surfaces, not just polygons. If two surfaces have different Euler characteristic, then you cannot find a bidirectional continuous mapping between them. Any such bijection must be discontinuous somewhere, which roughly means that arbitrarily close points are "torn apart" from each other.
- deleted 3y ago[deleted]
- iraqmtpizza 3y agodo they call them all spheres just to pretend that their work is relevant? I've heard from captain beyond that everything's a circle, but this is one step too far. a 100-dimensional non-uniform egg is not a sphere in any possible way. why is it not called an n-manifold or something like that
- Sniffnoy 3y agoI'm not sure what you're talking about. These are, in fact, n-dimensional spheres -- the set of points at unit distance from the origin in n+1 dimensions. (It's n+1 because, e.g., a sphere in 3 dimensions is intrinsically 2-dimensional.) An n-manifold would just mean any n-dimensional manifold. These are very particular n-dimensional manifolds, namely, spheres. Now of course, this is topology, so our equivalences are broad; but the thing these are all equivalent (homeomorphic) to is a sphere. Sure, you can take a more complicated shape that's equivalent to a sphere, but that complexity is incidental; the broad equivalences of topology let us ignore them. (Although, alternatively, they also let us turn the sphere into, say, a cube, if that's easier to think about, which often it is.)
- iraqmtpizza 3y agoOther than the article using the word sphere incessantly, I don't see how any of this is limited to spheres. I don't see even once how uniform distance plays into this except that the sphere is the simplest version of the sorts of things they're talking about. Your failure to banish my suspicions despite effort makes me that much more confident in my original conclusion. also a hypercube is not a cube--it's an n-cube. otherwise this is just lazy pop science rhetoric to get the kids excited about their field (and eventually suppress wages in mathematics with their newly-supplied labor, degree in hand). except not even science, so even less important I understand that these objects are topologically equivalent to n-spheres, but that doesn't make them n-spheres, let alone spheres proper. In fact, you point out that cubes and spheres are topologically equivalent despite zero spheres being cubes and zero cubes being spheres.
- kmill 3y ago
- Eduard 3y agowhat is this good for? please no knee-jerk 'this is pure mathematics, it doesn't need applicability' answers.
- kmill 3y agoThis is about understanding something about what goes on as you go rightward on the table at https://en.m.wikipedia.org/wiki/Homotopy_groups_of_spheres https://en.m.wikipedia.org/wiki/Homotopy_groups_of_spheres (under General Theory) There's an applications section in the Wikipedia article, but it's all to other parts of pure math. It's hard to summarize, but they've got to do with obstructions to untangling, unwrapping, or otherwise solving things to do with spaces.
- navels 3y agoWhen I was working in the field it sure as heck wasn't because of practical applications. The mathematics involved is beautiful.
- anArbitraryOne 3y agoJust like Feynman said: Physics is like watching internet porn. Sure, it may have useful results, like ad revenue, but that's not why we do it
- hollerith 3y ago>Feynman said: Physics is like watching internet porn. The Internet existed when Feynman died in 1988, but Internet porn did not, at least not watchable (video) Internet porn.
- anArbitraryOne 3y agoSurely, you must be joking
- plaguepilled 3y agoProving other theorems, which may themselves either prove further theorems or lead to direct applications. That's how the questions of "what to prove" often materialise.
- xeckr 3y ago>(think of a 100-dimensional sphere) Cheeky.
- Nevermark 3y agoSo … convex?
- unnah 3y agoAs the old joke goes: it's not hard at all, just think of an n-dimensional sphere and let n equal 100.
- quickthrower2 3y agoI though of a 100 dimensional vector in python with buzzing numbers
- billfruit 3y agoThe gossipy narrative style of the article is kind of jarring for an article on a topic like this. It took several paragraphs before it touched on the matter.
- delocalized 3y agoI always wonder what a popular science/math magazine would look like if it were oriented towards hackers. In this I mean people who have little background in the field but also the type of person who is used to bluntness and knows to RTFM. I would subscribe to one. Journal articles are often opaque to people who aren't already in the field, and popular science falls too often into the storytelling trap seen here.
- bawolff 3y ago> Journal articles are often opaque to people who aren't already in the field To be fair, manuals are often pretty opaque without the requisite background knowledge.
- smaddox 3y ago> I always wonder what a popular science/math magazine would look like if it were oriented towards hackers. In this I mean people who have little background in the field but also the type of person who is used to bluntness and knows to RTFM. Expensive.
- Nevermark 3y agoI would pay $1 per article whose title & byline interested me if I could count on the quality matching this [0] … … [0] https://news.ycombinator.com/item?id=37171553 https://news.ycombinator.com/item?id=37171553 (This comment reads faster with tail recursion.)
- bryanrasmussen 3y agoOK, assuming that there are enough people like you to make that a going concern now we just have to solve the problem of getting this level quality distributed through the population that wouldn't care for it enough to send it on to their friends, that is to say through the global network of humans with 1 in 10000 being one of the people willing to pay 1 dollar and the other 9999 people saying "what the hell, who cares" I think it may not work out.
- wwarner 3y agoWhat a great read. Exotic spheres are the dark energy of mathematics.
- dclowd9901 3y ago> It means that in very high dimensions (think of a 100-dimensional sphere) Sure, no problem, author.
- tomcam 3y agoI don’t know about you, but I start to get a little tentative at about 12 dimensions
- anArbitraryOne 3y agoEight, for me. Once you can fit packed spheres between each other in the norma "pyramid" configuration, my intuition breaks down. It should probably be true about four dimensions as well but I'm overconfident
- bryanrasmussen 3y ago>Infinitely more maps from spheres to telescopes means infinitely more maps between spheres themselves. The number of such maps is finite for any difference in dimension, but the new proof shows that the number grows quickly and inexorably. is it actually infinitely - or just a lot?
- r0uv3n 3y agoI think the article means that over all differences in dimension, the total number of missed maps is infinite.
- bryanrasmussen 3y agook - so what I'm wondering is what is the cardinality of missed maps? How big of an infinity is it?
- robinhouston 3y agoThe set of _all_ these maps is countable, so the number that were missed can only be countably infinite at most. (There are finitely many maps for each possible dimensional difference, and countably many possible dimensional differences.)
- pbhjpbhj 3y agoLost me at the end, but, don't inner-tubes have 2 holes (genus 2), topologically: one for inflation and one for the wheel to fit in. This makes them distinct from a torus (genus 1) and no homotopy exists between them. Clearly IANAM.
- superhuzza 3y agoThere is a distinction between a torus and a solid torus A torus is like an inner tube - an inner void and a big hole in the middle. A solid torus just has a big hole in the middle, like a donut. https://en.wikipedia.org/wiki/Solid_torus https://en.wikipedia.org/wiki/Solid_torus
- toth 3y agoYou probably are aware, but just to make it clear for others: topologically a torus has one hole (genus 1). The inner void is not considered a hole. Similarly, a 2-sphere (surface of solid sphere in 3d) has 0 holes (the inner void is not considered).
- adgjlsfhk1 3y agoto be slightly more technical what you're defining as "voids" are often considered 2-dimensional holes (as formalized by homology)
- toth 3y agoYou are right, but I am pretty sure they just meant the inner tube ignoring the puncture for inflation. Usually people just use donut/bagel for an illustration of a torus. Not sure why they used inner tube here - maybe to make it clear it is just the surface?
- shrx 3y agoA hollow inner tube with the puncture for inflation is topologically equivalent to a solid torus, if I understood correctly.
- ykonstant 3y agoThe following slides contain a more concrete description of the conjecture, its motivation and consequences: https://people.math.rochester.edu/faculty/doug/Talks/Glasgow-2022.pdf https://people.math.rochester.edu/faculty/doug/Talks/Glasgow...
- codeflo 3y agoI hate slide decks like these, where every page in the PDF contains one more bullet point than the last one. Maybe I'm particularly bad at this, but I spend way too long scanning each page for the new information. Is it impossible to configure LaTeX to only produce the final animation step as a completed page and skip the intermediate ones?
- cwzwarich 3y agoThe beamer class has a [handout] option, which at least attempts to do this (with some corner cases IIRC, but it's been a while).
- senkora 3y agoThe ideal way to read a deck like this, is to download the pdf and enter present mode on your pdf viewer so that you can click through each slide and immediately see where the new material has been added.