@@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.
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
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.
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 💯
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..
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.
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.
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
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
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 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…
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.
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.
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.
@@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. ;-)
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.
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
I had heard of this result, and it made no sense to me. I only understand it partially, but at least for a moment it seemed pretty clear. This is a magnificent example of great exposition!
2:10 You coul 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.
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.😅
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!
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. ;-)
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.
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
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 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?
@@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?
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...
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..."
with this model I am to believe that the red circle expands perpetually as dimension increases, but I thought that the sphere was of a specific size...
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.
To me it looks more like streched 3D, nothing like higher dimensions. I imagine HD like 3D objects or zones, existing in the classical 3D space, but with no possibilities to interact with them in any way. Something like "ghost" - it is near me, but i dont see it and i cant touch it. HD, if they exist, they are here with us, we just dont feel them - not quite exact as, but something similar to dark matter. And this is the best possible explanation to me. Besides, there is also another reason why i think this way but my English it too poor to explain this at 3.00 in the middle of the night. Regards.
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.
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.
Diagon the third declension at the 3rd position which is the second step, an equalization occurs. This was understood in antiquity. The understanding was in relation to material terms of societal decline. When defining for variation (declension) beyond the third position (stable, even, equal,) "materially" initiates process of decline. Material decline re-introducing the balance presented in "angle" in support of "Divine understanding" non-material. By sacred definition to move beyond the third declension is the even-tual realization that the center is the whole. "Knowledge is one point, of which human intelligence multiplies" "Even" This word is one of the greatest verities in terms of understanding from a be-ginning. "From even-ing to morning was the first day" This is not a material idea. When we even things by material definition, this material perception "even-ing" is followed by a "mourn" as seen in societal decline. The material de-cline be-ing the re-initialization of age re-set on "Divine angle" Yes, this one is "crazy" There is no greater lunacy then that which occurs through "The Beloved"
In 4-D, does the inner hypersphere have the same radius than the other 16 hyperspheres? I have this intuition because I know that the diagonal of a 4-D hypercube is equal to twice the side...
Yes, the diameter of the inner hypersphere can be found with (√(n) - 1)/2. In 4 dimensions, n=4, the diameter is .5, which is exactly the same as the outer hyperspheres.
![[柴犬ASMR]曼玉Manyu&小白Bai 毛发护理Spa asmr](http://i.ytimg.com/vi/0TsXQ7z2Dh4/mqdefault.jpg)
I was hoping for a 3D cut of the 4D case before lowering to the 2D cut.
That would* be cool.
3D cuts all around actually! Might as well use these amazing GPUs we have nowadays.
I was thinking the same thing, that would be helpful
It doesn't visualize higher dimensions, just finds a good 2D cut for a specific problem. Still extremely interesting
To me that's kind of a good way to visualize higher dimensions: find the relevant cuts to what you're interested in exploring
@@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
@@user-sl6gn1ss8p Yes exactly!
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.
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
Wow, this is a great way to visualize high-dimensional space!
Cary :D
Cary KitKat hoarder
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.
@@davidhand9721exactly, a "slice" of a 4d object would be a 3d object
@@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.
Bae wake up a new visualization of higher dimensions just dropped two years ago!!
😂😂😂❤
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.
The volume of read ball is proportional to (sqrt(D) -1)^D not to the power of 3
Isn't the diameter of the red ball (√D - 1)/2?
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?
@@pronounjow - I believe so. It's DEFINITELY not just (sqrt(D) - 1)
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 💯
I'm not convinced it's possible to imagine a 4D space all at once
Very nice. Your technique is more intuitive and satisfying than the spiky spheres in Matt Parker's Things to Make and Do in the Fourth Dimension.
Yeah the spikey sphere was bogus! Long diagonals is where it's at!
I think it's hypercubes that are "spiky" rather than hyperspheres.
I never had anyone explain visualizations of higher dimensional objects with respect to their diagonals. Great job.
Not going to lie, the crazy artistic interpretation wasn't half bad either.
Most intuitive way to understand this puzzle, and it's better then a 3blue1brown video so good job
the highest praise one could _possibly_ give a math youtuber
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..
We getting math UA-camr beef?
3blue 1brown has a video showing the same thing but he doesn't use it as a tool for visualization
nope sorry 3b1b is better
This is an amazing way to illustrate it... I never understand this until now!
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.
Mind blown. I can’t believe how that move from 2d to 3d made it seem so obvious.
This isn't a way of visualizing multiple dimensions, just a way of visualizing a 2-D slice of multiple dimensions. Still interesting.
Please put out more vids 🤙🏽looking fwd to next one 🙌🏽
Beautiful, thank you.
The explanation of something so 'intuitively' impossible is direct, easy to follow and - 'obvious' - once it's been pointed out! Thanks.
Wow, this is cool! I would love to see more higher dimentional objects and get an intuition for them in this way!
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.
I went into this expecting it to still not be intuitive, but this explanation feels entirely logical to me!
Great video! Thank you.
Great! Thank you 😊
Great video! And the title's right, this is the first video I've seen on this topic that actually helped me intuitively understand what's going on.
Beautiful!
Now I need the Ethics and subsequent Morals that proceed from such.
Really interesting and nice visualization, thanks !
Great work explaining this!
Wonderful. I really like it.
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
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
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)
Nicely done! 🌞
Ok, I will use this interesting property, when building my n-dimensional contraptions
This is sick as hell. I always knew there was some way to visualize this
You meant “slick as hell”?
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…
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.
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.
This is geniunly amazing
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.
it's all fun and games until you need to visualize a 1D infracube (a line)
I guess this makes some sense if you are already familiar with to topic. If not it just seems like "trust me bro"
I don't understand how increasing dimensions will only stretch the cube in one dimension.
Can we still call those "cubes"?
Cube is not stretched, stretched is only diagonal cut.
Exactly. This is not well done in my opinion, that this cross section represents not a cube, but a rectangular shape.
@@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. ;-)
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.
For a 10D hypercube, the diameter of the red circle is (sqrt(10)-1)/2. Roughly 1.08, so indeed bigger than a side of the 10D hypercube.
Very cool way to visualize it - it actually makes sense now.
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
Thank you for this video! I wish we could see the slice being rotated, next video idea? 😄
Fantastic.
I had heard of this result, and it made no sense to me. I only understand it partially, but at least for a moment it seemed pretty clear. This is a magnificent example of great exposition!
2:10 You coul 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.
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.😅
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!
Excellent
I wonder if this means then that higher dimensions are much worse at packing since so much space is left between the spheres?
Good video, I like the simple aproach. Wait for next topic :3
One hyper cube has hexagon hyper faces and cubes have an approach to infinite dimensions.
just so you know, the black frame until the first visualizatio0n made me think my playback was broken
Watching on phone in portrait mode😊
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. ;-)
They're taking a 2d cross-section of a higher dimensional shape
The Algorithm likes this video, and so do I.
Thanks. I think it would be fascinating to try using VR to visualize higher dimensions.
tavi.
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.
Great
I'm apparently your (7^3)rd subscriber. 😀
is it right that you'll have to travel the circle's length more if you were to get from one point to another using all axes
Impossible to actually visualise but pretty intuitive if you think about moving from 2D to 3D.
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.
Very simple, very effective
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
That’s about sphere packing not really this
Awesome
Seems like sq rt of 2 but in more dimensions
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.
Mentioning that this radius is 0 at 1d also makes sense.
She has a relaxing voice
So question, at what dimensionality is the center sphere the same radius as those enclosing it?
4
I still don't understand, how this red hypercube is so big and bigger than blue 😟
I can't believe I'm just finding this video now. Great visual explanation!
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?
Das ist sehr gut!
So at which dimension does the inner sphere exactly touch the boundary of the n-dimensional hyper cube?
Should be 4, shouldn’t it? Sqrt(N) - 1 = 1?
@@MikeGranby that formula has no correlation to this question whatsoever.
@@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?
Not getting the step between 3rd and 4th Ds. 😅
Everyone ask about 4d, but what about 0.5d?
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...
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..."
with this model I am to believe that the red circle expands perpetually as dimension increases, but I thought that the sphere was of a specific size...
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 feel robbed...
To me it looks more like streched 3D, nothing like higher dimensions.
I imagine HD like 3D objects or zones, existing in the classical 3D space, but with no possibilities to interact with them in any way. Something like "ghost" - it is near me, but i dont see it and i cant touch it.
HD, if they exist, they are here with us, we just dont feel them - not quite exact as, but something similar to dark matter.
And this is the best possible explanation to me.
Besides, there is also another reason why i think this way but my English it too poor to explain this at 3.00 in the middle of the night.
Regards.
godamnit show the 3d visualization of the 4d space
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.
4D because result of slice visualization in 4D. This was NOT what the title claims.
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? 🤞
I still don’t get it😢. I’ve been trying to visualize 4D for about a month now, to no avail. 😢😢
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.
Play 4D Golf and 4D Toys
So what is the actual shape of the 4D and 10D, all I see are diagonal cuts. Everything can be a simple visualized, look at that 3D one.
Diagon the third declension at the 3rd position which is the second step, an equalization occurs. This was understood in antiquity. The understanding was in relation to material terms of societal decline. When defining for variation (declension) beyond the third position (stable, even, equal,) "materially" initiates process of decline. Material decline re-introducing the balance presented in "angle" in support of "Divine understanding" non-material. By sacred definition to move beyond the third declension is the even-tual realization that the center is the whole. "Knowledge is one point, of which human intelligence multiplies" "Even" This word is one of the greatest verities in terms of understanding from a be-ginning. "From even-ing to morning was the first day" This is not a material idea. When we even things by material definition, this material perception "even-ing" is followed by a "mourn" as seen in societal decline. The material de-cline be-ing the re-initialization of age re-set on "Divine angle"
Yes, this one is "crazy" There is no greater lunacy then that which occurs through "The Beloved"
do you think higher dimensions are containers for lower dimensions?
In 4-D, does the inner hypersphere have the same radius than the other 16 hyperspheres?
I have this intuition because I know that the diagonal of a 4-D hypercube is equal to twice the side...
Yes, the diameter of the inner hypersphere can be found with (√(n) - 1)/2. In 4 dimensions, n=4, the diameter is .5, which is exactly the same as the outer hyperspheres.
how do we know, how many „spheres“ (if we even can call them that in higher dimensions) fit into a n-dimensional „cube“?
GIMME MOAAAR DIMENSIONS🤣
Very nice! A great way to show something very unintuitive