[HN Gopher] A high-dimensional sphere spilling out of a high-dim... ___________________________________________________________________ A high-dimensional sphere spilling out of a high-dimensional cube Author : EvgeniyZh Score : 204 points Date : 2022-01-17 17:05 UTC (5 hours ago) (HTM) web link (stanislavfort.github.io) (TXT) w3m dump (stanislavfort.github.io) | emsign wrote: | There's a key difference between how we perceive and think about | two dimensional space and 2+n dimensional space. If you want to | seal in one geometric object with other geometric objects like in | this example then in 2D it's sufficient to let the enclosing | objects touch each other at their widest points, but in higher | dimensions they have to touch each other at their narrower | points, overlapping into each other in many cases. And in the | first example which was the 2D example the outer objects touch | each other at their outmost points. In 2D it looks like a full | enclosure with no way for the inner circle to get through. But | our brains assume that the higher dimension versions will be 100% | sealing it off as well, because we saw it in the first example, | right? | | This is an example of a misleading first example. What actually | will happen and what you expect will happen are not the same, | because you got tricked by the first example. | smegsicle wrote: | and you can see the inner circle is much smaller than the | surrounding circles in 2d, the difference is a little less so | in 3d? which would continue until the so-called 'inner' circle | is friggn _huge_. | | I mean, everyone knows that everything is super far apart in | higher dimensions, it surprises me that it takes up to 10D | before he slips out-- perhaps more intuitive when i keep in | mind that n-spheres are the most compact shape? | scatters wrote: | I don't think it's correct to see the n-sphere as poking out of | the sides of the cube; the sphere has constant curvature so it | has no protrusions. Rather, the n-cube gets more and more spiky | so the constraining spheres get (relatively) smaller and further | away from the centre, so constrain the central sphere less and | less. | | The spikiness of n-cubes is apparent when you look at the (solid) | angle at each vertex; it starts constant (1/2 pi radian in 2D, | 1/2 pi steradian in 3D) but reduces thereafter at an increasing | rate (1/8 pi^2, 1/12 pi^2, 1/64 pi^3, etc.). | scythe wrote: | The radius of the sphere, assuming the cube has side length 1, | is (sqrt(D)-1)/2. It seems worthy of note that this radius | becomes arbitrarily large. | kgwgk wrote: | And the distance from a vertex of the unit cube to the center | is sqrt(D)/2, which also becomes arbitrarily large. | | https://www.math.ucdavis.edu/~strohmer/courses/180BigData/18. | .. | | > The result, when projected in two dimensions no longer | appears convex, however all hypercubes are convex. This is | part of the strangeness of higher dimensions - hypercubes are | both convex and "pointy." | AnimalMuppet wrote: | Isn't it more accurate to say that hypercubes are convex | and _their projections_ are pointy? | [deleted] | kgwgk wrote: | Yes. Hence the quotes, I guess. But hyperspheres are | convex and their projections are also convex. So it's not | really appropriate either to say that they are "spiky" as | in https://news.ycombinator.com/item?id=29969613 | thaumasiotes wrote: | > the (solid) angle at each vertex; it starts constant (1/2 pi | radian in 2D, 1/2 pi steradian in 3D) | | Unstudied curiosity: steradians seem to be defined in terms of | the point of a cone. But the corner of a cube is the | intersection of several (for a cube, 3) planes, not a cone. How | do you do the calculation of steradians? | wlesieutre wrote: | Cone is one scenario to think about, but what it really comes | down to is the fraction of the full sphere that your solid | angle includes. | | To make a 2D analogy, you can think of 2D angles as a portion | of a circle represented as a fraction of 2pi radians. Cut it | in half and you have pi radians, cut that in half (so a | quarter of the circle) and you have pi/2 radians, or 90 | degrees. | | That 90 degrees "quarter of a circle" example is looking at | it as the "point of a pie slice", but it's the same 90 degree | 2D angle as you have in the corner of a square. | | You can look at the corner of a cube the same way. The full | sphere of solid angle is 4pi, a hemisphere is 2pi. Now take | that hemisphere and cut it into quarters (1/8ths of a | sphere). Each of those quarters is pi/2 steradians, and the | solid angle at the center is the same solid angle represented | by 1/8th of the sphere is the same solid angle you have at | the corner of a cube. | | Or to put it another way, you could pack the corners of 8 | cubes around a point and it would leave no empty gaps, so the | corner of each cube is occupying 1/8th of a "full" 4pi | steradians. | thaumasiotes wrote: | Thanks, that was helpful in thinking about it. | shenberg wrote: | The spiky view of n-spheres is due to all of their volume | getting concentrate next to the axes - the proportion of volume | which is less than e away from an axis to the entire volume of | the sphere tends to 1 exponentially with n, which leads to the | spiky visualization as we imagine 3d but with more axes. | 6gvONxR4sf7o wrote: | The n-cube is spiky. The n-sphere is kinda defined by its | non-spikiness, because it is rotationally invariant. | kgwgk wrote: | But there is an alternative visualization where the sides of | the unit (hyper)cube are deformed because the location of the | vertices gets further and further away from the origin. Of | course neither is a true representation of the | hyperdimensional case. | mgraczyk wrote: | I don't think this is the right way to process this | intuitively. When you use words like "volume getting | concentrated" it sounds like there is some non-uniformity in | the sphere, but the non-uniformity is really in our intuition | about space. | | What's weird isn't the sphere, it's distance, and I think | that's easier to process. Going from a (1d) sidewalk to a | (2d) football field to a (3d) ocean, it's easier to see our | intuitions about distances slowly breaking down. | vba616 wrote: | >it sounds like there is some non-uniformity in the sphere | | How I am understanding this is that the non-uniformity is | in the cube, and I think it's very helpful in visualizing | it. | | I can imagine in 3d, the centers of the sides of the cube | being pulled in, so that it's kind of hyperbolic looking. | sdwr wrote: | Love this explanation, visualizing corner vs centre is | intuitive in a way higher dimensions usually aren't. | tgb wrote: | I'm not certain I know what you mean, but it doesn't seem to | work. If you mean to take epsilon neighborhoods in Euclidean | norm of the axes, then I'm skeptical that that contains most | of the mass of the sphere, since simulating random points on | a thousand dimensional sphere doesn't seem to give points | near these axes. | | If instead you mean most of the mass of the ball is in the | points which have all but one coordinate within epsilon of | zero, then the intuition doesn't follow. It's equally true | that you get most of the mass considering just points with | _all_ coordinates within epsilon. And for that you get that | the mass of the ball is concentrated in an epsilon cube at | the origin, which excludes exactly the spikes that you were | basing the intuition on. The weird thing in this thought | experiment is the epsilon cube not the sphere. For example | the epsilon cube contains points much further than epsilon | from the origin and so it 's maybe not surprising that it | contains most of the sphere. | trhway wrote: | >since simulating random points on a thousand dimensional | sphere doesn't seem to give points near these axes. | | that is kind of circular argument as "random" really | depends on the density measure underlying the chosen | sampling distribution. | kgwgk wrote: | If only there was some kind of natural density measure to | use in an Euclidean space R^D... | tgb wrote: | It's no more ill defined than volume itself (and less so | since it doesn't need an arbitrary scale). | momenti wrote: | https://news.ycombinator.com/item?id=3995615 | tgb wrote: | Perhaps you can be more specific since I don't see the | claim here being discussed there. | ColinWright wrote: | This phenomenon has been written up many times, and some of those | have been submitted here previously, with some discussion. For | those who might be interested to see those previous discussions, | here are two of them: | | https://news.ycombinator.com/item?id=12998899 | | https://news.ycombinator.com/item?id=3995615 | | Some of the comments here were made in those discussions, but | some of the comments on those discussions have not yet been made | here. | capableweb wrote: | > Some of the comments here were made in those discussions, but | some of the comments on those discussions have not yet been | made here. | | Many of the missing comments from those linked submissions have | also not been made here yet. Hope we'll see them soon. | it_does_follow wrote: | To add to this list: Richard Hamming includes a section on this | in his n-dimensional spaces talk from "The Art of Doing Science | and Engineering" lectures [0]. Stripe Press also recently re- | published a beautiful copy of the print version of these | lectures [1]. | | 0. https://youtu.be/uU_Q2a0S0zI?t=1716 | | 1. https://press.stripe.com/the-art-of-doing-science-and- | engine... | growt wrote: | At university I initially chose math as a minor. I think it was | this problem (without the pretty pictures) where I decided that | math was not for me. | rwmj wrote: | Just think of the high dimension n-cube like a spiky sea urchin. | It has 2^n spikes, and the spheres live in those spikes near the | ends. The central sphere is large because it extends out to those | spheres, extending outside the sea urchin's "body". | feoren wrote: | But ... it's not. It's not concave anywhere. If you draw a line | from any point of the n-cube to any other point, it never | passes outside the body of the cube. Perhaps your model gives | better intuition in "curse of dimensionality" cases like this | one, but it's clearly worse in other ways, right? It's simply | not at all an accurate description of the shape. | thadk wrote: | Doesn't Alicia Boole Stott's ability indicate that solid | intuitions are plausible though? | https://www.askaboutireland.ie/reading-room/life- | society/sci... | feoren wrote: | Maybe? I'm not claiming there's no way to have a good | intuition about 4D space -- in fact articles like this make | me want to figure out how to achieve such a thing. But it | seems likely to me that even if your brain is somehow | capable of visualizing 4D things, it would be just as weird | to move to 5D as it is for normal people to move to 4D. Did | Stott have any special intuition about 5+D? And we're | talking about making that cognitive jump _five_ more times | to get to 10D. | | However, it's clear the "starfish" intuition is simply not | accurate. That's not what N-cubes look like. The point of | this post is that we _should_ have cognitive dissonance | when we try to think about 10-cubes, because it 's _weird_ | that (A) the "inner sphere" pokes out of a shape that is | (B) convex everywhere. You can resolve the cognitive | dissonance easily by simply ignoring or rejecting B -- | sure, it's not weird that such a sphere would poke out of a | starfish. But _you are wrong_. It 's _not_ a starfish! It | 's convex everywhere! So you can't say "why do y'all have | cognitive dissonance about this?" | sdwr wrote: | It is accurate, just not completely accurate. You only | get cognitive dissonance if you try to resolve it all the | way.. stack multiple imperfect intuitions to approximate | the real thing. | wruza wrote: | It depends on how you define "concaveness". A cube is | concaver than a square _in a sense_. The travel from the | center of a unit square to its side takes 0.5, and to its | vertex ~0.7. For a cube, 0.5 and ~0.87. For a 100d cube, 0.5 | and 5. (For completeness, for 1d cube it 's 0.5, 0.5). Of | course that is not a true concaveness, but it gives a nice | sense of inside distances, especially with spheres, which are | defined as equidistant. | | We are just used to project a cube in a way that prevents to | see its linearly-spatial configuration (a projection messes | up lengths), but if you preserve these lengths and "flatten" | them instead, a cube will flatten out to a sort of a | shuriken. | function_seven wrote: | > _It depends on how you define "concaveness". A cube is | concaver than a square in a sense._ | | Yup. Even a regular 3D cube (and 2D square) has concave | faces if you're viewing it from a polar perspective. As I | stand in the center of the cube, measuring distances from | me to the surface, I'll see that measurement follow a | concave pattern. | | Yeah, I know that's not the definition of concavity or | whatever, but when relating a sphere to a cube, and trying | to get an intuition of higher-dimensional spaces, I think | it helps to look at it from the sphere's perspective rather | than a Cartesian one. | brummm wrote: | There is a very clear definition of convexity, as can be | seen here [1] on Wikipedia. Nothing to discuss about with | regards to definition. | | [1] https://en.wikipedia.org/wiki/Convex_set | wruza wrote: | Maybe there is a better word to describe this idea? | sdwr wrote: | It works if you aren't rigid about your mental | representations. Pull an inverse wittgenstein on the | intuitions. Instead of stool, chair, recliner all being | instances of the general category "chair" -- sea urchin, | cube, dodecahedron are all partially accurate descriptions of | the specific 10D cube. | 6gvONxR4sf7o wrote: | There's a certain way in which a cube "feels sharper" or | "feels spikier" than a square. Trying to formalize that, you | can compare the edge of a 3d box where two faces meet to the | point where three faces meet. I'd rather step on the two-face | edge than the three-face corner, and there's definitely a | sense in which the cube is spikier. | | It seems reasonable to extend that same intuition to n-D | sharpness/spikiness in an accurate way. Adding an extra | "face" just chops more off making those vertices sharper and | sharper, at least relative to the high dimensional space | around it. | feoren wrote: | I think that's a great insight; I especially like comparing | the sharpness of an two-face edge to a three-face corner; | we could expect stepping on a seven-face corner to be | slightly worse than stepping on a six-face. I think that | "sharpness" idea surely must be related to this phenomenon. | However, one should be careful not to let that "increasing | sharpness" idea lead to the mental image of a concave | shape, especially not a sea urchin. That would be a false | resolution to this "paradox" and that image of a N-cube | would lead to all sorts of other incorrect ideas, e.g. | regarding where the volume of the shape is. | 6gvONxR4sf7o wrote: | Maybe the right intuition is a sea urchin that is, due to | high dimensional unintuitive properties, still convex. | It's almost entirely extremely pointy corners, and yet | they're "magically" connected to each other in an | entirely convex manner. | fghorow wrote: | Is there some obvious way (that I seem to be missing) to see that | the centrally inscribed D-sphere _must_ touch all of the other | spheres in high dimensions? | | That's probably a stupid question, but while that fact is | intuitively obvious for D={2,3} -- as this problem tries to | demonstrate -- higher dimensions are unintuitively WEIRD. | SamReidHughes wrote: | Symmetry. | fghorow wrote: | OK, Another reference is [1], that agrees with the result given | by the OP. | | I'm not trying to do research here, I'm just boggling at the | unintuitive result, and trying to see if there might be a flaw | in the chain of logic. The fact that this is "well known" is | enough to scare me off from barking up this particular tree. | | [1] https://www.math.wustl.edu/~feres/highdim | nealabq wrote: | You can argue by symmetry that if the central sphere touches | one of the corner spheres, it must touch them all. And it must | touch one because otherwise you would increase it's radius | until it did. | Dylan16807 wrote: | Well we can calculate the touch point for one sphere, and we | know that it would overlap that sphere if it was radius | sqrt(D). | | And all the other spheres are simple symmetrical mirrors, so | how could it not touch all of them at the same time it touches | one? That should scale to an arbitrary number of dimensions, | right? | cantagi wrote: | I also found it very weird, but here's my intuition. | | There are 2^k > 512 spheres stuck to eachother across k-d | (pretend k=9). The line from the center to the point where the | inner sphere touches one of outer spheres has to shortcut | through all k dimensions to get from the center to the sphere. | | This distance has been massively inflated due to the number of | dimensions. But the distance to the edge of the box hasn't been | inflated - it's just constant, so the inner sphere breaks out. | spenczar5 wrote: | Substep 6 is not obvious to me. It is not clear that the point | where the inner sphere and the space-filling spheres intersect | _must_ be along the line from the center of the cube to the | center of each sub-cube. | | To put it another way, it's not obvious to me that the point of | contact between the inner pink circle and the outer black circle | is along the green line. | jjgreen wrote: | Both spheres are symmetric around this line, so a point on | intersection anywhere but on the line would give you a circle | of intersection (and so a disk, by convexity) ... | spenczar5 wrote: | How do you know they are both symmetric around this line? The | rest of your argument makes sense to me, yeah. | unnah wrote: | The spheres are symmetric around the line because the line | goes through the center points of the spheres. | hervature wrote: | Not obvious for higher dimensions or even D=2? Surely you agree | the center of both circles (regardless of dimension) occur | along the line from the center of the cube to one of the | corners. Therefore, if you just radially grow these spheres | they must touch for the first time along this line. To be | clear, this is nothing special about the cube, if you draw a | line between the center of any two circles, the first time they | touch will be somewhere along this line. In this case, the | inner circle is obviously along the diagonal and it doesn't | take much to see that the outer circles are as well by their | construction. Therefore, the diagonal is the line that connects | the centers. | spenczar5 wrote: | I don't agree that the center of both circles needs to lie | along that line. The space-filling ones do by definition, but | I don't see why the center one has to be on that line. It | seems like it must in D=2, of course, but I couldn't prove | that it must for D=9, or even that it is unique. | ted_dunning wrote: | The line goes from the center of the cube to the corner. | The central sphere is concentric with the cube. Therefore | the center of the central sphere is on the line from the | center to the corner. In fact, it the center of that sphere | is on any line from the center of the cube to anywhere. | jeeceebees wrote: | I think this is a property spheres. It seems to me that any two | spheres that are touching have a straight line from one center | to the other center exactly through the point of contact. Try | thinking of just two spheres and adding more in step-by-step. | | Then the result follows because all the spheres are defined as | centered on the cube/sub-cubes respectively. | spenczar5 wrote: | The inner sphere is not defined as centered on the cube; it | is defined as touching all the other spheres. | | That said, there is a symmetry argument that if it were | centered anywhere else, something is wrong. But that only | works if there is only one _unique_ sphere that touches all | the other spheres, which is also not obvious to me in higher | dimensions. | Dylan16807 wrote: | You can go ahead and define it as centered on the cube. | That still demonstrates the strange nature of high- | dimensional spheres even if there wasn't a unique solution | for touching all the other spheres. | spenczar5 wrote: | Aha! Right, this is pretty convincing to me. Thanks! | hprotagonist wrote: | relatedly, euclidean distance is a shitty metric in high | dimension. | AnimalMuppet wrote: | How so? And, what is a _better_ metric, and why? | hprotagonist wrote: | cosine similarity's a lot better. | | https://stats.stackexchange.com/questions/99171/why-is- | eucli... | CrazyStat wrote: | Cosine similarity is not a distance metric. | contravariant wrote: | Well it is on the unit sphere, but then it's equivalent | to the euclidean metric... | srean wrote: | Do I spot a geometer here ? You are indeed right but its | not something that is well known. | anon_123g987 wrote: | For example: | | _The Mahalanobis distance is a measure of the distance | between a point P and a distribution D, introduced by P. C. | Mahalanobis in 1936. It is a multi-dimensional generalization | of the idea of measuring how many standard deviations away P | is from the mean of D. This distance is zero for P at the | mean of D and grows as P moves away from the mean along each | principal component axis. If each of these axes is re-scaled | to have unit variance, then the Mahalanobis distance | corresponds to standard Euclidean distance in the transformed | space. The Mahalanobis distance is thus unitless, scale- | invariant, and takes into account the correlations of the | data set._ | | https://en.wikipedia.org/wiki/Mahalanobis_distance | Dylan16807 wrote: | If all your dimensions are equal to each other then it | gives the same result as Euclidean distance? I don't think | this counts as better, then. | CrazyStat wrote: | Mahalanobis distance is just a way of stretching Euclidian | space to achieve a certain sort of isotropy (it normalizes | an ellipsoid to the unit sphere). It is built on top of | Euclidian distance and is not an alternative to it. | anon_123g987 wrote: | Euclidian distance works well in 2D and 3D as special | cases. I would say Mahalanobis distance is its | generalization (yes, built on top of it), which works | better in the multidimensional (multivariate) case. | srean wrote: | Mahalanobis distance isn't that different from euclidean | distance at all as far as effects of dimensions is | concerned it just applies a stretch, rotation or more | accurately a linear transformation to the space. | | In short, much that I love Mahalanobis distances' many | properties it does zilch for dimensionality. | CrazyStat wrote: | No. Mahalanobis distance is not an alternative to | Euclidian distance because it's not even measuring the | same kind of distance. The are incommensurate, both | figuratively and literally: Mahalanobis distance is | unitless while Euclidian distance is not. | | Euclidian distance measures the distance between two | points, while Mahalanobis measures the distance between a | distribution (canonically multivariate normal) and a | point. Mahalanobis distance is not a generalization if | Euclidian distance, it's an altogether different concept | of distance that doesn't even make sense without talking | about a distribution with mean and covariance matrix. | anon_123g987 wrote: | What has more seeds, an apple or a fruit? | CrazyStat wrote: | What's a better fruit, an apple or an apple pie? | | Like Mahalanobis distance, apple pie is not a fruit and | is not a generalization of an apple. | srean wrote: | I agree about your larger relevant point but the | following that you say is bit of a red herring | | > Euclidean distance measures the distance between two | points, while Mahalanobis measures the distance between a | distribution (canonically multivariate normal) and a | point | | In a discussion about metric and metric spaces we dont | care about those things, its abstracted out and | considered irrelevant. All that matters is that we have a | set of 'things' and a distance between pairs of such | things that satisfies the properties of being a distance | (more precisely, properties of being a metric). | | @CrazyStat (I cannot respond to your comment so leaving | it here) | | I think you overlooked | | > things that satisfies the properties of being a | distance (more precisely, properties of being a metric). | | that I wrote. Of course it has to satisfy the properties | of being a metric. The red herring, as far as | dimensionality is concerned, is the complaint that | Mahalanobis is defined over distributions while Euclidean | is over points. | | The part about MD that you get absolutely right is its | nothing but Euclidean distance in a space that has been | transformed by a linear transformation. MD (the version | with sqrt applied) and ED aren't that different, | especially so in the context of dimensionality | | @CrazyStat response to second comment. | | It indeed isnt, its just Euclidean distance under linear | transformation. I was just quoting you, you had said | | > while Mahalanobis measures the distance between a | distribution | | My point was even it is defined for distributions its not | really relevant. | | > Mahalanobis "distance" is more closely related to a | likelihood function than to a true distance function. | | That's a subjective claim, and open to personal | interpretation. Mathematically MD is indeed a metric | (equivalently a distance) and it does show up in the log | likelihood function. Mahalanobis was a statistician, but | MD is a bonafide distance in any finite dimensional | linear space, with possible extensions to infinite | dimensional spaces by way of a positive definite kernel | function (or equivalently, the covariance function of a | Gaussian process) | CrazyStat wrote: | MD isn't defined over distributions, though. There are | perfectly good distance metrics defined over | distributions, but MD isn't one of them. It's a | "distance" between one distribution and one point, not | between two distributions or two points. | | Mahalanobis "distance" is more closely related to a | likelihood function than to a true distance function. | CrazyStat wrote: | A function that measures the distance between two | different classes of 'things' (distribution and point, in | this case) is necessarily not a distance metric. It | trivially fails to satisfy the triangle inequality, | because at least one of d(x,y), d(x,z), d(y,z) will be | undefined--no matter how you choose x, y, z you'll end up | either trying to measure the distance between two points | or the distance between two distributions, neither of | which can be handled. | | This is not a red herring, it's a fundamental issue. | amitport wrote: | Which distance would you chose? And to what purpose? | nealabq wrote: | For the orthoplices, the kissing sphere in the middle pokes | through the facets in dimension 12. | | I don't know when that happens with the simplices. I assume the | middle sphere pokes thru before it does with the hypercubes, | since the simplices are pointier. | gjm11 wrote: | I think that isn't the case. The simplices are pointier, which | means that the "corner spheres" don't go so far into their | corners. | | If I've done my calculations right, putting n+1 n-spheres in | the corners of an n-simplex with unit sides so that they're | tangent to one another gives them a radius of 1 / | (sqrt(2n(n+1)) + 2), and then if you put a sphere in the middle | tangent to all those it has radius [sqrt(2n/(n+1)) - 1] times | this. | | So for very large n, the "corner spheres" have radius of order | 1/n, and the "centre sphere" has radius about sqrt(2)-1 times | the radius of the "corner spheres", and both of these -> 0 as n | -> oo. | | (But! "If I've done my calculations right" is an important | condition there. I make a lot of mistakes. If you actually care | about the answer then you should check it.) | woopwoop wrote: | Vitali Milman apparently drew high-dimensional convex bodies as | "spiky" to try to get at this intuition. So the n-dimensional | cube in this case would look like a starfish, with the balls | inscribed in the subdivided cubes way out in the tentacles. When | you draw it this way, of course the middle ball is not contained | in the cube (it is hard to reason precisely about this picture, | it does not encode an obvious precise analogy). | jjgreen wrote: | It is disturbing to find one's intuition failing in higher | dimensions. | nabla9 wrote: | It fails in multiple ways. | | For example the volume (hypervolume) gets concentrated close to | the surface of the sphere when dimension grows. For example, if | you have symmetric multidimensional probability distributions | around the zero it becomes weird. | kgwgk wrote: | What does become weird? | malux85 wrote: | exciting* | Jeff_Brown wrote: | Yes! Let's list more! | | I've got a couple, maybe (depends on your intuition i guess): | | In 4d a (topological) sphere can be knotted. | | Hyugens's principle: When a wave is created in a field in | N-dimensional space, if N is even, it will disturb an ever- | expanding region forever (think a pebble hitting a pond's | surface) whereas if N is odd the wavefront will propagate | forever but leave no disturbances in its wake (think of a | flashbulb, or of someone shouting in an infinite space full of | air but no solids to echo off of). | datameta wrote: | Potentially a typo in there - one of the cases must be odd? | Jeff_Brown wrote: | You were right, it's fixed now, thanks! | smegsicle wrote: | i can see why you can't tie a circle in a knot, but why can't | you tie a regular sphere into a knot? | Jeff_Brown wrote: | I should perhaps have said "a (topological) sphere can _be_ | a knot ". Corrected, thanks! | contravariant wrote: | > In 4d you can tie a sphere in a knot. | | Arguably you can do that in 3D, if you accept the horned | sphere as a knot. Though I suppose that does raise the | question of what you are willing to call a sphere. | | Regardless it's an embedding of a sphere that cannot be | deformed into a unit sphere so I think the analogy holds. | feoren wrote: | Measure a group of humans on N traits and take the individual | average of each trait. For surprisingly small N (think | 10-ish, but obviously depending on your group size), it's | highly likely that no human in your group (or even in | existence) falls within 10% of the average in every trait. | This is roughly equivalent to the statement that less and | less of the volume of an N-sphere is near the center as N | increases. | | Sometimes called "the flaw of averages". Of course I learned | about this from another HN post recently: | | https://www.thestar.com/news/insight/2016/01/16/when-us- | air-... | jjgreen wrote: | You want a list? https://mathoverflow.net/questions/180846/ | Jeff_Brown wrote: | aw hells yes | codegladiator wrote: | https://www.youtube.com/watch?v=mceaM2_zQd8 | macilacilove wrote: | In lower dimensional projections the inner sphere will be so big | that it overlaps with the smaller spheres. So much so that it | intersects with the enclosing cube too. It will not develop the | weird spike protein things like in the illustration in any | projection. | f0xtrot wrote: | I have my doubts that anything changes by adding another | dimension, does the distance really change from 2d to 3d? I'm no | good at activley imagining larger dimensions than that tho. | | None the less, it's a fun thought excercise! Thanks | h2odragon wrote: | I understand some of these words. Then there's another voice | saying "this explains turbulence" but it can't explain further. | dan_mctree wrote: | "This means that as the dimension grows, the central sphere will | grow in radius," | | Am I missing something? It to me that only r*D grows as we | increase the dimension, not r by itself. Since we don't seem to | get r > 2a for any D, I don't really get the conclusion that it's | sticking out of the cube | phantasilide wrote: | Does anyone have further references to pieces that examine this | property? I read the linked paper from Strogatz which showed a | similar geometry for the basins of coupled oscillators. Though it | was interesting it did not provide more insight. | TuringTest wrote: | These are the alien geometries on other dimensions that lead to | madness in Lovecraft's work, right? | legohead wrote: | Do dimensions even really exist? | srean wrote: | What do you mean by exist ? Do real numbers exist ? | 0x264 wrote: | Oh yes, they do :) | legohead wrote: | Don't know why I got downvoted, it was a serious question. | | Can you give an example of how we've made use of the 4th, | 5th, 6th dimension, etc.? | [deleted] | gorloth wrote: | Higher dimensional geometry can show up in lower | dimensional problems. This numberphile video | (https://www.youtube.com/watch?v=6_yU9eJ0NxA) involves a | puzzle about throwing darts at a dart board which is solved | by using the volumes of 4+ dimensional spheres. | f0xtrot wrote: | doesn't 4th just include time with a 3d object. Would | things like waves/pulses fall under that? 5th seems to be | used in the Kaluza-Klein theory[0] for gravitation and | electromagnetism. | | [0] | https://en.m.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory | | edit: I agree, I don't know why you got down voted. It was | a thought provoking question. | cs702 wrote: | Yes. As the number of dimensions n increases above 3, the | interior angles at each vertex of an n-cube get smaller and | smaller[a], while the n-cube's hypervolume gets more and more | concentrated near its hypersurface, which is itself a | _hypervolume_ of n-1 dimensions.[b] | | Lines and areas are different animals; we cannot reason about | them apples-to-apples; we know this intuitively. Areas and | volumes are different animals too; we cannot reason about them | apples-to-apples; we know this intuitively. Similarly, | n-dimensional objects and (n+1)-dimensional objects are different | animals; we cannot reason about them apples-to-apples. | | As human beings, we find it so difficult to reason "visually" | about higher dimensional spaces, in part, I believe, because our | puny little brains have spent a lifetime learning to model three | dimensions (with a fourth dimension, time, flowing only in one | direction). | | -- | | [a] See this comment by scatters: | https://news.ycombinator.com/item?id=29969181 | | [b] See this old thread for intuitive explanations about how and | why this happens with n-spheres as we increase the number of | dimensions n: https://news.ycombinator.com/item?id=15676220 | denton-scratch wrote: | Back in the 70's, Martin Gardner published an article in his | Mathematical Games column in the SciAm, about visualising | rotating hyperspheres and hypercubes. The way I remember it, his | imaginary friend Dr. Morpheus (or something) had shown him colour | animations of these rotating objects. There were a couple of | stills in the article. The 4D objects were of course projected | down to 2D. | | I've played with an animation of a wireframe hypercube that you | could rotate around different axes. It was quite mind-boggling. | But Gardner particularly raved about the mind-altering effect of | viewing a rotating hypersphere. I've always wanted to view that, | but I never heard anything about it again. It seems to me that a | 2D projection of a hypersphere must look to all intents and | purposes like a sphere. | | Does anyone know where that article might be archived? Or where I | can view an animation of a rotating hypersphere? | strgrd wrote: | https://www.youtube.com/results?search_query=rotating+hypers... | Lots of animations come up on YouTube. ___________________________________________________________________ (page generated 2022-01-17 23:00 UTC)