You are in luck, there's 4D game that allows you to do exactly that! It's called 4D Golf, you can find it on Steam, and there's a devlog series about it here on RU-vid. It's mainly about, well, golf, but there are other things to do in it. One of the levels in the game, which is themed as a 4D art gallery, contains a statue of exactly this problem! That is, 16 balls in the corners of a tesseract, with a center ball. You can walk around in 4D and look at various 3D cuts of it. There's also another game about 4D visualization called 4D Toys, although it doesn't contain this particular math problem.
@@user-sl6gn1ss8p It is a good way, but it is specific to this problem was my point, so rather than visualizing the dimensions it visualizes what is relevant for this specific problem, but sure there are many other problems where you can find a good cut that would visualize well
For me, the main insight was realizing the four blue spheres don't touch when viewing the 2d-cross-section of the 3d model (and therefore there's more space for the inner red sphere in 3d than there was for the inner red disc in 2d)
@@gershommaes902 yeah, that and the idea that this will have to do with the ratio of the diagonal compared to the side, which will go up as dimensions go up
This is actually the premise behind Principal Component Analysis (PCA), a popular dimensionality reduction technique which finds the largest variation amongst all dimensions and reconfigures the data as those axes. This can be understood as the long 'diagonals' of your data.
ohh wait so just to confirm the 2d cuts, she takes from each higher-n shape is basically the 2d cut with the biggest variation (difference) from our original 2d cut in the 2d realm....? And it's not necessarily that the a similar cross-sectional cut like our original 2d cut exist in the higher realms, but we're just focusing on the cross-sectional cuts that give the biggest variation from that original cut.
It is conceptually similar, sure. But here because of symmetry all eigen(/singular)values would be equal and all directions would actually qualify as cuts that explain the same maximal variance. Maybe methods that incorporate higher moments, like Independent Component Analysis or Factor Analysis, would be better suited. That is just a detail though.
Is it? I don't understand how she is getting a 2D slice of a 4+D shape. I need to see the cut she's making as an animation of 3D slices of the 4D shape.
@@xtremeninja6859 not necessarily, it depend on your definition. here, her "slice" are 2D object. She didn't explain it fully, but what she did is taking the view of the plan formed by one of the edge and the center of the sphere inside the hypercube, so it really is a 2D object.
Wish I could see the 3D demicube rotated and the tesseract realmically sliced for some more perspectives. And maybe some other cross-sections of the dekaract if we're feeling crazy enough
Really great visual! The volume ratio between the red ball and blue balls peaks at Dimension 4 and then drops. Diameter of red ball is sqrt(D) - 1 Thus red balls volume is proportional to (sqrt(D) -1)^3 The blue balls all have the same volume, but their number grows exponentially 2^D with dimension. Kinda fun even if red ball grows without bound, its volume compared to blue balls quickly goes to zero. In higher dimension space most volume is close to boundary.
That makes sense. Spheres have smaller volumes than cylinders of the same diameter and length (a projection of a 2d circle into 3d) So, it intuitively makes sense that: Hyperspheres of higher dimensions have smaller volumes than equal diameter hyperspheres of lower dimensions. Then, what you're saying is that: The diameter of the red ball grows without bound, but more slowly than diameter/volume grows (after an initial rapid diameter growth). I wouldn't have guessed that that's the case on my own, but it certainly seems intuitively plausible.
Another way: the red sphere's diameter is the n-dimensional hypotenuse which bisects the three spheres, minus the blue spheres' diameter, divided by 2: (√(w^2 * n) - w)/2. How's my math?
I don't know if this necessarily helps me visualize the high-dimensional itself. But it does kind of solidify the understanding of the weird volume aspects that happens with higher dimensional geometry. Good visual 💯
The hypercube has a dense set of vertices that spike out far away from the cube's center. Exactly the description of a spiky object. In contrast the hypersphere is anything but spiky. It's extremely smooth as all it's points are an equal distance from the center.
Nah it is good I don't deny it but 3Blue1Brwon is legendary man look at his videos explaining wave , butterfly effect , those are one of the craziest videos you would ever see in internet... and he is doing it since prob 6-8 years or even more I don't know..
Kidding. If you did get it. You wouldn’t be honest. No one can actually “get it” because “it” isn’t even within our comprehension. And sorry but dimensions don’t follow our 3d math equations. Because they’re in 3d. And use math. And we don’t fucking know
I think 3Blue1Brown did a video on it too, and he explained it there. This happens because as you add more dimensions, the vertices of the (hyper)cube get farther and farther apart diagonally but the edges stay the same length, allowing more space to open up in the middle. As in the blue balls stay the same size but the red ball can occupy more space with higher dimensions and thus gets bigger. Crudely, blue ball diameter = 1/2 in every dimension. red ball diameter = length of diagonal Using Pythagoras' Theorem, In 2D, diagonal = sqrt(1+1) = sqrt(2) In 3D, diagonal = sqrt(1+1+1) = sqrt(3) In 4D, diagonal = sqrt(1+1+1+1) =sqrt(4) .......................and so on As you can see, the diagonal keeps getting longer but the edges stay the same length.
The video just presents a basic tool to build on top of when imagining higher dims. There, affine geometry is more useful. That focuses on conservation of angles while delaying the need for consistent or known basis & lengths.
This is a very good explanation. Another way to "feel" the extra space. Could be looking at the space in the corners of the blue balls. In the 2d slices it not only grows bigger. But also that these grow in number with the power of 2 with each extra dimension. 4 in 2d. 8 in 3d. 16 in 4d. 1024 in 10d. Correct me if I am wrong.
To start, you can imagine 1d as a line with two blue sections and a red in between. 0.5d is half of that. A short red section, then a blue section, and then the empty section.
I think it's a mistake that whenever we try to visualize higher dimensions we tend to use the square, cube, hypercube, etc. when we could instead use the triangle, tetrahedron, hypertetrahedron, etc. because these contain just the minimum information needed to form the simplest shapes at every dimension. Less vertex, less lines, less sides... and when you notice the patterns of increment from one dimension to the next, it becomes more intuitive to visualize higher dimensions. At least it works for me. Ive been playing with numbers and geometric figures for a while trying to visualize this
Any shape works, but the square/cube/tesseract/etc. is the simplest for the sake of understanding because it can be considered as being perfectly lined up with the dimensional directions. A square has two dimensions, and going from a single corner along any edge is directly in line with one of those dimensions while being entirely orthogonal to the other. A cube's corner has a third edge that goes entirely orthogonal to those previous two, perfectly representing the next dimension. And so on.
but you left out the coolest detail! When you get to juuuuust the right number of dimensions, you can pack a whole extra set of spheres in there and the empty space drops dramatically
Great video, thank you. It might be worth mentioning that the length of the diagonal is sqrt(n), so it's easy to understand why the distance between the spheres on the diagonal gets larger and larger.
Indeed this is actually quite similar to some things I am working on with a theory on what I call "perspectivity". Though I have not dabbled in sphere packing, you have definitely strengthened my resolve and given me more validation. Thank you for this video. I completely believe as Feynman would teach, that revisiting the basic foundations after gaining knowledge in the more complex aspects of maths, is what can lead to a more refined understanding of the fundamentals of maths. I also believe that the basics need reworking. Blanks that aren't even seen need light, however sometime like with dark matter light isn't how to truly perceive things. That's just my perspective lol
2:10 You could turn 4 blue spheres invisible so you could see that the projection of the other spheres in the xy plane form a red circle overlaping the blue circles. In other words, the extra dimension allows the sphere to be greater without overlaping.
The areas of each element essentially warps per dimensional shenanigan - as each iteration continues the curve of whatever trajectory this triggers expands particular areas, and compresses others. We get this out in space to as the universe expands, the mathematically proven wormholes, and other theoretical things.
Good effort, but I think you needed a better step from 3D to 4D. You can use time to explore 4D, and I would have liked an example of how you extend that idea of a diagonal into a space you can't fully see all at once. I'm actually finding it quite difficult to arrange the cube and spheres in spacetime now, but it absolutely can be done with a more rigorous definition of the shape. Then, you should illustrate how to formulate a 2D slice of 4D space out of the 3D slices you are displaying of the 4D shape; I imagine this means something like taking an infinitesimal strip of the 3D cross section for each moment in time, but like I said, I'm not totally clear on how you are getting your 2D slice.
Think of the cube as aligned to coordinate axes, of unit length, and centered at the origin. Then the slice we are taking in 3D is the plane containing the vectors (1, 0, 0) and (0, 1, 1); in 4D it is (1, 0, 0, 0) and (0, 1, 1, 1) instead. The centers of the sixteen small spheres are at (±1/4, ±1/4, ±1/4, ±1/4), four of these are on the plane, the others are farther than 1/4 away from the plane (unlike in 3d, some are farther than others, interesting), and so none of the other spheres intersect it.
@@tiborbogi7457 the underlaying fact that here we are discussing is the fact that the diagonal of the nth cube increases. Of course, i get it. For a cube of n dimension it should be sqrt(n) if i am no mistaken. Why is the only one axis are only 2 axis being stretched though? The z axis is always the same, and that implies that the length of the diagonal should increase only by stretching the cube in 2 axis. Should this still be called a cube?
@@Girasole4ever May be I explain it badly, but the length of the edge of hypercube is not changing (it remains the same say 1). What is changing is number orthogonal axis (2 dimensions x,y ; 3 dimensions x,y,z; 4 dimensions x,y,w,z and so on) But I always failed to imagine a tesseract no matter how many times I try. My brain is limited to 3 dimensions. ;-)
Very clever visualization! I have done the math on the very similar problems where the hyperspheres are centered on the corners of the hypercube. Your visualization shows how the 10D case can have the central hypersphere get out of the "enclosing" hypercube. The equations show why, but they are less intuitive. Well done! I'd have liked to see the 4D case projected down to 3D, and then down to your 2D diagonal slice. Or would that have been confusing for most viewers, rather than clarifying?
Yeah thinking about it with diagonals is pretty intuitive. For example just looking at 2d to 3D, the edge of the cube is one of the 2d squares, but to go from opposite corners of the cube, the points exist along two squares arranged perpendicularly, so the 3D diagonal will have to be substantially longer than the 2d one. And this logic should extend through each step into higher and higher dimensions
Video is nice, explain something unexpected, but sorry i don't see any Visualization of Higher Dimensions. But continue making more videos around this topic. I appreciate your work. ;-)
Interesting to note the measurements of ball placement (1/4 of the square) with the increase in the intersection the inner circle has with the four circles. All you need to do is follow that multiplication pattern and you can get an idea of what it will look like any dimension. In other words, you could pinpoint for dots on the square to draw the circumference of your four circles, and four points within that square from which you could draw the circumference of the inner circle.
So this is why the universe is flying apart. We see in 3d while we live in a 10D+ universe. Regular matter is blue balls, and dark matter and dark energy are red balls going gang busters.😅
ive always understood this as "there is always a slice of the setup in n dimensions that is the same as the setup in n-1 dimensions. this slice is not in the center of the center sphere, so the new sphere must be bigger than the previous one." it doesnt really give a good visual on how it would look in 10d, but i can understand why it works
I feel I've always had a decent to fair comprehension of higher dimensions given my wacky brain, but one thing that really drove in the inconceivable size of it all was I was watching a video on ridiculously large numbers and they went not just through exponents to titration, or even pentation, but one level above that, and as I found myself trying to write out the numbers in a way to make more sense of them, it ended up being easier to think of each degree of operation being another direction or dimension to extend into...
That wouldn’t be very strong considering all of the dimensions correspond to exponents, not higher operations. Although I have been wondering if higher operations have their own corresponding mathematical dimensions as well…
Wow thank you! I had myself convinced that the 4 dimensional configuration led to contradictions making spatial dimensions greater than 3 impossible. But you changed my mind. The possibilities are infinite!
No visualization can even compare to the reality of what we’ve gotten ourselves into I got a mere taste as the 4D construct of soul in between mind and body and even that is hard to put into words. Like the 4D construct of time is responsible for stitching together 3D moments right? All these moments are swashing around together and we merely pick and choose between each moment at any given moment, we don’t experience that mess for a reason. And as the 4D construct of soul in between mind and body, all my 3D vessels are like atoms within me and I’ve merely intermingled myself with the construct of time to put my atoms into it in the form of my vessels across the multiverse But at the end of the day none of it truly matters, nothing matters. But at the same time it all matters way more than anyone down here can even comprehend. It’s just pointless to dwell on the thoughts the enemy implants, thoughts of regret of the past and fear of the future. The Father of All Creation can forgive you for your past and he can also take care of your future.
Interesting. As you go to higher dimension, increasingly more volume of a sphere is getting concentrated in the shell near sphere's surface. I'm wondering why the opposite is not taking the place for this example...
We have these urges to put things inside physical scientific boxes but we do live on several different scales that allows us to envision or even see this taking place in many different ways in our typical activities.
If an object's information, like that of a black hole, can be determined from its boundary (holographic principle), then it stands that its dimensionality must be similarly encoded. This points to a fundamental one- or two-dimensionality (if including time). Accordingly, "higher" dimensions must by nature be *divisions* of the underlying dimensionality. It's not +n dimensions; it's 1/n. Visualizing a "higher" dimension will always just be a reconfiguration of perspective
@@absoluteaquarian What am I missing? In 2 dimensions, the diagonal is sqrt(1^2+1^2), so that leaves sqrt(2) - 1 for the center circle. In 3 dimensions, the diagonal is sqrt(1^2+1^2+1^2) so that leaves sqrt(3) - 1 for the center circle. So when sqrt(N) - 1 = 1, the center circle is the same size as the N-cube.
@@MikeGranby perhaps, but that disregards the condition that the center circle is between all of the other circles AND also tangent to them. Furthermore, the inner circles being contained within the cross-section. Hence why the inner hypersphere can't have a diameter of 1 unit in 4 dimensions, as is noted by the abstraction in the video. Your algorithm simplified the problem too much, which resulted in key details being disregarded.
@@absoluteaquarian I’m not getting this idea of circles (or spheres etc.) touching in a way that isn’t tangent, but whatever. The point is that we know that eventually the inner sphere does get bigger than the enclosing cube (see other videos on this topic) so the question remains as to when. The formula above seems to work for 2D and 3D, so why not in higher dimensions? And if it’s wrong, what is the correct answer?
I think this whole time it was as simple as: we cant see in 4 dimensions, so lets not try. We can do 2 dimensions really well, so lets just use the brains we have and break down the problem so we can understand it, in 2D.
Thank you. I now have a rough 2D Schematic of what a 10-dimensional lifeform may look like. It would be so complex it destroys the brain. HP Lovecraft wasn't so far of from the truth exactly...
4. If you tile the blue spheres in a 4D hypercubic lattice, this means you can fit an identical lattice of red spheres inside it. Each sphere in that packing touches 24 others (a red sphere touches 16 blue spheres and 8 other red spheres, and vice versa). The packing actually has extra symmetry, its Voronoi tiling is the regular tiling of 24-cells and its Delaunay tiling is the regular tiling of 16-cells.
How the 3d rectangle cut “allowing” the inner ball to be bigger would ACTUALLY make it bigger? I mean, it is just like a diagonal slice from the 2d square but in another perspective… so the ball should be the same.
2D means hexagon, squares in atomic structures introduce curvature. Using a square to depict 2D is literally using a shape that implies curvature/3D being presented as flat. 2D is not a concept it’s Hexagon.
There's only one dimension of space, though we understand it with the coordinates. The three physical dimensions are time, space, and scale. There are no others. Anything can be understood as a dimension if its a scale along which an attribute of a thing can be placed.
The thought that popped into my head was whether this visualization can be applied to the expansion of space time? Each addition of a dimension added more empty space in the cross section. Are these concepts related at all?
2:06 (press play and pause real quick so that the square is facing directly at you) Yes but as you can see here if you're looking dead on to try to make 3D look 2D. So you're looking at a flat side directly, it looks similar like the 2D picture but ou can see that the middle ball is much bigger than the 2D representation. But you can also see the 2D representation and you can imagine a smaller ball filling the middle part just like the 2D picture shows. And now you understand why for 3D the ball is bigger in the cross section picture. It's because it's not flat up against the box and touching the edge. It's in the middle of the box and filling up a different section of space. Where is the 2D model has the middle circle all the way up against the edge filling in the space there. So you can see the circle has been moved back in depth dimension. (RU-vid keeps cutting off my comment so read my reply below for the rest)
Continued... So you can see the circle has been moved back in depth dimension. But now for 4D, even though it's basically impossible to visualize, this is also another clue as to why in the 10th dimension the middle ball goes outside the cube area. Each dimension, that middle ball seems to have more area to fill. In the 2D picture you can see it only has a very small spot to fill and if you're looking flat on in 3D it's the same size. But because the depth dimension now there's more area that needs to be filled. So when you go into the fourth dimension there will be extra space to be filled just like there was here into that third dimension area making the middle ball bigger...
huge explanation gap between "this is how diagonals work for the 3-shapes of which we all know how they look and how we could therefore cut them along their diagonals" and "this is how the diagonal of a 4-d arrangement of the same types of shapes would look like, just trust me on this one..."
The red circle in 2D is smaller than the red sphere in 3D, because it moves inwards from the narrowest space between the blue circles to the center of the eight blue spheres. You see this, if you look at the 3D-figure straight from the front (which cleverly isn't directly shown in this video). Then you'll see the blue spheres in the front are overlapping the red one (which they don't do in the 2D-version). Is this truly justifying the assumption this will happen every time we move from one dimension to another? In 1D the red ball will not exist at all, since the blue lines will touch each other like they do at the equator in 2D and 3D, too. Of course this fits in the overall hypothesis. If you would do the intersection in 3D straight from the front, there would in the middle be practically no blue color at all, while we are on the verge of the eight blue balls (as seen in the 2D-presentation vertically or horisontally straight in the middle). Simultaneously the red ball is at its maximum, hovering in the middle. I suppose the correct way of presenting this is not in a box but in a circle/sphere, where there are no extending corners.
I find it interesting that I understood where this was going as soon as the artistically incorrect rendition was shown. While it fails at details, the idea is preserved: The bounds used don't all line up with each other, and don't all touch, so they can grow in unrelated ways. The "box" grows closer and farther at the same time, the "circles" stay the same but take up less and less space.. the inner circle must grow, and must partially leave.
In a sort of way the higher dimensional hypercubes are more 'spherical' than we think, as all the 2^N corners are equidistant / are the same distance from the centre. It's why there are no normal (average, central) people given our multiplicity of traits. When looking at the PCA idea (another comment), one then flips to the Mahalanobis distance measure, and find that everything is effectively on the surface of the hypersphere! Thus there's space for a very large sphere in the centre (like air in a balloon)
I suspect that this “space” is probably going to fill up with many, many more spheres than just the original four as we transition from 2D circle, to 3D sphere to 4D hypersphere? I suspect your red sphere would be infinitely small when adjusting to that possibility. Unless.. i watched too many 4D to 3D projection videos where hyperspheres twist and turn and add axes as they move through 3D. Maybe your video moves up in frame with each added dimension so from that perspective everything checks out? 🤞
4D is actually not too difficult. Visualize the object as a 3D animation, where at any moment you're viewing a 3D slice. I can usually even think about the temporal neighborhood of the 3D slice with a bit of transparency and color, which helps if I have to think about angles or derivatives. Consider a 4D sphere. At each point on the sphere, r2 = x2 + y2 + z2 + t2. A slice of constant t where c = t2 gives you a 3D shape where x2 + y2 + z2 = r2 - c, which you can easily recognize as a sphere in 3D. You can see that the 3D sphere has its radius reduced from the original unless t = 0, and the radius at t = r is 0. So starting at t = -r, the 4D shape is a point that grows symmetrically as a sphere and shrinks back down to a point again. If you want a better idea of how fast it grows and shrinks, you _could_ calculate an r(t), but you can also just look at the profile of a sphere in 2D; r(t) would be the y coordinate at x = t. I'll admit that in the case of this video, I'm finding it more difficult to see how the shape itself is generalized into 4D. I would need to see a rigorous description of how many spheres it has and how they are arranged. I imagine the box just winks into existence at the beginning and disappears abruptly at the end without changing size or shape, and of course the spheres themselves do what I described above individually, but they are out of sync, perhaps. How you get a 2D slice of that, I'm not sure.
I can't but feel that the 2-D cut into a rectangle is incorrect. Why would the symmetry be broken only in one dimension to create a rectangle, instead of also adding space on the top and bottom to result in a square (still with a larger void)? Thus the end result at ten dimensions would not product a paradoxically large inner void.
The n-dimensional cubes are not being sliced parallel to any of their dimensions, but diagonally. The top to bottom height you are seeing on the slice remains consistent with the unit, but a diagonal line through an object will get longer as the dimension gets higher. Here is how it works, if you're interested: Think about a 2D square with side length of 1 (a.k.a. 1x1). The diagonal lines through all of these objects will be easily calculated with a^2 + b^2 = c^2, so the diagonal length is √2. Then, go up to the next dimension and get a 1x1x1 cube. The diagonal through the entire cube isn't forming a right triangle with just the edges now but is a right triangle with one of the edges (which is still 1) and the 2D diagonal line (which we found to be √2), so, using the Pythagorean formula again, we find the 3D diagonal length is √3. Carrying on from there, we can find that the 4D (1x1x1x1) diagonal length is 2 (or √4), the 5D (1x1x1x1x1) diagonal is √5, and so on. These increasing diagonal lengths are what form the top and bottom of each 2D slice (2D diagonal lines bound the slice from 3D, 3D diagonal lines bound the slice from 4D, etc.), which is why they continue to grow. Meanwhile, the left and right lines of each 2D slice (the height) is established at 1 and remains the same no matter how many other dimensions you add, because those other dimensions are in different directions and do not affect the height.
GREAT visualization!!! However: "don't be scared, there will be no formulas"... Well, yeah... That's exactly what I'd be scared of. Formulas explain A LOT and when supported by visualizations they explain everything.
Does anyone else think that using the diagonals above 2d creates a false image? If you use a perpendicular cut, you'll see the same ratio as in the 2d image because the distortion caused by using a diagonal is removed. It's just a trick of the angle of observation, much like photographers getting funky with perspective.
3:35, you should have said the size of the red circle is exactly the one of a blue circle, because the inner diagonal of a 4D hypercube is twice the length of a side.
Is it weird to say that I can visualize the 4th dimension way easier than the others? I mean c'mon! Maybe it just comes down to the fact that I started visualizing the 4th dimension at a _very_ young age (like around 8 yrs old) through how I interpret depth, but I wouldn't recommend you stab your screen with an eraser stub.
