FINALLY! A Good Visualization of Higher Dimensions

[@phisgr]

I was hoping for a 3D cut of the 4D case before lowering to the 2D cut.

[@carykh]

Wow, this is a great way to visualize high-dimensional space!

[@tinycatzilla]

Cary :D

[@briananeuraysem3321]

Cary KitKat hoarder

[@davidhand9721]

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]

​@@davidhand9721exactly, a "slice" of a 4d object would be a 3d object

[@simeonsurfer5868]

@@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.

[@TrimutiusToo]

It doesn't visualize higher dimensions, just finds a good 2D cut for a specific problem. Still extremely interesting

[@user-sl6gn1ss8p]

To me that's kind of a good way to visualize higher dimensions: find the relevant cuts to what you're interested in exploring

[@TrimutiusToo]

@@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

[@gershommaes902]

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)

[@user-sl6gn1ss8p]

@@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

[@gershommaes902]

@@user-sl6gn1ss8p Yes exactly!

[@FireyDeath4]

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

[@theusaspiras]

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.

[@safa-uc1mk]

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.

[@DamaKubu]

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.

[@dmitripogosian5084]

The volume of read ball is proportional to (sqrt(D) -1)^D not to the power of 3

[@pronounjow]

Isn't the diameter of the red ball (√D - 1)/2?

[@haph2087]

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.

[@The-KP]

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?

[@meinbherpieg4723]

I never had anyone explain visualizations of higher dimensional objects with respect to their diagonals. Great job.

[@Henry3.1415]

Most intuitive way to understand this puzzle, and it's better then a 3blue1brown video so good job

[@dylanherrera5395]

the highest praise one could _possibly_ give a math youtuber

[@xninja2369]

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..

[@bitslay]

We getting math KZbinr beef?

[@JustAFrame]

3blue 1brown has a video showing the same thing but he doesn't use it as a tool for visualization

[@enya_yurself]

nope sorry 3b1b is better

[@stevethecatcouch6532]

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.

[@sinisternightcore3489]

Yeah the spikey sphere was bogus! Long diagonals is where it's at!

[@chriswilson1853]

I think it's hypercubes that are "spiky" rather than hyperspheres.

[@imacds]

Not going to lie, the crazy artistic interpretation wasn't half bad either.

[@2the431]

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 💯

[@njdotson]

I'm not convinced it's possible to imagine a 4D space all at once

[@eryqeryq]

This is an amazing way to illustrate it... I never understand this until now!

[@Lucidthinking]

Beautiful, thank you.

[@VanVlearMusic]

Bae wake up a new visualization of higher dimensions just dropped two years ago!!

[@4stim0]

😂😂😂❤

[@vladyslavkorenyak872]

Wow, this is cool! I would love to see more higher dimentional objects and get an intuition for them in this way!

[@bernardofitzpatrick5403]

Please put out more vids 🤙🏽looking fwd to next one 🙌🏽

[@matematicke_morce]

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.

[@X3MgamePlays]

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.

[@SOBIESKI_freedom]

Beautiful!

[@lodewijk.]

I went into this expecting it to still not be intuitive, but this explanation feels entirely logical to me!

[@adandap]

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.

[@joeybasile1572]

Fantastic.

[@davidwright8432]

The explanation of something so 'intuitively' impossible is direct, easy to follow and - 'obvious' - once it's been pointed out! Thanks.

[@KarlDeux]

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.

[@KarlDeux]

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.

[@maynardtrendle820]

Nicely done! 🌞

[@zazem4835]

Good video, I like the simple aproach. Wait for next topic :3

[@officiallyaninja]

This is sick as hell. I always knew there was some way to visualize this

[@stevenschilizzi4104]

You meant “slick as hell”?

[@wbwarren57]

Great video! Thank you.

[@raimundomuthemba766]

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.

[@Girasole4ever]

I don't understand how increasing dimensions will only stretch the cube in one dimension. Can we still call those "cubes"?

[@tiborbogi7457]

Cube is not stretched, stretched is only diagonal cut.

[@nicholascurran1734]

Exactly. This is not well done in my opinion, that this cross section represents not a cube, but a rectangular shape.

[@Girasole4ever]

@@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?

[@tiborbogi7457]

@@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. ;-)

[@overthebrick4407]

Really interesting and nice visualization, thanks !

[@humanperson2375]

This is geniunly amazing

[@curtishorn1267]

So question, at what dimensionality is the center sphere the same radius as those enclosing it?

[@GODDAMNLETMEJOIN]

4

[@tedsheridan8725]

Very cool way to visualize it - it actually makes sense now.

[@doomofthedestiny8065]

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...

[@sonicwaveinfinitymiddwelle8555]

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

[@chuckhammond5892]

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

[@russianbear54]

I still don’t get it😢. I’ve been trying to visualize 4D for about a month now, to no avail. 😢😢

[@davidhand9721]

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.

[@FireyDeath4]

Play 4D Golf and 4D Toys

[@cannot-handle-handles]

I'm apparently your (7^3)rd subscriber. 😀

[@wombleofwimbledon5442]

Now I need the Ethics and subsequent Morals that proceed from such.

[@gracicot42]

Thank you for this video! I wish we could see the slice being rotated, next video idea? 😄

[@rubenkossen3498]

So at which dimension does the inner sphere exactly touch the boundary of the n-dimensional hyper cube?

[@MikeGranby]

Should be 4, shouldn’t it? Sqrt(N) - 1 = 1?

[@absoluteaquarian]

@@MikeGranby that formula has no correlation to this question whatsoever.

[@MikeGranby]

@@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.

[@absoluteaquarian]

​@@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.

[@MikeGranby]

@@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?

[@PriitKallas]

Great work explaining this!

[@morphtek]

godamnit show the 3d visualization of the 4d space

[@tiborbogi7457]

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. ;-)

[@nahometesfay1112]

They're taking a 2d cross-section of a higher dimensional shape

[@kianushmaleki]

Wonderful. I really like it.

[@Ykulvaarlck]

just so you know, the black frame until the first visualizatio0n made me think my playback was broken

[@andrewferguson6901]

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

[@fredselvaggio1435]

That’s about sphere packing not really this

[@BlazeMakesGames]

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

[@Salara2130]

I guess this makes some sense if you are already familiar with to topic. If not it just seems like "trust me bro"

[@philipoakley5498]

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)

[@CoachS2]

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?

[@nnoxie.a]

it's all fun and games until you need to visualize a 1D infracube (a line)

[@THEL05]

Excellent

[@efjay3183]

Awesome

[@usptact]

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...

[@Duiker36]

The Algorithm likes this video, and so do I.

[@andrechaos9871]

Ok, I will use this interesting property, when building my n-dimensional contraptions

[@tangentfox4677]

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.

[@igalbitan5096]

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...

[@SgtSupaman]

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.

[@nice3294]

Very simple, very effective

[@rodrigoappendino]

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.

[@richarddeese1087]

Thanks. I think it would be fascinating to try using VR to visualize higher dimensions. tavi.

[@pronounjow]

I can't believe I'm just finding this video now. Great visual explanation!

[@davidi.levine6253]

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!

[@timothysmudski1058]

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!

[@apophisxo4480]

To be completely honest...I still don't get it :(

[@Cecilia-ky3uw]

Yeah

[@vivaselementum]

Higher dimensional spaces are simply spiky-er... spikier. It can contain much more (hyper-) volume than a 3D space would.

[@maskedvillainai]

It’s simple actually. The largest circumference of the shake is the highest peak average. The lowest solids is the smallest.

[@maskedvillainai]

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

[@PeterFamiko-lw8ue]

Great

[@chronik_sword1244]

how do we know, how many „spheres“ (if we even can call them that in higher dimensions) fit into a n-dimensional „cube“?

[@cabudagavin3896]

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...

[@munarong]

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.

[@nartoomeon9378]

Hmm, in other words, in 10-dimensional space, we can't fit one sphere inside a hypercube that touches other spheres with diameter 1/2 because it has a diameter greater than 1. I wonder what dimension has this latter case? Is it possible?

[@honeymak]

do you think higher dimensions are containers for lower dimensions?

[@grysby]

Everyone ask about 4d, but what about 0.5d?

[@zit1999]

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? 🤞

[@neochris2]

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

[@SgtSupaman]

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.

[@maxmudita5622]

Wut the fudge!? Diagonals ??? Principal component analysis? Dimensionality explosion! Which dimension is larger 1st or 10th?

[@uwuowo7775]

What is the width of the 2D cut? Is it (n-1)^(1/2) for n=dimensions, or did i make a mistake

[@SgtSupaman]

That is correct. The height is 1, the diagonal is √n, so the width will be √(n-1) or (n-1)^(1/2).

[@uwuowo7775]

@@SgtSupaman im actually good at mental arithmetic then

[@nicholascurran1734]

Why is it that the higher dimensional sections revisited as 2d plane crosscuts stretch only the x axis and not the y? What does a slice of a hypercube look like? Or a slice of a hypersphere? It seems to me it can take on many different appearances depending on the start and finish point of the cut. It is presumptuous to think that representing higher dimensional shapes in 2d yields this particular configuration, and frankly feels very much like "cherry picked" statistics where the conclusion is being supported by a biased representation. Please correct this if I'm mistaken. I'm open to hearing how, outside of imagination ("on paper"), this makes sense.

[@sinisternightcore3489]

-- Why is it that the higher dimensional sections revisited as 2d plane crosscuts stretch only the x axis and not the y? The y axis is just a side length, which is 1 regardless of how many dimensions you add. The x-axis shows a (hyper)diagonal. A square has two diagonals, which both have length sqrt(2). A cube has these diagonals too, on its six sides, but it also has the diagonal going through the cube connecting any two opposite points, and that has a length of sqrt(3). In hypercubes you can find even longer "diagonals" even though the lengths of the sides don't increase. -- It seems to me it can take on many different appearances depending on the start and finish point of the cut. Correct. -- feels very much like "cherry picked" statistics where the conclusion is being supported by a biased representation. We are only looking at a very specific slice, but the rectangle sill marks the boundary of the hypercube in that plane. We only need to find one slice where the circle intersects the rectangle to conclude that the hypersphere intersects the hypercube. EDIT: In fact we carefully select a slice which contains the points where the hyperspheres touch. If we chose any random slice, most likely the hypersphere slices wouldn't touch (tangentially) and it would be tough to figure out where to draw them.

[@nicholascurran1734]

@@sinisternightcore3489 thanks for the explanation. I am still curious, if you happen to know more, about the relationship between the cubes and spheres as dimensions grow. The slice along the hyper diagonal, producing circles instead of oblongs, is only able to display that portion of the shapes, which in my mind appears as a limited display of what would occur (based on my, potentially lack of, understanding). I visualize a higher dimension not containing the same "tesselation" as shown here. These slices assume the same pattern of shape distribution among space regardless of higher and lower dimensions, don't they? Blocks stack on top of each other, do hypercubes stack on each other the same way? Would a jar of marbles and a jar of hyperspheres "stack" or "tesselate" the same way? My intuition says no, unless both situations only differ in appearance, in which case they are the same, relative to the observer and the inhabited dimension. Since there are equations showing that space in other dimensions, and the way shapes interact in those spaces, appear different when viewing them on a different plane, then they can't be the same. Maybe future games with these mathematics and vr will help clear up the differences. For now, I can say while these videos are entertaining, they appear to lack something of substance, even if I can't quite place what it is, or adequately describe why it seems off to me. Cheers!

[@sinisternightcore3489]

@@nicholascurran1734Not sure if it helps but I'll try: Stacking in higher dimensions works the same way in principle but there are differences. For example if a square can contain four circles, then a cube of the same side length can contain eight spheres of the same diameter. And a 4D hypercube can fit 16 hypershperes and so on. There is more space as you add dimensions. There is no distorting or phasing through each other, or disappearing going on if that's what you're thinking about. These notions arise from other methods of projecting or visualising higher dimensional space. This video is also by no means a mathematical proof. But understand that if you take a 2D slice of an any dimensional thing, that slice obeys the rules of 2D math.

[@nicholascurran1734]

@@sinisternightcore3489 hmm... I thought a hypercube is 6 conjoined cubes (each face of hypercube being a cube itself), which would make for 48 spheres + 6 centered spheres. Maybe that's where I'm missing something to "see" it? If a 4d hypercube holds 16 spheres + 2 centered spheres, how are they positioned such that the long diagonal slice only contains 4 circles + 1 centered circle? I appreciate your responses, and perhaps my questions could be better worded.

[@nicholascurran1734]

@@sinisternightcore3489 for clarity, some of the visualization I get, which involves transformations, is from a vr interaction of hypercube dynamics. Maybe seeing one of those videos can show you what I'm visualizing, and understand where I'm going wrong?

[@jasonligon5937]

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.😅

[@dustinfrost2603]

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

[@JoakimfromAnka]

Das ist sehr gut!

[@jareknowak8712]

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.

[@TypoKnig]

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?

[@ethanjensen7967]

She has a relaxing voice

[@RolandPihlakas]

Is this "10-D inner circle being bigger than the cube" somehow related to why string theory has 10 or 11 dimensions of space, if I remember correctly?

[@zswu31416]

I don't think so. String theory has that many dimensions because you *need* that particular number of dimensions for certain symmetries to work out nicely. It has nothing to do with this particular problem, and the 10 is just a coincidence.

[@JavierMarineroRamos]

Is there a number of dimensions for which the blue and the red circle are the same size?

[@SgtSupaman]

Yes, in 4 dimensions. The blue hyperspheres have a diameter of .5, which is exactly what the red hypersphere's diameter will be in the 4th dimension. (√(4) - 1)/2

[@REMdonor]

YESSS

[@atreidesson]

Mentioning that this radius is 0 at 1d also makes sense.

[@safa-uc1mk]

2:21 why did you cut from that angle of the cube instead of the angle where we can see one face of the cube flatly ..if that makes sense.

[@safa-uc1mk]

brainstorming: my guess is to show that such a space does exist where the 2d cross-sectional cut would make it appear as though the middle spear is bigger..?! now wondering why that changes though when you look from different angles for a cube..

[@safa-uc1mk]

Hmm now I'm thinking that your 2d cuts of the different n-spaces don't account for the different 2d cuts that exist in higher n-spaces T-T kind of lost

[@kazedcat]

​@@safa-uc1mkThe diagonal is longer than the side. As you go up in dimension the ratio between the main diagonal and the side gets larger. If you cut along the main diagonal you get the longest distance within a hypercube.

[@SgtSupaman]

A slice that is parallel with one of the faces would be a 2-dimensional image where the circles are smaller parts of the spheres and the outer circles would never touch the inner circle and the sides of the square at the same time. You would have to overlay multiple slices to get the full diameter of each sphere, which would make it seem like they are overlapping. That isn't nearly as helpful in illustrating how a sphere that never physically intersects the other spheres ends up reaching outside the containing shape. The diagonal slice maintains the image with the full diameter of each sphere to prove they never intersect.

[@peamutbubber]

Impossible to actually visualise but pretty intuitive if you think about moving from 2D to 3D.

[@1cool]

556th subscriber

[@thedeathcake]

I thought a cut into a 4D object results in a 3d object?

[@avi12]

Fun fact: Aug 22 was my 24th birthday I wish KZbin had suggested this video that day

[@Oxxyjoe]

I have difficulty understanding how this discussion about the diagonal translates to the inner hypersphere actually exceeding the bounds of the hypercube, and I will explain why I feel like this doesn't add up. It does follow that as you go up to 4 dimensions, the diagonal is longer than in 3, but when I look at this wider rectangle which represents this, I have to ask: When you rotate the object back to facing one side rather than the diagonal, it ought to return to looking like a cube. Yet I do not see how it would return to being cubed shape while the inner blue hyperspheres now seem too small to fill the hypercube. You see what I mean? The object represented by this rectangle doesn't look like a rotated cube anymore.

[@rlindeque]

It just happens to work out nicely for diagonal slices of the hypercube because the hyperspheres are packed into corners. If you don't slice diagonally through the corners you won't see the shapes touching at their boundaries anymore, and that makes sense.

[@davidzaydullin]

you could've also shown 1D example

[@DontfeelNienfeeler]

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.

[@davidhand9721]

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.

[@terdragontra8900]

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.

[@davejacob5208]

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..."

[@stevehines7520]

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"