Strange Spheres in Higher Dimensions - Numberphile
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Date: 2022-04-08
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Comments and reviews: 9
John
Wait. In the 3rd dimension, the distance between 2 spheres diagonally is no longer the radius of the sphere but the radius of a cut somewhere lower than halfway the sphere, so the actual radius can be bigger.
So it not that the sphere is getting spikier. its just that the radius in 3 dimensions is only a cut at a point that is -lower- than halfway in 4d, so again there's a more optimal place for the radius to be centered in. The 4d sphere is still equidistant at all points from the centre.
Just as a 3d sphere no matter how you cut it with a 2d plane always looks like a circle, the same is the case for 4d, no matter how you cut it out in a 3d frame, it will always appear as a sphere.
So actually, it's not that spheres are spikey in higher dimensions, it's that spheres become less efficient to stack in higher dimensions allowing bigger gaps for other spheres.
The same way that 2D spheres are more efficiently stacked, than 3d spheres.
I think the experiment doesn't actually suggest the end conjecture.
In fact all it seems to tell me is 4d and higher d spheres are actually so smooth in a way, that they leave even more space wasted.
Comparing the ratio of area of a square to a circle vs the ratio of volume of cube to a sphere maybe shows the same thing?
I'm too lazy to do it right now: b
reply
Wait. In the 3rd dimension, the distance between 2 spheres diagonally is no longer the radius of the sphere but the radius of a cut somewhere lower than halfway the sphere, so the actual radius can be bigger.
So it not that the sphere is getting spikier. its just that the radius in 3 dimensions is only a cut at a point that is -lower- than halfway in 4d, so again there's a more optimal place for the radius to be centered in. The 4d sphere is still equidistant at all points from the centre.
Just as a 3d sphere no matter how you cut it with a 2d plane always looks like a circle, the same is the case for 4d, no matter how you cut it out in a 3d frame, it will always appear as a sphere.
So actually, it's not that spheres are spikey in higher dimensions, it's that spheres become less efficient to stack in higher dimensions allowing bigger gaps for other spheres.
The same way that 2D spheres are more efficiently stacked, than 3d spheres.
I think the experiment doesn't actually suggest the end conjecture.
In fact all it seems to tell me is 4d and higher d spheres are actually so smooth in a way, that they leave even more space wasted.
Comparing the ratio of area of a square to a circle vs the ratio of volume of cube to a sphere maybe shows the same thing?
I'm too lazy to do it right now: b
reply
WereElf
You can imagine 4d spheres as spheres, that change their radius as time passes. One can draw this setup in 4d by doing a clip of the 3d model with the spheres changing their radius appropriately. The interesting thing is that when the surrounding spheres are at their maximum radius (1, the middle one's radius is 0. And vice versa - when the middle one's radius is equal to 1, the other spheres have a radius of 0.
At first I tried to draw few frames of this setup in 2d, but I noticed that the circles were overlapping. But when I moved to 3d - the problem fixed itself: P
At no point of time do I imagine higher dimensional spheres as -spiky- tho. And 6d is the last dimension, that can be somewhat represented in a 3d engine in a somewhat understandable/organized manner. Perhaps someone can push their luck with a 7d representation, if they make 3d animations, but it will be a clutter xP
So seeing the -spiky- spheres is a long shot: (
reply
You can imagine 4d spheres as spheres, that change their radius as time passes. One can draw this setup in 4d by doing a clip of the 3d model with the spheres changing their radius appropriately. The interesting thing is that when the surrounding spheres are at their maximum radius (1, the middle one's radius is 0. And vice versa - when the middle one's radius is equal to 1, the other spheres have a radius of 0.
At first I tried to draw few frames of this setup in 2d, but I noticed that the circles were overlapping. But when I moved to 3d - the problem fixed itself: P
At no point of time do I imagine higher dimensional spheres as -spiky- tho. And 6d is the last dimension, that can be somewhat represented in a 3d engine in a somewhat understandable/organized manner. Perhaps someone can push their luck with a 7d representation, if they make 3d animations, but it will be a clutter xP
So seeing the -spiky- spheres is a long shot: (
reply
Koen
N-spheres are of course not really spiky, they are perfectly symmetric wrt any rotation axis, plane (or other linear subspace) going through their centre.
The mean distance of a random point within an n-dimensional sphere to the centre is n/(n+1) times their radius, so with increasing dimension, n-balls 'consist more and more' of their outer regions.
Here's a nice puzzle: An n-dimensional sphere is inscribed has a circumscribed n-dimensional cube (such that it touches all its walls. It also has an inscribed cube all of whose vertices all touch the sphere.
How many times would the smaller fit in the bigger cube? For which n is this a rational number? What is this number for n=100? Don't google it!
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N-spheres are of course not really spiky, they are perfectly symmetric wrt any rotation axis, plane (or other linear subspace) going through their centre.
The mean distance of a random point within an n-dimensional sphere to the centre is n/(n+1) times their radius, so with increasing dimension, n-balls 'consist more and more' of their outer regions.
Here's a nice puzzle: An n-dimensional sphere is inscribed has a circumscribed n-dimensional cube (such that it touches all its walls. It also has an inscribed cube all of whose vertices all touch the sphere.
How many times would the smaller fit in the bigger cube? For which n is this a rational number? What is this number for n=100? Don't google it!
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Jussi
I hope I calculate everything right.
A cube that inscribes four unit diameter spheres would be 2-3 = 8 in volume.
A unit diameter sphere is Pi/6 - d-3 = 0, 5236 and all four are therefore 2, 0944 in volume.
8 - 2, 0944 = 5, 9056 which is the remaining space in the cube.
A sphere that encloses that much volume would be 2, 24262 diameter.
is there actually a dimension (possibly imaginary, non-integer etc) where the volume of the central sphere equals the remaining space of the cube?
And would one of those spiky 3+ dimensional spheres possibly be like a spherically concaved diamond shape with its tips where the packing spheres kiss?
reply
I hope I calculate everything right.
A cube that inscribes four unit diameter spheres would be 2-3 = 8 in volume.
A unit diameter sphere is Pi/6 - d-3 = 0, 5236 and all four are therefore 2, 0944 in volume.
8 - 2, 0944 = 5, 9056 which is the remaining space in the cube.
A sphere that encloses that much volume would be 2, 24262 diameter.
is there actually a dimension (possibly imaginary, non-integer etc) where the volume of the central sphere equals the remaining space of the cube?
And would one of those spiky 3+ dimensional spheres possibly be like a spherically concaved diamond shape with its tips where the packing spheres kiss?
reply
ratoim
A book of math riddles had the question how a rider could legally enter a bus with a 2 meter fishing rod when the maximum dimension of any carry-on was 1. 5 meters. The solution was to place the pole in a shallow box of dimension 1. 5 meters on two sides, thus adhering to the rules. The solution further elaborated that the pole could be placed in a cubical box of 1. 2 meters on a side, again adhering to the rules. It then concluded the rider could carry a pole of any length desired as long as the rider could procure a legally sized box in high enough dimensions.
Not too difficult to pack away a sphere in the same manner.
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A book of math riddles had the question how a rider could legally enter a bus with a 2 meter fishing rod when the maximum dimension of any carry-on was 1. 5 meters. The solution was to place the pole in a shallow box of dimension 1. 5 meters on two sides, thus adhering to the rules. The solution further elaborated that the pole could be placed in a cubical box of 1. 2 meters on a side, again adhering to the rules. It then concluded the rider could carry a pole of any length desired as long as the rider could procure a legally sized box in high enough dimensions.
Not too difficult to pack away a sphere in the same manner.
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Kashish
I have a question. Can you help?
can you plz help me with problem where I am trying to find 2 points closest to a point say 'a' in a multi-dimensional data space where there are clusters of multi-dimensional data points
I am trying to make a version of k-nearest neighbour with weights for each data point
I want u to imagine it like in 3D space bunch of clusters of points are floating and each cluster is a group so when I put a point in that space an expanding sphere comes out of that point and stops as it comes across 2 data points of different groups, I just want to know how can it be done in multiple dimensions
reply
I have a question. Can you help?
can you plz help me with problem where I am trying to find 2 points closest to a point say 'a' in a multi-dimensional data space where there are clusters of multi-dimensional data points
I am trying to make a version of k-nearest neighbour with weights for each data point
I want u to imagine it like in 3D space bunch of clusters of points are floating and each cluster is a group so when I put a point in that space an expanding sphere comes out of that point and stops as it comes across 2 data points of different groups, I just want to know how can it be done in multiple dimensions
reply
Rick
It's not that hard to imagine actually. You see each sphere is kissing in 1 axis. In 2d the sphere kisses in x & y. In 3d the sphere kisses in x, y & z. So in 4d the spheres kisses x, y, z & w. If we look at how the spheres in 3d fits in from a 2d perspective, we see that the spheres are overlapping. In fact, due to how our eyes work we never seen the full picture on how the spheres fits inside in 3d. So 4d there will be even more overlapping because they have an extra dimension to fit in. That's why the sphere becomes larger.
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It's not that hard to imagine actually. You see each sphere is kissing in 1 axis. In 2d the sphere kisses in x & y. In 3d the sphere kisses in x, y & z. So in 4d the spheres kisses x, y, z & w. If we look at how the spheres in 3d fits in from a 2d perspective, we see that the spheres are overlapping. In fact, due to how our eyes work we never seen the full picture on how the spheres fits inside in 3d. So 4d there will be even more overlapping because they have an extra dimension to fit in. That's why the sphere becomes larger.
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Zion
I reconcile this in my mind by looking at how in 2D, it's impossible to put a circle of any size into the box because of the padding blocks the outside completely, but in 3D, there is a little bit of space on the sides, so it logically follows that the sides of higher dimensional cubes would have even more space open, and the sphere only has to grow toward the faces, orthogonally, whereas the padding spheres expand away from it toward the ever-furthering corners (i. e. sqrt(2, sqrt(3, sqrt(4, etc.
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I reconcile this in my mind by looking at how in 2D, it's impossible to put a circle of any size into the box because of the padding blocks the outside completely, but in 3D, there is a little bit of space on the sides, so it logically follows that the sides of higher dimensional cubes would have even more space open, and the sphere only has to grow toward the faces, orthogonally, whereas the padding spheres expand away from it toward the ever-furthering corners (i. e. sqrt(2, sqrt(3, sqrt(4, etc.
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Funkenstein
It seems pretty easy to demonstrate that there are no higher dimensions. Just find two spheres that can't touch each other on any side. It works when trying to prove three dimensions in a two dimensional world, afterall, seeing as how a sphere is just different sizes of circles depending on where its intersecting.
So, since higher-dimension spheres should be interacting with us in some fashion at different stages of its orientation, it should be pretty easy to find such spheres, no?
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It seems pretty easy to demonstrate that there are no higher dimensions. Just find two spheres that can't touch each other on any side. It works when trying to prove three dimensions in a two dimensional world, afterall, seeing as how a sphere is just different sizes of circles depending on where its intersecting.
So, since higher-dimension spheres should be interacting with us in some fashion at different stages of its orientation, it should be pretty easy to find such spheres, no?
reply
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