Dimensions 1, 8 & 24 have universally optimal configurations; 3 is a zoo
May 30, 2019 4:51 PM Subscribe
Prof. Maryna Viazovska, head of Number Theory at École polytechnique fédérale de Lausanne (EPFL), with co-authors Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko, recently proved that the E8 root lattice and the Leech lattice (Wikipedia x2) are universally optimal among point configurations in Euclidean spaces of dimensions 8 and 24, respectively ... which is a strong form of robustness not previously known for any configuration in more than one dimension (PDF, Arxiv.org), expanding upon Viazovska's prior solutions to 8- and 24-dimensional sphere packing, which had been studied since at least 1611 (Quanta Magazine).
More context on the recent publication from Quanta Magazine:
More context on the recent publication from Quanta Magazine:
The points could be an infinite collection of electrons, for example, repelling each other and trying to settle into the lowest-energy configuration. Or the points could represent the centers of long, twisty polymers in a solution, trying to position themselves so they won’t bump into their neighbors. There’s a host of different such problems, and it’s not obvious why they should all have the same solution. In most dimensions, mathematicians don’t believe this is remotely likely to be true.
But dimensions eight and 24 each contain a special, highly symmetric point configuration that, we now know, simultaneously solves all these different problems. In the language of mathematics, these two configurations are “universally optimal.”
The sweeping new finding vastly generalizes Viazovska and her collaborators’ previous work. “The fireworks have not stopped,” said Thomas Hales, a mathematician at the University of Pittsburgh who in 1998 proved (Princeton University, Annals of Mathematics) that the familiar pyramidal stacking of oranges is the densest way to pack spheres in three-dimensional space.
Eight and 24 now join dimension one as the only dimensions known to have universally optimal configurations. In the two-dimensional plane, there’s a candidate for universal optimality — the equilateral triangle lattice — but no proof. Dimension three, meanwhile, is a zoo: Different point configurations are better in different circumstances, and for some problems, mathematicians don’t even have a good guess for what the best configuration is.
I understand the words, and get the gist, but I don’t have a fundamental grasp of what is meant, mathematically, by “a sphere in the 8th dimension”.
A sphere is a three-dimensional object, yes? But somehow, it is mathematically projected, or extrapolated, or re-defined, into an 8th dimensional frame of reference. Which, again, because of my lack of mathematical intuition, is meaningless to me.
So, I’m applauding this theoretical breakthrough, but recognizing I don’t really understand why it’s so important.
posted by darkstar at 5:24 PM on May 30, 2019 [3 favorites]
A sphere is a three-dimensional object, yes? But somehow, it is mathematically projected, or extrapolated, or re-defined, into an 8th dimensional frame of reference. Which, again, because of my lack of mathematical intuition, is meaningless to me.
So, I’m applauding this theoretical breakthrough, but recognizing I don’t really understand why it’s so important.
posted by darkstar at 5:24 PM on May 30, 2019 [3 favorites]
I'll admit to being confused by what is under discussion here.
The quoted excerpt says "Thomas Hales ... proved that the familiar pyramidal stacking of oranges is the densest way to pack spheres in three-dimensional space".
Then it says, "Dimension three, meanwhile, is a zoo: Different point configurations are better in different circumstances".
The Quanta article says, in its headline, "A Ukrainian mathematician has solved the centuries-old sphere-packing problem in dimensions eight and 24."
There appear to be two different issues at stake, given what I am going to call the "contradictory" text on the third dimension, but I don't know what. Presumably sphere packing is a "circumstance". What then are some other "circumstances"...?
I ask this publicly for the edification of anyone else like me who, by this, is confused! P.S. I also searched on "universal optimality" and didn't find anything introductory on math.
posted by sylvanshine at 5:25 PM on May 30, 2019 [1 favorite]
The quoted excerpt says "Thomas Hales ... proved that the familiar pyramidal stacking of oranges is the densest way to pack spheres in three-dimensional space".
Then it says, "Dimension three, meanwhile, is a zoo: Different point configurations are better in different circumstances".
The Quanta article says, in its headline, "A Ukrainian mathematician has solved the centuries-old sphere-packing problem in dimensions eight and 24."
There appear to be two different issues at stake, given what I am going to call the "contradictory" text on the third dimension, but I don't know what. Presumably sphere packing is a "circumstance". What then are some other "circumstances"...?
I ask this publicly for the edification of anyone else like me who, by this, is confused! P.S. I also searched on "universal optimality" and didn't find anything introductory on math.
posted by sylvanshine at 5:25 PM on May 30, 2019 [1 favorite]
Sorry, now that I've made my joke, this is cool as hell. My math education and knowledge is just high enough to grasp the magic of the conclusion without being able to parse any of the the process they took to get there. But, certainly, the lay expectation would be that configuration/packing would only get more complex as dimensions increased, rather than getting all excited for a few dimensions and then chilling out again.
Am I correct in my recollection that none of the prominent comprehensive string theories posit that we secretly exist in 8 or 24 dimensional space?
posted by 256 at 5:27 PM on May 30, 2019 [1 favorite]
Am I correct in my recollection that none of the prominent comprehensive string theories posit that we secretly exist in 8 or 24 dimensional space?
posted by 256 at 5:27 PM on May 30, 2019 [1 favorite]
I don’t have a fundamental grasp of what is meant, mathematically, by “a sphere in the 8th dimension”
A circle is a two dimensional sphere. If you use the rather broken analogy of time as a fourth dimension, then a dot that grew into a sphere at a uniform speed and then receded back into being a dot would be a four dimensional sphere. In n-dimensional space, a sphere is just a contiguous body where every point on its surface is equidistant from its geometric centre.
posted by 256 at 5:31 PM on May 30, 2019 [10 favorites]
A circle is a two dimensional sphere. If you use the rather broken analogy of time as a fourth dimension, then a dot that grew into a sphere at a uniform speed and then receded back into being a dot would be a four dimensional sphere. In n-dimensional space, a sphere is just a contiguous body where every point on its surface is equidistant from its geometric centre.
posted by 256 at 5:31 PM on May 30, 2019 [10 favorites]
A sphere is a three-dimensional object, yes? But somehow, it is mathematically projected, or extrapolated, or re-defined, into an 8th dimensional frame of reference. Which, again, because of my lack of mathematical intuition, is meaningless to me.
My 8D math is rusty but first consider a circle (that is, a 2D sphere). Suppose it has radius 1 (a unit circle, in math parlance). It's the set of all points in the plain (aka 2D space) that are exactly one unit from the center. Similarly a unit sphere is the set of all points in 3D space (a,b,c) exactly one unit away from the center. So extrapolating up, and 8D (hyper-)sphere is just the set of all points described by a 8-variable vector (a,b,c,d,e,f,g,h) that are 1 unit away from the center.
posted by axiom at 5:33 PM on May 30, 2019 [13 favorites]
My 8D math is rusty but first consider a circle (that is, a 2D sphere). Suppose it has radius 1 (a unit circle, in math parlance). It's the set of all points in the plain (aka 2D space) that are exactly one unit from the center. Similarly a unit sphere is the set of all points in 3D space (a,b,c) exactly one unit away from the center. So extrapolating up, and 8D (hyper-)sphere is just the set of all points described by a 8-variable vector (a,b,c,d,e,f,g,h) that are 1 unit away from the center.
posted by axiom at 5:33 PM on May 30, 2019 [13 favorites]
Hyperspheres are the most boring hyper-object, because every projection of them onto three dimensions is a sphere.
The main subject here is beyond my understanding but sounds absolutely bonkers
posted by vibratory manner of working at 5:34 PM on May 30, 2019 [4 favorites]
The main subject here is beyond my understanding but sounds absolutely bonkers
posted by vibratory manner of working at 5:34 PM on May 30, 2019 [4 favorites]
Will it create luggage space?
posted by clavdivs at 5:37 PM on May 30, 2019 [12 favorites]
posted by clavdivs at 5:37 PM on May 30, 2019 [12 favorites]
Yes, but only for 8-dimensional oranges.
posted by darkstar at 5:39 PM on May 30, 2019 [16 favorites]
posted by darkstar at 5:39 PM on May 30, 2019 [16 favorites]
A sphere is a three-dimensional object, yes? But somehow, it is mathematically projected, or extrapolated, or re-defined, into an 8th dimensional frame of reference. Which, again, because of my lack of mathematical intuition, is meaningless to me.
Not sure if this will help, but think of a circle and sphere as instances of the same idea in 2- and 3-coordinate space: a locus of points at equal distance from a center. What that locus looks like geometrically depends on the type of space it's realized in, including the notion of distance pertaining to that space! In taxicab geometry, for example, a "circle" looks like a diamond (a rotated square); this is the region of points you can drive to within a fixed distance on a grid of roads. (I can't dig up a link right now, but the Walk Score site will draw you maps of where you can get to on transit in a fixed amount of time from a fixed starting point; these "circles" aren't even necessarily connected!) On a chessboard, a circle of radius 1 for a knight might consist of up to eight squares the knight can move to, and a circle of radius 2 would consist of all the squares it can reach in two moves.
A sphere in 8-dimensional space is a set of "points", which are really just 8-tuples of numerical coordinates, where distance is defined in a way analogous to the Euclidean distance formula. (That formula has a term with x's in it and a term with y's in it; imagine a version that looks similar, but with 8 terms referencing 8 coordinates.) Distance can be used as a measure of various things -- it might quantify how "different" two points are, or how many simple operations it takes to get from one to another (think of the knight on the chessboard). An example of where this comes up is in digital communications, where you want all possible "signals" to be different enough from each other that they can be distinguished even with a little bit of interfering noise. There, sphere packing translates to maximizing your alphabet of possible signals while maintaining the desired amount of contrast between them.
If you recognize that there are various uses for systems with eight variables, then you can appreciate why these ideas might be useful without necessarily being able to "see" it geometrically. But the powerful thing here is that these rather abstract notions of distance can be conceived of geometrically, giving extra insight to those who have learned the mental heuristics.
There appear to be two different issues at stake, given what I am going to call the "contradictory" text on the third dimension, but I don't know what. Presumably sphere packing is a "circumstance". What then are some other "circumstances"...?
It's a question of what you want to optimize. In classical sphere packing, you're trying to maximize the density -- essentially the number of spheres packed into a given amount of space -- but there might be problems where you want to optimize something else. An example in the article mentions optimizing configurations of particles so that energy is minimized (something nature does on its own, that we'd like to understand).
posted by aws17576 at 5:49 PM on May 30, 2019 [15 favorites]
Not sure if this will help, but think of a circle and sphere as instances of the same idea in 2- and 3-coordinate space: a locus of points at equal distance from a center. What that locus looks like geometrically depends on the type of space it's realized in, including the notion of distance pertaining to that space! In taxicab geometry, for example, a "circle" looks like a diamond (a rotated square); this is the region of points you can drive to within a fixed distance on a grid of roads. (I can't dig up a link right now, but the Walk Score site will draw you maps of where you can get to on transit in a fixed amount of time from a fixed starting point; these "circles" aren't even necessarily connected!) On a chessboard, a circle of radius 1 for a knight might consist of up to eight squares the knight can move to, and a circle of radius 2 would consist of all the squares it can reach in two moves.
A sphere in 8-dimensional space is a set of "points", which are really just 8-tuples of numerical coordinates, where distance is defined in a way analogous to the Euclidean distance formula. (That formula has a term with x's in it and a term with y's in it; imagine a version that looks similar, but with 8 terms referencing 8 coordinates.) Distance can be used as a measure of various things -- it might quantify how "different" two points are, or how many simple operations it takes to get from one to another (think of the knight on the chessboard). An example of where this comes up is in digital communications, where you want all possible "signals" to be different enough from each other that they can be distinguished even with a little bit of interfering noise. There, sphere packing translates to maximizing your alphabet of possible signals while maintaining the desired amount of contrast between them.
If you recognize that there are various uses for systems with eight variables, then you can appreciate why these ideas might be useful without necessarily being able to "see" it geometrically. But the powerful thing here is that these rather abstract notions of distance can be conceived of geometrically, giving extra insight to those who have learned the mental heuristics.
There appear to be two different issues at stake, given what I am going to call the "contradictory" text on the third dimension, but I don't know what. Presumably sphere packing is a "circumstance". What then are some other "circumstances"...?
It's a question of what you want to optimize. In classical sphere packing, you're trying to maximize the density -- essentially the number of spheres packed into a given amount of space -- but there might be problems where you want to optimize something else. An example in the article mentions optimizing configurations of particles so that energy is minimized (something nature does on its own, that we'd like to understand).
posted by aws17576 at 5:49 PM on May 30, 2019 [15 favorites]
Warning, pedantry ahead!
The ordinary vanilla sphere we all know and love is the 2-sphere. The surface of the Earth is a 2-sphere, its surface is two-dimensional, i.e. latitude and longitude are enough to specify a point. The complete Earth is a ball in 3-D.
A 3-sphere is the three-dimensional "surface" of a four-dimensional object (a 4-D ball). And a circle is a 1-sphere.
posted by phliar at 5:52 PM on May 30, 2019 [12 favorites]
The ordinary vanilla sphere we all know and love is the 2-sphere. The surface of the Earth is a 2-sphere, its surface is two-dimensional, i.e. latitude and longitude are enough to specify a point. The complete Earth is a ball in 3-D.
A 3-sphere is the three-dimensional "surface" of a four-dimensional object (a 4-D ball). And a circle is a 1-sphere.
posted by phliar at 5:52 PM on May 30, 2019 [12 favorites]
visualizing extra spacial dimensions is pointless since it's mostly impossible.
the best explanation i've come across is from 3blue1brown (whose other videos folks in this thread will probably REALLY enjoy)
posted by arbitrarycode at 5:53 PM on May 30, 2019 [10 favorites]
the best explanation i've come across is from 3blue1brown (whose other videos folks in this thread will probably REALLY enjoy)
posted by arbitrarycode at 5:53 PM on May 30, 2019 [10 favorites]
it will actually create vast amounts of luggage space because, while the unit cube (your hypothetical suitcase) has volume 1 in all higher dimensions, the volume of the sphere which sits inside it touching each of its faces, that volume approaches 0 as the dimension of the space increases without limit.
posted by jamjam at 5:59 PM on May 30, 2019 [1 favorite]
posted by jamjam at 5:59 PM on May 30, 2019 [1 favorite]
(Partially to answer my own question, I didn't realize there were two different Quanta articles. I clicked on the first one, but another one, which I didn't see, is the one quoted. The full article there seems to explain my question. I'd delete it but prob too late.)
posted by sylvanshine at 6:07 PM on May 30, 2019
posted by sylvanshine at 6:07 PM on May 30, 2019
it will actually create vast amounts of luggage space because, while the unit cube (your hypothetical suitcase) has volume 1 in all higher dimensions, the volume of the sphere which sits inside it touching each of its faces, that volume approaches 0 as the dimension of the space increases without limit.
But does this mean I will or won't have to pay to check a second bag now?
posted by Insert Clever Name Here at 6:09 PM on May 30, 2019
I think I'll wait for this to come out as a movie.
posted by dances_with_sneetches at 6:16 PM on May 30, 2019 [2 favorites]
posted by dances_with_sneetches at 6:16 PM on May 30, 2019 [2 favorites]
Sure, jamjam, but where would you put your luggage? Let's say your hypothetical hypersuitcase is being stored inside an n-dimensional ball. If the edge length of the cube remains fixed, the radius of the ball will increase without limit as the dimensions do. So: more luggage space inside, but it will never fit in the higher-dimensional overhead bin.
posted by phooky at 6:22 PM on May 30, 2019 [1 favorite]
posted by phooky at 6:22 PM on May 30, 2019 [1 favorite]
I was told we wouldn't have to do math.
posted by Joe in Australia at 6:25 PM on May 30, 2019 [5 favorites]
posted by Joe in Australia at 6:25 PM on May 30, 2019 [5 favorites]
Sphere packing is not only relevant for airline travel....
Error-correcting codes are pretty fundamental -- we just could not have modern electronic doo-dads without error-correction. So what it is error correction? If you need to send N bits over a noisy channel (all real-world channels are noisy), you would be rather put-out if any of those bits got changed... so what you do is send a few extra bits (usually called parity), cleverly calculated so that the receiver can recognise and correct some number of errors.
So you have an N-dimensional space of data, and you add M bits of parity -- now you have a M+N dimensional space of messages. There are 2^(M+N) points there, of which 2^M are the only valid ones (since a message has M bits). So we'd like to arrange 2^M points in that (M+N)-dimensional space such that those points are maximally separated. (Errors move the data-points around in the space.)
Therefore the fact that the 24-D Leech Lattice is amazingly well packed means there is a really excellent error-correcting code with M+N=24. This is the Golay code, and was used in communicating with Voyager. (N happens to be 4 -- it can correct 3 corruptions and detect an additional 4th.)
posted by phliar at 6:34 PM on May 30, 2019 [27 favorites]
Error-correcting codes are pretty fundamental -- we just could not have modern electronic doo-dads without error-correction. So what it is error correction? If you need to send N bits over a noisy channel (all real-world channels are noisy), you would be rather put-out if any of those bits got changed... so what you do is send a few extra bits (usually called parity), cleverly calculated so that the receiver can recognise and correct some number of errors.
So you have an N-dimensional space of data, and you add M bits of parity -- now you have a M+N dimensional space of messages. There are 2^(M+N) points there, of which 2^M are the only valid ones (since a message has M bits). So we'd like to arrange 2^M points in that (M+N)-dimensional space such that those points are maximally separated. (Errors move the data-points around in the space.)
Therefore the fact that the 24-D Leech Lattice is amazingly well packed means there is a really excellent error-correcting code with M+N=24. This is the Golay code, and was used in communicating with Voyager. (N happens to be 4 -- it can correct 3 corruptions and detect an additional 4th.)
posted by phliar at 6:34 PM on May 30, 2019 [27 favorites]
Error correcting codes are also related to the "snake in the box" problem which I'm not sure that's really relevant, I just love the name.
posted by Pyry at 6:48 PM on May 30, 2019 [5 favorites]
posted by Pyry at 6:48 PM on May 30, 2019 [5 favorites]
Well, duuuuuh.
posted by Saxon Kane at 6:50 PM on May 30, 2019
posted by Saxon Kane at 6:50 PM on May 30, 2019
My brother is a topologist. He once tried to explain to me the kinds of things he worked on. Rule number one: don't try to picture it.
posted by nickmark at 8:02 PM on May 30, 2019 [6 favorites]
posted by nickmark at 8:02 PM on May 30, 2019 [6 favorites]
Dimension three, meanwhile, is a zoo: Different point configurations are better in different circumstances, and for some problems, mathematicians don’t even have a good guess for what the best configuration is.
Probably this one.
posted by some little punk in a rocket at 9:43 PM on May 30, 2019 [1 favorite]
Probably this one.
posted by some little punk in a rocket at 9:43 PM on May 30, 2019 [1 favorite]
it will actually create vast amounts of luggage space because, while the unit cube (your hypothetical suitcase) has volume 1 in all higher dimensions, the volume of the sphere which sits inside it touching each of its faces, that volume approaches 0 as the dimension of the space increases without limit.
Oh, so a bag of holding? Why didn't they say that in the first place?
posted by Avelwood at 11:15 PM on May 30, 2019 [2 favorites]
Oh, so a bag of holding? Why didn't they say that in the first place?
posted by Avelwood at 11:15 PM on May 30, 2019 [2 favorites]
If you'll permit me a table . . . one of the interesting and even slightly bizarre things about moving up into higher dimensions is what happens to the relationship between the size of the sphere and the size of the cube just large enough to hold the sphere.
For example, in 2 dimensions you can think of a circle radius 1 (area = pi = 3.14) and the square that just fits over that circle (length of side=2, so area = 2x2 = 4). So you can see the circle is a little smaller in radius and there are four of these little "shoulder" areas where the square has area that is not inside the circle.
So the 2-sphere is almost as big as the 2-cube.
In 3 dimensions, a sphere of radius one has volume about 4.19 while the cube that just encloses it has volume 8 (side length is 2, so volume is 2x2x2). So . . . now there are EIGHT little "shoulders", one for each corner of the cube, where the cube has some volume that the sphere doesn't.
So the 3-sphere is barely HALF as big as the 3-cube.
So you get the idea that as you keep going up the dimensions, there will be even more of those "shoulders" and, say, the 6-sphere or 21-sphere will probably be quite a bit smaller than the corresponding "cube" that just encloses it.
But--below is the continuation of this calculation up to 24. What happens is even more dramatic:
And . . . it only gets worse from there.
posted by flug at 11:41 PM on May 30, 2019 [24 favorites]
For example, in 2 dimensions you can think of a circle radius 1 (area = pi = 3.14) and the square that just fits over that circle (length of side=2, so area = 2x2 = 4). So you can see the circle is a little smaller in radius and there are four of these little "shoulder" areas where the square has area that is not inside the circle.
So the 2-sphere is almost as big as the 2-cube.
In 3 dimensions, a sphere of radius one has volume about 4.19 while the cube that just encloses it has volume 8 (side length is 2, so volume is 2x2x2). So . . . now there are EIGHT little "shoulders", one for each corner of the cube, where the cube has some volume that the sphere doesn't.
So the 3-sphere is barely HALF as big as the 3-cube.
So you get the idea that as you keep going up the dimensions, there will be even more of those "shoulders" and, say, the 6-sphere or 21-sphere will probably be quite a bit smaller than the corresponding "cube" that just encloses it.
But--below is the continuation of this calculation up to 24. What happens is even more dramatic:
- The volume of the n-sphere peaks at just over 5 and then it actually starts to get smaller (!).
- The volume of the n-cube that just encloses it continues to grow exponentially as a power of 2.
- Pretty soon the n-sphere is just a tiny, tiny little speck of dust in the middle (mostly, sort of) of the n-cube that just barely encloses it.
TABLE - Dimension vs n-Sphere vs n-Cube VolumesBy the time you get to dimension 24, the 24-cube has about 8.7 billion times the volume of the 24-sphere it just barely encloses.
Column 1 = dimension
Column 2 = volume of the n-sphere of that dimension, radius 1
Column 3 = volume of the n-cube of that dimension that can just barely enclose the n-sphere of radius 1
1 2.00 2
2 3.14 4
3 4.19 8
4 4.93 16
5 5.26 32
6 5.17 64
7 4.72 128
8 4.06 256
9 3.30 512
10 2.55 1024
11 1.88 2048
12 1.34 4096
13 0.91 8192
14 0.60 16384
15 0.38 32768
16 0.24 65536
17 0.14 131072
18 0.08 262144
19 0.05 524288
20 0.03 1048576
21 0.01 2097152
22 0.01 4194304
23 0.00 8388608
24 0.00 16777216
And . . . it only gets worse from there.
posted by flug at 11:41 PM on May 30, 2019 [24 favorites]
"Apparently Maryna Viazovska gave a series of talks on this at IHES" [1,2,3,4,5,6] i wonder what chalk she uses :P
also btw, apparently, some progress on the riemann hypothesis!
posted by kliuless at 12:12 AM on May 31, 2019 [1 favorite]
also btw, apparently, some progress on the riemann hypothesis!
posted by kliuless at 12:12 AM on May 31, 2019 [1 favorite]
Another fairly weird fact about the volumes of the unit spheres (spheres of radius 1) is, that if you add up the volumes of all the unit spheres in even dimensional spaces, which means summing an infinite series, it does have a finite value, and that value is eπ.
posted by jamjam at 12:25 AM on May 31, 2019 [13 favorites]
posted by jamjam at 12:25 AM on May 31, 2019 [13 favorites]
Halp! I'm trapped inside a porcupine. It's dark, cold, and vast yet very cramped and lonely.
posted by zengargoyle at 4:07 AM on May 31, 2019 [1 favorite]
posted by zengargoyle at 4:07 AM on May 31, 2019 [1 favorite]
Dimension three, meanwhile, is a zoo: Different point configurations are better in different circumstances, and for some problems, mathematicians don’t even have a good guess for what the best configuration is.
My uneducated guess: this is a feature, not a bug. We live in the most interesting dimension.
posted by Termite at 5:26 AM on May 31, 2019
My uneducated guess: this is a feature, not a bug. We live in the most interesting dimension.
posted by Termite at 5:26 AM on May 31, 2019
You don’t want to fly with 8-dimensional luggage because a single day’s change of clothes will put you over the weight limit.
posted by ardgedee at 6:34 AM on May 31, 2019
posted by ardgedee at 6:34 AM on May 31, 2019
So the surface of a 1-dimensional sphere is two points, I guess, and the whole volume of it is a line segment.
Spin that segment around its center and you form a 2-dimensional circle.
Spin the circle around its center (on another axis) and you form a 3-dimensional sphere.
Spin the sphere around its center on... another... axis...? and my brain hurts
Do this enough times and you have CASE NIGHTMARE GREEN
posted by Foosnark at 7:15 AM on May 31, 2019 [2 favorites]
Spin that segment around its center and you form a 2-dimensional circle.
Spin the circle around its center (on another axis) and you form a 3-dimensional sphere.
Spin the sphere around its center on... another... axis...? and my brain hurts
Do this enough times and you have CASE NIGHTMARE GREEN
posted by Foosnark at 7:15 AM on May 31, 2019 [2 favorites]
A lot of your more progressive, and smaller-handed, metal bands are clutching invisible 8-dimensional oranges these days.
posted by Wolfdog at 7:28 AM on May 31, 2019
posted by Wolfdog at 7:28 AM on May 31, 2019
The great thing about having an 8-dimensional orange is you can cut it up into a couple standard 3-dimensional oranges and you have a 2-dimensional orange left over, otherwise known as an orange slice, which I suggest further cutting in four and using to garnish a lovely set of mixed drinks.
posted by meinvt at 7:35 AM on May 31, 2019 [1 favorite]
posted by meinvt at 7:35 AM on May 31, 2019 [1 favorite]
> the best explanation i've come across is from 3blue1brown yt (whose other videos folks in this thread will probably REALLY enjoy)
I've been reading Matt Parker's book (Things to Make and Do in the Fourth Dimension) to/with my son - it's fantastic! I really recommend his videos on the Numberphile youtube channel. In particular, here's a Matt Parker video that addresses packing spheres. (“There is a trick you can use in mathematics called… not worrying about it.”)
posted by RedOrGreen at 8:51 AM on May 31, 2019 [1 favorite]
I've been reading Matt Parker's book (Things to Make and Do in the Fourth Dimension) to/with my son - it's fantastic! I really recommend his videos on the Numberphile youtube channel. In particular, here's a Matt Parker video that addresses packing spheres. (“There is a trick you can use in mathematics called… not worrying about it.”)
posted by RedOrGreen at 8:51 AM on May 31, 2019 [1 favorite]
If you'll permit me a table . . . one of the interesting and even slightly bizarre things about moving up into higher dimensions is what happens to the relationship between the size of the sphere and the size of the cube just large enough to hold the sphere.
I thought you were going to talk about Escaping Spheres, which is bizarre in just the opposite way!
posted by aws17576 at 10:15 AM on May 31, 2019 [2 favorites]
I thought you were going to talk about Escaping Spheres, which is bizarre in just the opposite way!
posted by aws17576 at 10:15 AM on May 31, 2019 [2 favorites]
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posted by 256 at 5:22 PM on May 30, 2019 [7 favorites]