My attempt to explain the degrees of freedom for orientation working from the fact that a n dimensional creature could manipulate a (n+1) dimensional object through all of that object's degrees of rotational freedom: A 2D creature (the flat fish) can rotate a 3D object by using its 1 degree of freedom of its own rotation and its 2 degrees of freedom from position for a total of 3. A 3D creature (the human) can rotate a 4D object by using its own 3 degrees of rotation and the 3 from position for a total of 6. A 4D creature can rotate a 5D object by using its own 6 degrees of rotation and the 4 from position for a total of 10. A 5D creature can rotate a 6D object by using its own 10 degrees of rotation and the 5 from position for a total of 15. (and so on.)
Another way: in 2D you can rotate on the xy plane, in 3D you can rotate on the xy, xz, and yz planes, in 4D you can rotate on the xy, xz, yz, xw, yw, and zw planes, etc
@@columbus8myhw If you wanted to unchain yourself from the origin, then you could say that you can rotate an ND object around any N-1D subspace. 2-2 = 0, so in 2D, you can choose any point (0D subspace) to rotate around. In 3D, 3-2 = 1, so your axes are any line. 4D yields the mind bending planar axes of rotation since 4-2 = 2. Interestingly, the number of degrees of freedom for an ND rotation around an arbitrary center is the same as the number of degrees of freedom for an N+1D rotation around the origin.
Bro, that eye tracking with your VR mask is trippy. I love it. 1:05 it even captured the subtle angle of shifting your eyes from a near target (the globe) to a farther target (the camera lense).
Time is our broken perception of the 4th dimension. We here (living in a 3D experience) can’t perceive the entirety of 4D reality, so we break it down into 3D phenomena that change over time. So, time literally is the 4th spatial dimension, but spread out as a temporal experience. Time is the point where 3D intersects with a 4D object.
This is so unbelievably amazing! Thank you so much for making this video. I've witnessed impossible for 3 dimension objects while using DMT. this is so fascinating, after that experience.
I thought of either: * the set of independent rotations as being the set of independent 2D subspaces, giving C(D,2), D choose 2 * in D dimensions, we can think of the hypersphere as being prametrized by D-1 angles. But also any point in the sphere looks locally like R^(D-1), so it has as extra rotational degrees of freedom, as many as the space in D-1 dimensions. Let's call the number of rotional degrees of freedom in D dimension, M(D). Then we have M(D) = D-1 + M(D-1). Because in D=2, we have M(2)=1, then M(D)=sum_{i=1}^{D-1} i = D*(D-1)/2, the triangular numbers which are the same as C(D,2)
The fish and globe shadow was something I had seen before but I never found a good mental understanding for this until now: giving the fish the shadow as it's method of rotation is essentially mapping the 3 degrees of freedom of the fish to the 3 rotation degrees of freedom of the sphere. The fish can use any of it's 3 degrees to manipulate the 3 degrees of the sphere
Since rotation always happens within a plane, and a plane is always two-dimensional, the number of rotational degrees of freedom is equal to the number of planes that can be created by choosing any two of the axes within the higher dimensional space (the axes themselves represent translational degrees of freedom). So, the number of rotational degrees of freedom for a space with dimension d is given by (d choose 2) which can be written as d! / 2(d - 2)! from the binomial formula. Another thing many commentors have noticed is that the rotational degrees of freedom in dimension d is equal to the sum of the translational and rotational degrees of freedom in dimension d - 1, which seems like a strange coincidence. I believe I have an intuitive explanation of that as well. The first obvious fact is that the higher dimensional space inherits the rotational degrees of freedom from the lower dimensional space. In 2D there is a single plane of rotation (the XY-plane). 3D space inherits this plane and adds two additional planes (the YZ-plane and the ZX-plane), but why exactly the number two? Why the exact number d - 1? Well, when we go up a dimension, we add another axis. This additional axis can then be paired up exactly once with each already existing axis to form a new rotational degree of freedom. For example, when we add the z-axis, we form two additional planes of rotation precisely because there were already two existing axes to form the planes with. Similarly, moving from 3D to 4D by adding a w-axis, this new axis can be paired up with the existing x,y and z axes to form new rotational degrees of freedom, resulting in 6 rotational degrees of freedom. And so on, for higher dimensions.
In this representation it could be worth mentioning that 4 dimensions enable two simultaneous but independent rotations, one along one 2D coordinate plane and the other along the plane of the remaining 2 dimensions. Thus there are no "axes" of rotation but rather planes. Considering the 3D surface of the 4D sphere, longitudes and latitudes could also work better if we visualize them as a grid of cells made of intersecting surfaces rather than lines. The system (or one of the possible systems) would have two orthogonal "equators" lined with prism-shaped cells while all the other cells are distorted cubes.
4D rotations don't correspond to linear coordinate axes, but you can argue that you still have "axes of rotation" that just happen to be planes rather than lines. This kind of thinking helps when not limiting yourself to rotations around the origin and also allows for more complex "rotations", like a 4D rotation-like transformation that's actually "around" a sphere.
spoiler alert | 2D rotation is the only rotation that has one degree of freedom, therefore, if a sphere can have its projections make X different two-dimensional rotations, it has X*1=X degrees of freedom, and to make a two-dimensional rotation, take a straight line and rotate it in the place of another line, leaving unchanged all the lines perpendicular to the rotation dimension, then taking perpendicular projection planes you can have lines to rotate that are perpendicular to each other, so you must select all the sets of two perpendicular lines , that having X dimensions there are X perpendicular lines,, with quick calculations, the number of sets being equal to the number of rotations, comes (x*(x-1))/2 degrees of freedom for X dimensions sphere
The “number degrees of freedom” of rotation, is referring to the dimensionality of the rotation group, right? So, in 2D, the group is SO(2) (the circle group), and in 3D it is SO(3), and in n-D, SO(n) an element of SO(n) is specified by giving an ordered orthonormal basis of n-D space (such that the determinant you get is 1 rather than -1, but that requirement won’t change the dimensionality) specifying the first vector, there’s an n-1 dimensional choice there (the minus one is because the vector has to be of a specified length, so we remove a factor that is from scaling), and then the what remains is to pick an orthonormal basis for the subspace orthogonal to the vector chosen first. So, f(n) = (n-1) + f(n-1) And f(0)=0 so, f(1) = 0, f(2)=1, f(3)=3, f(4)=6, Yeah, these are the triangular numbers, so, the dimensionality of SO(n) is n(n-1)/2 . So, in particular, the dimensionality of SO(5) is 10.
Absolutely fantastic video! The visualization is amazing and your explanation is very clear! Rotation is essentially a 2D operation that occurs in a plane. The second dimension is the first dimension where rotations can occur, because we have two coordinates, x and y. In the third dimension, we add another coordinate direction, z, and so we get two more possible rotations, in xz, and in yz. Add another direction perpendicular to the other three and we arrive at the fourth dimension where there are three more rotations possible: xw. yw, and zw. I love it that you can have two rotations occurring simultaneously in the fourth dimension that do not affect each other, like xw and yz at the same time. Thanks for posting this video and explanation!
Each new dimension adds (d-1) rotations because you can rotate through that new dimension across each one of the previous dimensions. 6 dimensions gives 10+5=15, 7 dimensions gives 15+6=21, and so on.
you can figure out degrees of freedom seeing how many extra planes will exist. For example, if a being is in the x, y plane to get the extra planes from the 3 dimensions the extra axes are y,z and x,z. So if for example we are adding an extra coordinate f the extra planes added from 3 dimensions to 4 dimensions would be x,f; y,f and z,f.
2:20 The projection of the sphere’s shadow onto the 2D plane is identical to the Steinmetz diagram of electromagnetism. Apollonian circles, each pair of which meets at 90 degrees to one another, ie like being inside a torus. This is also what the Yin Yang symbol means. It’s a symbolic representation of a 4D sphere, or a 3D torus, as a 2D image.
A 4D sphere and a 3D torus aren’t the same? Also, do you mean a 3-sphere which lives in 4D space, or do you mean a 4-sphere? And, by a 3D torus, do you mean a solid torus (a three dimensional manifold with boundary isomorphic to S^1 x S^1 ) or do you mean S^1 x S^1 x S^1 ? If you glue two solid torii together at their boundary in a way that exchanges which direction around the boundary is contractable, then I believe you get the 3-sphere. Maybe that’s what you were getting at? (If you glue it the other way, uh, I think you get the product S^1 x S^2 ?)
@@drdca8263 A 3D torus is a projection of a 4D sphere. In toplogical terms, I mean a 3-sphere, and I do not mean solid (which would be more appropriately called a "ball").
@@hermes_logios pi_1(a 3D torus) is non-trivial, but pi_1(S^3) is trivial. Like, there are non-contractible loops in the 3D torus (regardless of whether you mean D^2 cross S^1, or S^1 cross S^1 cross S^1, or S^1 cross S^1 ) , but no non-contractible loops in S^3 . But, any loop in the image of a projection should be the image of some loop in the domain of the projection, like, if you have some path in the image of the projection, and take a point in the preimage of some point in that path, there should be a path in the domain of the projection which goes through that point, and when composed with the projection, gives the path in the image of the projection (assuming that it is a linear projection, as in this video, not just a more general continuous surjection which is a one-sided inverse of some inclusion) and also assuming, that the space that is being projected, is convex... Oh wait, the 3-sphere isn’t convex, hold on.... Wait I have to go do something, I’ll say more later... But I think the argument I have here will work with some modification Edit: ok, meeting was canceled. Ok, so, while the 3-sphere as embedded in R^4 isn’t convex, for any two points on it, if they aren’t antipodal, there is a unique shortest path between them. If we take a loop in the image of S^3 under the projection, then... the preimage of that loop will have at least one point in the 3-sphere above each point in the curve in the image of the 3-sphere, and, ah, specifically each point in the loop should either have 1 or 2 points in its preimage, and, yes, we can indeed locally trace the path, though perhaps going around the loop could produce a path from one preimage of the base point to the other... But then, going around again would lead to the first one again, so, any loop in the projection, *when gone around twice*, is the image of some loop in S^3, And any loop in S^3 is contractible, and the image of the contraction of the loop will be a contraction of the image of the loop, and therefore, for any loop in the projection of S^3 (or any sphere) under a like, linear projection onto a “screen” one dimension down, like in this video, will have the repeating of that loop twice, be contractible. But! In the torus, there are loops which cannot be contracted, even after you repeat them any number of times. Therefore, the image of S^3 under a projection of the kind shown, is not the torus.
xy 1, xy xz, yz 3, xy xz yz wx wy wz 6, ...with 5 coordinates we have 10 different ways to pick a pair of coordinate axis xy xz yz wx wy wz vx vy vz vw. Nice work!
Reason I came here is to understand this better... when a hyper cube is rotated it looks like it's propagating ... so I got to thinking what does a sphere look like spinning.. does it also look like it propergates like a hypercube the I remembered the two ways you can rotate a hypercube and realised its not two ways but the combination of both or it would would not be rotation. I'm beginning to think that light lies on the boundary of two and 3 dimensional space and black holes are onthe boundary of 3 and 4th dimensional space and the rotation of an atom is 4th dimentional
Not a formal explanation, just my intuition. In 2D, you can rotate from one dimension onto the other, so there is only one pairing (degree of freedom). In 3D, you rotate a dimension onto one of the other two dimensions, that’s 2, than there’s a pairing between the remaining two dimensions, that’s 1 more, for a total of 3 degrees of freedom. In 4D, you can rotate a dimension onto one of the other three dimensions, that’s 3, then of the three remaining dimensions, one of the remaining dimensions can rotate onto the other two, that’s 2 more, and the remaining two can rotate onto each other, that’s 1 more, for a total of 6 degrees of freedom. So with with each additional number of dimensions, the additional dimension can rotate onto the previous dimensions, which there are n-1 of, and all degrees of freedom from the lower dimensions are preserved.
you are forgetting that the fish would only be able to perceive its world as a one dimensional line. due to the nature of the 2 dimensional fish, nothing can ever be behind or in front of the fish. it would only be able to interact with and view the shadow of the globe by grabbing the out most edges, and only seeing the shadow as a line changing in vertical and horizontal sizes, and various edges and smooth parts of the outermost line of the shadow. the fish wouldn't be able to grab the shadow from whatever position it wants nor view the shadow from an above perspective as shown in the demonstration
1 + 2 + .. (n-1), where n is the number of dimensions. Its simple, the number of degrees of freedom is the number of parameters to describe orientation of coordinate system. The end of the first axis of coordinate system is on n-dimensional sphere (n-1 parameters), the end of the second axis then has to be on the n-1 dimensional sphere and so on.
The reason is simple. We just have to count the number of "planes" in a given space. Suppose we are in 4d space, then, there are 4 axis, and each axis can be combined with the other 3 remaining axis, which would result in 4x3=12 possibilities. But we have to divide the result by 2 because of the repeated permutation.
Nice, but that's not what I was looking for. It's said that a 4d creature can mirror 3d objects, so I'm looking for a computer animation that shows how that would look to us.
You can work it out by thinking about what a 2D flatlander would see if you were to "mirror" one of their friends by rotating them in the plane they live in. First, you'd choose a line in their plane that you were going to mirror them around. Second, you'd rotate them 180 around that line. Their friend would see the line part of them stay in place - so, if the flatlander had their liver and spleen on that line, as you rotate them, their friend would see a 1D cross section of their liver and spleen appear to stay where it was while they were rotated. Before you start the rotation, their friend would see them; during, they'd see only the cross section; and at the end, they'd see their friend reflected, with that 1D cross section the only part that has stayed in place. So what's this like for 3D creatures rotated by a 4D creature? You work it out. Clue: in 4D, you do not rotate around lines, you rotate "around" planes.
The thing that always confuses me with these 2D analogies is that we do not really see what the fish sees - we see a top-down view of the fish's world. The fish only sees in 1D, so the shadows of the 3D sphere, as confusing as it looks for us, looks just like random 1D garbage to the fish. In other worlds, the citizens of Flatland, the 3D sphere moving through their plane will not look like a circle that appears, grows, shrinks and disappears.. it will just look like a line... as practically everything will. So in that vein, can we even see the 3D shadow of a 4D sphere, or are we seeing 2D "garbage" of an actual 3D shadow of a 4D sphere?
This is a good point but you are slightly wrong in an important way! Imagine looking at the Eiffel tower from a distance, and holding up a small model of the Eiffel tower next to it so that they appear to be the same size. In this picture, the two things you're seeing look very similar. In a photograph, you might even be able to cleverly make them seem identical. But your brain can usually pretty easily work out how far away something is. It is able to do this because your brain is extremely well-adapted to: EXTRACTING A 3D MODEL OF THE WORLD FROM 2D THINGS. Even this video is technically 2D! It's just pixels on a screen! And yet you're able to conceptualize that room I was in in 3D! A fish that evolved in a 2D universe would only get a 1D picture of the world - it would indeed just see a line if it looked at a circle. BUT, its vision would PROBABLY be well-evolved for the task of noticing curvature, i.e. noticing that some parts of the line are closer to it than other parts. This is just like how you can look at a snooker ball and perceive that it is a sphere, not a circle/disc. So the fish DOES get a 2D idea of its world, just as you get a 3D idea of our world. Of course, it doesn't see the whole thing. If one thing is in front of another from its point of view, the fish can't see the hidden thing. So, when you "look down on" the 2D world, you do get a better view of it. You can see the shape of every object, and you can even see things which are, from the fish's point of view "contained inside" other things. So when a 4D creature "looks down on" a 3D world containing an object like your body, it can see... :)
@@hamish_todd how can we see the 2 dimension in reality? Its only exist in our mind isnt it? Im not sure if we are able to interact just as the 4d unable to understand our primitive way ....i mean ,what 4d getting from 3d its a very different story in its own world ,its not something developing or add to the previous knowledge, its a whole new thing. Nothing from the 3d world makes sense for the 4d and vica versa ......am i making sense? :)
Assuming the jumps for the degrees of freedom continue in this pattern 1->3->6->10 the solution should be the number of dimension d and its corresponding degree of freedom f added together to get out the degrees of freedom for the next higher dimension so you get for 2d+1f(2d)=3 degrees. 3d+3f(3d)=6. 4d+6f(4d)=10, 5d+10f(5d)=15 Therefore my guess for the 6th dimension would be 21 degrees of freedom. This patter can be described by the formula : f=(d^2-d)/2
Based on your shadow manipulation, I'm guessing the principle is: the number of rotational degrees of freedom in n dimensions is equivalent to the number of rotational + translational degrees of freedom in n-1 dimensions. Because the number of translational degrees of freedom increases by 1 every time, the rotational degrees of freedom is described by the triangular numbers.
The headset is very large so the cover had to be large too! If you look at for example 0:56 you see it poking out underneath Yeah the head steals the show, but nothing to be done 🤷
Well I have an idea of my own. The fish, since it lives on 2 dimensions would have its perspective only be a line. So if you put the shadow of a sphere in, it would look like lines appearing and disappearing. And so with that in mind, wouldn’t the shadow of a 4D sphere look like a bunch of geometric planes morphing and twisting?
Ohhhh, this is a good point! Note how what the fish sees in its world is a bunch of lines (it can see how far away they are from it, so it can get an idea of their 2D shape, just as you can get an idea of the 3D bunch of twisted wires with your eyes). I COULD have covered the spheres that I showed to the fish with POINTS, then it'd just see a bunch of points moving around, obviously. I chose lines because that lets me have nice things like the world map. You're right that you COULD have stick a bunch of planes on the surface of a sphere in 4D. Then you could project that down to 3D and see a bunch of morphing and twisting, yes! That's how eg this nice video was made ruclips.net/video/cermfDnqQ5M/видео.html I chose to make it simple for myself and just have lines / curves on both the 3D sphere and the 4D sphere :) one reason is that it's easier to see how the kind of morphing you get when you bring them down to a lower dimension is similar
Hi there. I have thinking alot about fourth dimension. And most of videos shows us example on 2d dimension creatures. But no one told us that two demensional creatures sees in one dimesion. Like we have 2D screens, we also can see only 2D pictures, we can't see 3D, we just imagine it because of how it looks. to see 2D structures at full shape, we need to be 1 dimension higher, and we are. we can see 2D shapes all at once, but 2D creatures can't. Same with 3d Shapes. we can only see "shadow" of outer form, we can't see inside, to do that we need to be fourth dementional creatures, but we are not. We using X-rays and wireframe for 3d-apps, but we always see 2d image of 3D object. And to go thurther to see 4D at once we need to be 5D creatures (o.o') but yeah 4D creatures view is 3D and we can use VR for it. But we will see only shadow of the object, not the object itself... (btw this is my thoughts about 4D)
You are correct! There is a funny joke about this in Futurama ruclips.net/video/JSq-uYkPYNo/видео.html here's another good video ruclips.net/video/NZFxQXe7LMM/видео.html see also "Flatland" A human eye only forms a 2D picture, yes. You can only see the outsides of things, you can't see the inside and outside at. It helps that we have two eyes though, just as it would help a 2D creature if they had 2 eyes!
@@hamish_toddnice joke xD, but it really shows how 2d creatures sees things) And yes, two points of view can tell us the deph of the objects that we see, but isn't this a result of brain activity? I'm wandering is there a way to teach brain see 4D objects? And another thing, How to make view of 4D object. Because shadow it is only a shadow in 3D, not a 4D object itself. And we can see that shadow again only in 2D. I think about using VR to make our brain learn how 4D object looks like. Because in video we can't feel deph of the object. Also I saw only three ways to represent 4D objects: 1. Shadow, like in your video; 2. 3D slice of 4D object; 3. Projection of 4D object to 3d space. And I think third is a way to go. And rotation is used to be like in your video. Texturing is anothe question...
@@TheRedKorsar Alas our eyes evolved in this universe, you can never *really* see a 4D object, you can only use the tricks :) Your methods 1 and 3 are the same. One other method is to put a 3D creature in a 3D shape that is twisted around in the higher-dimensional space in some way. Here is a video that does that ruclips.net/video/_5l8v6Gn2sE/видео.html and there are videos on the same channel which twist the 3D space in different ways.
@@hamish_todd woah, yeah non Euclidian space is a way to go :D looks so great! Also 1 and 3 metod are kind of same but not exactly. By shadow I mean that it will be point light source, and for all slices that will be like parallel light source... With first we can get 3d infinite shadow, but not with parallel casting.
@@TheRedKorsar Sure. The jargon for those is: Projection from a point: "point projection". Projection with parallel lines: "orthographic projection". Essentially both can be done with either shadows or with cameras!
You should’ve used our globe in 4d so we could see some familiar things in our everyday lives to see how they behave in the 4d. If that’s even possible.
It isn't lol. The "surface" of a 4d hypersphere is actually an entire 3D sphere including the interior. If you only mapped the surface of the surface (so to speak), you wouldn't be able to see most of the object.
What would you like to know? :) I have a little presentation on how I did the 4D stuff that is available to my patrons www.patreon.com/hamish_todd/creators You can find the code in this directory github.com/hamishtodd1/hamishtodd1.github.io/tree/master/mixedReality - code for the spheres is in twoSphereExploration.js and threeSphereExploration.js
No depth information at all! The video is always behind. It helps that I have a 3D model of my controller floating around. I need to make sure my arm doesn't get too close to my helmet though, it looks bad at eg 7m24
That eye tracking works so well lool
My attempt to explain the degrees of freedom for orientation working from the fact that a n dimensional creature could manipulate a (n+1) dimensional object through all of that object's degrees of rotational freedom:
A 2D creature (the flat fish) can rotate a 3D object by using its 1 degree of freedom of its own rotation and its 2 degrees of freedom from position for a total of 3.
A 3D creature (the human) can rotate a 4D object by using its own 3 degrees of rotation and the 3 from position for a total of 6.
A 4D creature can rotate a 5D object by using its own 6 degrees of rotation and the 4 from position for a total of 10.
A 5D creature can rotate a 6D object by using its own 10 degrees of rotation and the 5 from position for a total of 15.
(and so on.)
interesting.
Great explanation
This lines up well with the second diagonal of Pascal's Triangle.
Another way: in 2D you can rotate on the xy plane, in 3D you can rotate on the xy, xz, and yz planes, in 4D you can rotate on the xy, xz, yz, xw, yw, and zw planes, etc
@@columbus8myhw If you wanted to unchain yourself from the origin, then you could say that you can rotate an ND object around any N-1D subspace. 2-2 = 0, so in 2D, you can choose any point (0D subspace) to rotate around. In 3D, 3-2 = 1, so your axes are any line. 4D yields the mind bending planar axes of rotation since 4-2 = 2.
Interestingly, the number of degrees of freedom for an ND rotation around an arbitrary center is the same as the number of degrees of freedom for an N+1D rotation around the origin.
Moral of the story: If you control someone's shadow, you control them entirely!
Bro, that eye tracking with your VR mask is trippy. I love it. 1:05 it even captured the subtle angle of shifting your eyes from a near target (the globe) to a farther target (the camera lense).
oh wow, mixed reality presentations are cool
Hypersphere look like magnetic field
The head thing earned my subscription.
Is it there so that we have eyes to look at and focus on while yours are in VR land?
Yup!
Time is our broken perception of the 4th dimension. We here (living in a 3D experience) can’t perceive the entirety of 4D reality, so we break it down into 3D phenomena that change over time. So, time literally is the 4th spatial dimension, but spread out as a temporal experience. Time is the point where 3D intersects with a 4D object.
This is so unbelievably amazing! Thank you so much for making this video. I've witnessed impossible for 3 dimension objects while using DMT. this is so fascinating, after that experience.
4:26 Looks like magnetic field lines.
I wonder if 4d people have 3d games and for them it is like our 2d games
I thought of either:
* the set of independent rotations as being the set of independent 2D subspaces, giving C(D,2), D choose 2
* in D dimensions, we can think of the hypersphere as being prametrized by D-1 angles. But also any point in the sphere looks locally like R^(D-1), so it has as extra rotational degrees of freedom, as many as the space in D-1 dimensions. Let's call the number of rotional degrees of freedom in D dimension, M(D). Then we have M(D) = D-1 + M(D-1). Because in D=2, we have M(2)=1, then M(D)=sum_{i=1}^{D-1} i = D*(D-1)/2, the triangular numbers which are the same as C(D,2)
1, 3, 6, 10 Are the triangular numbers!
The fish and globe shadow was something I had seen before but I never found a good mental understanding for this until now: giving the fish the shadow as it's method of rotation is essentially mapping the 3 degrees of freedom of the fish to the 3 rotation degrees of freedom of the sphere. The fish can use any of it's 3 degrees to manipulate the 3 degrees of the sphere
Since rotation always happens within a plane, and a plane is always two-dimensional, the number of rotational degrees of freedom is equal to the number of planes that can be created by choosing any two of the axes within the higher dimensional space (the axes themselves represent translational degrees of freedom).
So, the number of rotational degrees of freedom for a space with dimension d is given by (d choose 2) which can be written as d! / 2(d - 2)! from the binomial formula.
Another thing many commentors have noticed is that the rotational degrees of freedom in dimension d is equal to the sum of the translational and rotational degrees of freedom in dimension d - 1, which seems like a strange coincidence. I believe I have an intuitive explanation of that as well.
The first obvious fact is that the higher dimensional space inherits the rotational degrees of freedom from the lower dimensional space. In 2D there is a single plane of rotation (the XY-plane). 3D space inherits this plane and adds two additional planes (the YZ-plane and the ZX-plane), but why exactly the number two? Why the exact number d - 1?
Well, when we go up a dimension, we add another axis. This additional axis can then be paired up exactly once with each already existing axis to form a new rotational degree of freedom. For example, when we add the z-axis, we form two additional planes of rotation precisely because there were already two existing axes to form the planes with.
Similarly, moving from 3D to 4D by adding a w-axis, this new axis can be paired up with the existing x,y and z axes to form new rotational degrees of freedom, resulting in 6 rotational degrees of freedom. And so on, for higher dimensions.
I would love more videos like this! Really enjoyable and with a unique presentation.
In this representation it could be worth mentioning that 4 dimensions enable two simultaneous but independent rotations, one along one 2D coordinate plane and the other along the plane of the remaining 2 dimensions. Thus there are no "axes" of rotation but rather planes. Considering the 3D surface of the 4D sphere, longitudes and latitudes could also work better if we visualize them as a grid of cells made of intersecting surfaces rather than lines. The system (or one of the possible systems) would have two orthogonal "equators" lined with prism-shaped cells while all the other cells are distorted cubes.
4D rotations don't correspond to linear coordinate axes, but you can argue that you still have "axes of rotation" that just happen to be planes rather than lines. This kind of thinking helps when not limiting yourself to rotations around the origin and also allows for more complex "rotations", like a 4D rotation-like transformation that's actually "around" a sphere.
spoiler alert | 2D rotation is the only rotation that has one degree of freedom, therefore, if a sphere can have its projections make X different two-dimensional rotations, it has X*1=X degrees of freedom, and to make a two-dimensional rotation, take a straight line and rotate it in the place of another line, leaving unchanged all the lines perpendicular to the rotation dimension, then taking perpendicular projection planes you can have lines to rotate that are perpendicular to each other, so you must select all the sets of two perpendicular lines , that having X dimensions there are X perpendicular lines,, with quick calculations, the number of sets being equal to the number of rotations, comes (x*(x-1))/2 degrees of freedom for X dimensions sphere
The “number degrees of freedom” of rotation, is referring to the dimensionality of the rotation group, right?
So, in 2D, the group is SO(2) (the circle group), and in 3D it is SO(3), and in n-D, SO(n)
an element of SO(n) is specified by giving an ordered orthonormal basis of n-D space (such that the determinant you get is 1 rather than -1, but that requirement won’t change the dimensionality)
specifying the first vector, there’s an n-1 dimensional choice there (the minus one is because the vector has to be of a specified length, so we remove a factor that is from scaling),
and then the what remains is to pick an orthonormal basis for the subspace orthogonal to the vector chosen first.
So, f(n) = (n-1) + f(n-1)
And f(0)=0
so, f(1) = 0, f(2)=1, f(3)=3, f(4)=6,
Yeah, these are the triangular numbers,
so, the dimensionality of SO(n) is n(n-1)/2 .
So, in particular, the dimensionality of SO(5) is 10.
this rotated 4d ball looks like electromagnetic field
Beautifully presented, thank you
Is there any evidence of higher spacial dimensions in our universe?
Don’t check reply if you don’t wanna get spoiled
They vr really is good visual. I'd use this video in schools.
Absolutely fantastic video! The visualization is amazing and your explanation is very clear!
Rotation is essentially a 2D operation that occurs in a plane. The second dimension is the first dimension where rotations can occur, because we have two coordinates, x and y. In the third dimension, we add another coordinate direction, z, and so we get two more possible rotations, in xz, and in yz. Add another direction perpendicular to the other three and we arrive at the fourth dimension where there are three more rotations possible: xw. yw, and zw. I love it that you can have two rotations occurring simultaneously in the fourth dimension that do not affect each other, like xw and yz at the same time.
Thanks for posting this video and explanation!
Each new dimension adds (d-1) rotations because you can rotate through that new dimension across each one of the previous dimensions. 6 dimensions gives 10+5=15, 7 dimensions gives 15+6=21, and so on.
The rule is (n)(n+1)/2
The sequence has the exact same rule as triangular numbers.
@@Tshego2000n(n-1)/2 surely?
For 2D space, there is one dimension of rotation, not 3.
Great conceptualizing , really amazing work dude! you may have already adjusted, but a new microphone would help the audio. Keep it up!
Are you actually doing eye tracking, or are you snapping the gaze to the object closest to your center of vision?
Either way it's an impressive effect!
The second one!
you can figure out degrees of freedom seeing how many extra planes will exist. For example, if a being is in the x, y plane to get the extra planes from the 3 dimensions the extra axes are y,z and x,z. So if for example we are adding an extra coordinate f the extra planes added from 3 dimensions to 4 dimensions would be x,f; y,f and z,f.
I love the fish so cute
2:20 The projection of the sphere’s shadow onto the 2D plane is identical to the Steinmetz diagram of electromagnetism. Apollonian circles, each pair of which meets at 90 degrees to one another, ie like being inside a torus.
This is also what the Yin Yang symbol means. It’s a symbolic representation of a 4D sphere, or a 3D torus, as a 2D image.
A 4D sphere and a 3D torus aren’t the same?
Also, do you mean a 3-sphere which lives in 4D space, or do you mean a 4-sphere?
And, by a 3D torus, do you mean a solid torus (a three dimensional manifold with boundary isomorphic to S^1 x S^1 ) or do you mean S^1 x S^1 x S^1 ?
If you glue two solid torii together at their boundary in a way that exchanges which direction around the boundary is contractable, then I believe you get the 3-sphere. Maybe that’s what you were getting at?
(If you glue it the other way, uh, I think you get the product S^1 x S^2 ?)
@@drdca8263 A 3D torus is a projection of a 4D sphere. In toplogical terms, I mean a 3-sphere, and I do not mean solid (which would be more appropriately called a "ball").
@@hermes_logios pi_1(a 3D torus) is non-trivial, but pi_1(S^3) is trivial.
Like, there are non-contractible loops in the 3D torus (regardless of whether you mean D^2 cross S^1, or S^1 cross S^1 cross S^1, or S^1 cross S^1 ) , but no non-contractible loops in S^3 .
But, any loop in the image of a projection should be the image of some loop in the domain of the projection,
like, if you have some path in the image of the projection, and take a point in the preimage of some point in that path, there should be a path in the domain of the projection which goes through that point, and when composed with the projection, gives the path in the image of the projection (assuming that it is a linear projection, as in this video, not just a more general continuous surjection which is a one-sided inverse of some inclusion)
and also assuming, that the space that is being projected, is convex...
Oh wait, the 3-sphere isn’t convex, hold on....
Wait I have to go do something, I’ll say more later...
But I think the argument I have here will work with some modification
Edit: ok, meeting was canceled.
Ok, so, while the 3-sphere as embedded in R^4 isn’t convex, for any two points on it, if they aren’t antipodal, there is a unique shortest path between them.
If we take a loop in the image of S^3 under the projection, then...
the preimage of that loop will have at least one point in the 3-sphere above each point in the curve in the image of the 3-sphere,
and, ah, specifically each point in the loop should either have 1 or 2 points in its preimage,
and, yes, we can indeed locally trace the path,
though perhaps going around the loop could produce a path from one preimage of the base point to the other...
But then, going around again would lead to the first one again,
so, any loop in the projection, *when gone around twice*, is the image of some loop in S^3,
And any loop in S^3 is contractible, and
the image of the contraction of the loop will be a contraction of the image of the loop,
and therefore, for any loop in the projection of S^3 (or any sphere) under a like, linear projection onto a “screen” one dimension down, like in this video,
will have the repeating of that loop twice, be contractible.
But! In the torus, there are loops which cannot be contracted, even after you repeat them any number of times.
Therefore, the image of S^3 under a projection of the kind shown, is not the torus.
Very very nice presentation!
xy 1, xy xz, yz 3, xy xz yz wx wy wz 6, ...with 5 coordinates we have 10 different ways to pick a pair of coordinate axis xy xz yz wx wy wz vx vy vz vw. Nice work!
Please put a camera on your reflection map. It's freaking me out.
Email me a picture of your face looking intrigued :D
Wooow, very impressive
Greetings from Mexico,🇲🇽
And... Maybe the fórmula of sucesion is: (n-1)n/2
Welcome to ASTROLEA!
It's interesting that the shadow of a 4 Dimensional Sphere, looks very much like the magnetic field.
3d object has 2d vision with depth proception. Disadvantage: has to guess what is behind an object.
2d object with 1d vision?
I can just imagine 4 dimensional beings just looking down on 3 dimensional beings talkign about them like fish.
Reason I came here is to understand this better... when a hyper cube is rotated it looks like it's propagating ... so I got to thinking what does a sphere look like spinning.. does it also look like it propergates like a hypercube the I remembered the two ways you can rotate a hypercube and realised its not two ways but the combination of both or it would would not be rotation. I'm beginning to think that light lies on the boundary of two and 3 dimensional space and black holes are onthe boundary of 3 and 4th dimensional space and the rotation of an atom is 4th dimentional
Not a formal explanation, just my intuition. In 2D, you can rotate from one dimension onto the other, so there is only one pairing (degree of freedom). In 3D, you rotate a dimension onto one of the other two dimensions, that’s 2, than there’s a pairing between the remaining two dimensions, that’s 1 more, for a total of 3 degrees of freedom. In 4D, you can rotate a dimension onto one of the other three dimensions, that’s 3, then of the three remaining dimensions, one of the remaining dimensions can rotate onto the other two, that’s 2 more, and the remaining two can rotate onto each other, that’s 1 more, for a total of 6 degrees of freedom. So with with each additional number of dimensions, the additional dimension can rotate onto the previous dimensions, which there are n-1 of, and all degrees of freedom from the lower dimensions are preserved.
I'm a math instructor so I immediately got the correct answer of 10, without the faintest idea of why it would be the correct answer.
Instructor?
@@dmitri2011 Yeah, it's like a low grade tutor.
you are forgetting that the fish would only be able to perceive its world as a one dimensional line. due to the nature of the 2 dimensional fish, nothing can ever be behind or in front of the fish. it would only be able to interact with and view the shadow of the globe by grabbing the out most edges, and only seeing the shadow as a line changing in vertical and horizontal sizes, and various edges and smooth parts of the outermost line of the shadow.
the fish wouldn't be able to grab the shadow from whatever position it wants nor view the shadow from an above perspective as shown in the demonstration
1 + 2 + .. (n-1), where n is the number of dimensions. Its simple, the number of degrees of freedom is the number of parameters to describe orientation of coordinate system. The end of the first axis of coordinate system is on n-dimensional sphere (n-1 parameters), the end of the second axis then has to be on the n-1 dimensional sphere and so on.
How did you make that augmented reality it’s so cool
The number of degrees of freedom is n choose 2 because rotation happens in 2 out of the n dimensions.
You use the white board!
I guess The number of rotational degrees of freedom would be n*(n-1)/2 which is equal to the sum of k with k varying from 1 to n, with step 1.
The reason is simple. We just have to count the number of "planes" in a given space. Suppose we are in 4d space, then, there are 4 axis, and each axis can be combined with the other 3 remaining axis, which would result in 4x3=12 possibilities. But we have to divide the result by 2 because of the repeated permutation.
Nice, but that's not what I was looking for. It's said that a 4d creature can mirror 3d objects, so I'm looking for a computer animation that shows how that would look to us.
You can work it out by thinking about what a 2D flatlander would see if you were to "mirror" one of their friends by rotating them in the plane they live in. First, you'd choose a line in their plane that you were going to mirror them around. Second, you'd rotate them 180 around that line. Their friend would see the line part of them stay in place - so, if the flatlander had their liver and spleen on that line, as you rotate them, their friend would see a 1D cross section of their liver and spleen appear to stay where it was while they were rotated. Before you start the rotation, their friend would see them; during, they'd see only the cross section; and at the end, they'd see their friend reflected, with that 1D cross section the only part that has stayed in place.
So what's this like for 3D creatures rotated by a 4D creature? You work it out. Clue: in 4D, you do not rotate around lines, you rotate "around" planes.
This is the best explanation I've seen!
I fidn the mask distracing
Rotations in n dims = n C 2 (n choose 2)
1, 3, 6, 10, 15, 21, 28, 36..........?
The thing that always confuses me with these 2D analogies is that we do not really see what the fish sees - we see a top-down view of the fish's world. The fish only sees in 1D, so the shadows of the 3D sphere, as confusing as it looks for us, looks just like random 1D garbage to the fish. In other worlds, the citizens of Flatland, the 3D sphere moving through their plane will not look like a circle that appears, grows, shrinks and disappears.. it will just look like a line... as practically everything will.
So in that vein, can we even see the 3D shadow of a 4D sphere, or are we seeing 2D "garbage" of an actual 3D shadow of a 4D sphere?
This is a good point but you are slightly wrong in an important way!
Imagine looking at the Eiffel tower from a distance, and holding up a small model of the Eiffel tower next to it so that they appear to be the same size.
In this picture, the two things you're seeing look very similar. In a photograph, you might even be able to cleverly make them seem identical.
But your brain can usually pretty easily work out how far away something is. It is able to do this because your brain is extremely well-adapted to: EXTRACTING A 3D MODEL OF THE WORLD FROM 2D THINGS. Even this video is technically 2D! It's just pixels on a screen! And yet you're able to conceptualize that room I was in in 3D!
A fish that evolved in a 2D universe would only get a 1D picture of the world - it would indeed just see a line if it looked at a circle. BUT, its vision would PROBABLY be well-evolved for the task of noticing curvature, i.e. noticing that some parts of the line are closer to it than other parts. This is just like how you can look at a snooker ball and perceive that it is a sphere, not a circle/disc.
So the fish DOES get a 2D idea of its world, just as you get a 3D idea of our world. Of course, it doesn't see the whole thing. If one thing is in front of another from its point of view, the fish can't see the hidden thing. So, when you "look down on" the 2D world, you do get a better view of it. You can see the shape of every object, and you can even see things which are, from the fish's point of view "contained inside" other things.
So when a 4D creature "looks down on" a 3D world containing an object like your body, it can see... :)
@@hamish_todd how can we see the 2 dimension in reality? Its only exist in our mind isnt it? Im not sure if we are able to interact just as the 4d unable to understand our primitive way ....i mean ,what 4d getting from 3d its a very different story in its own world ,its not something developing or add to the previous knowledge, its a whole new thing. Nothing from the 3d world makes sense for the 4d and vica versa ......am i making sense? :)
Ooh! Ooh! I know! For dimensions D, the number of rotational axes is C(D, 2) for combinatorial function C.
The rule is (n)(n+1)/2
Look up the quaterion system from Sir William Rowan Hamilton, he described the 4th dimension as beeing spacial.
i^2= j^2= k^2= ijk = -1
Assuming the jumps for the degrees of freedom continue in this pattern 1->3->6->10 the solution should be the number of dimension d and its corresponding degree of freedom f added together to get out the degrees of freedom for the next higher dimension so you get for 2d+1f(2d)=3 degrees. 3d+3f(3d)=6. 4d+6f(4d)=10, 5d+10f(5d)=15 Therefore my guess for the 6th dimension would be 21 degrees of freedom. This patter can be described by the formula : f=(d^2-d)/2
Based on your shadow manipulation, I'm guessing the principle is: the number of rotational degrees of freedom in n dimensions is equivalent to the number of rotational + translational degrees of freedom in n-1 dimensions. Because the number of translational degrees of freedom increases by 1 every time, the rotational degrees of freedom is described by the triangular numbers.
The number of degrees of freedom is given by the triangular numbers...exactly why,I'm not sure.
That head thing is unimaginably distracting. Perhaps a Zoro style mask would get the job done?
The headset is very large so the cover had to be large too! If you look at for example 0:56 you see it poking out underneath
Yeah the head steals the show, but nothing to be done 🤷
@@hamish_todd I wasn't aware you were wearing a headset, my apologies! Although, I admit I'd rather see the real headset.
@@Dark_Jaguar Look in my channel for "Lecture inside virtual reality" and see what you think!
2D = 1 degree + 2 is 3D 3 degrees and +3 is 4D 6 degrees and +4 is 5D 10 degrees so i think 6D has 15 degrees of rotation.
Im baked and this really helped understand 4d. Great explanation
Is the formula, the previous dimension + previous degrees of freedom?
Very cool
Lol
kann younot maka 4d tig in vr ???
Nice
I like to imagine we cant even imagine how it looks, so 4d is even more awesome
Well I have an idea of my own. The fish, since it lives on 2 dimensions would have its perspective only be a line. So if you put the shadow of a sphere in, it would look like lines appearing and disappearing. And so with that in mind, wouldn’t the shadow of a 4D sphere look like a bunch of geometric planes morphing and twisting?
Ohhhh, this is a good point!
Note how what the fish sees in its world is a bunch of lines (it can see how far away they are from it, so it can get an idea of their 2D shape, just as you can get an idea of the 3D bunch of twisted wires with your eyes).
I COULD have covered the spheres that I showed to the fish with POINTS, then it'd just see a bunch of points moving around, obviously. I chose lines because that lets me have nice things like the world map.
You're right that you COULD have stick a bunch of planes on the surface of a sphere in 4D. Then you could project that down to 3D and see a bunch of morphing and twisting, yes! That's how eg this nice video was made ruclips.net/video/cermfDnqQ5M/видео.html
I chose to make it simple for myself and just have lines / curves on both the 3D sphere and the 4D sphere :) one reason is that it's easier to see how the kind of morphing you get when you bring them down to a lower dimension is similar
Hi there. I have thinking alot about fourth dimension. And most of videos shows us example on 2d dimension creatures. But no one told us that two demensional creatures sees in one dimesion. Like we have 2D screens, we also can see only 2D pictures, we can't see 3D, we just imagine it because of how it looks. to see 2D structures at full shape, we need to be 1 dimension higher, and we are. we can see 2D shapes all at once, but 2D creatures can't. Same with 3d Shapes. we can only see "shadow" of outer form, we can't see inside, to do that we need to be fourth dementional creatures, but we are not. We using X-rays and wireframe for 3d-apps, but we always see 2d image of 3D object. And to go thurther to see 4D at once we need to be 5D creatures (o.o') but yeah 4D creatures view is 3D and we can use VR for it. But we will see only shadow of the object, not the object itself... (btw this is my thoughts about 4D)
You are correct! There is a funny joke about this in Futurama ruclips.net/video/JSq-uYkPYNo/видео.html here's another good video ruclips.net/video/NZFxQXe7LMM/видео.html see also "Flatland"
A human eye only forms a 2D picture, yes. You can only see the outsides of things, you can't see the inside and outside at. It helps that we have two eyes though, just as it would help a 2D creature if they had 2 eyes!
@@hamish_toddnice joke xD, but it really shows how 2d creatures sees things)
And yes, two points of view can tell us the deph of the objects that we see, but isn't this a result of brain activity? I'm wandering is there a way to teach brain see 4D objects? And another thing, How to make view of 4D object. Because shadow it is only a shadow in 3D, not a 4D object itself. And we can see that shadow again only in 2D.
I think about using VR to make our brain learn how 4D object looks like. Because in video we can't feel deph of the object. Also I saw only three ways to represent 4D objects:
1. Shadow, like in your video;
2. 3D slice of 4D object;
3. Projection of 4D object to 3d space.
And I think third is a way to go. And rotation is used to be like in your video. Texturing is anothe question...
@@TheRedKorsar Alas our eyes evolved in this universe, you can never *really* see a 4D object, you can only use the tricks :)
Your methods 1 and 3 are the same.
One other method is to put a 3D creature in a 3D shape that is twisted around in the higher-dimensional space in some way. Here is a video that does that ruclips.net/video/_5l8v6Gn2sE/видео.html and there are videos on the same channel which twist the 3D space in different ways.
@@hamish_todd woah, yeah non Euclidian space is a way to go :D looks so great!
Also 1 and 3 metod are kind of same but not exactly. By shadow I mean that it will be point light source, and for all slices that will be like parallel light source...
With first we can get 3d infinite shadow, but not with parallel casting.
@@TheRedKorsar Sure. The jargon for those is:
Projection from a point: "point projection".
Projection with parallel lines: "orthographic projection".
Essentially both can be done with either shadows or with cameras!
Amazing, love the video format! Subbed
You should’ve used our globe in 4d so we could see some familiar things in our everyday lives to see how they behave in the 4d. If that’s even possible.
It isn't lol. The "surface" of a 4d hypersphere is actually an entire 3D sphere including the interior. If you only mapped the surface of the surface (so to speak), you wouldn't be able to see most of the object.
hi mate i really need to know how do you drag and grab objects like this. Could you help me please?
What would you like to know? :)
I have a little presentation on how I did the 4D stuff that is available to my patrons www.patreon.com/hamish_todd/creators
You can find the code in this directory github.com/hamishtodd1/hamishtodd1.github.io/tree/master/mixedReality - code for the spheres is in twoSphereExploration.js and threeSphereExploration.js
What is the software you are using here for the fish and the AR head? Something you wrote?
Yup! Code here github.com/hamishtodd1/hamishtodd1.github.io/tree/master/mixedReality
R'yleh R'yleh R'yleh!!!
I'm going to cry
Beautifully done!
What are you using to capture depth information from the camera? Your AR is very impressive basically no z fighting issues
No depth information at all! The video is always behind. It helps that I have a 3D model of my controller floating around. I need to make sure my arm doesn't get too close to my helmet though, it looks bad at eg 7m24
real nice !
Love it!
I can't find any reference to that 4D object you mention at 7:35 (the hock? hot? hoch? vibration) -- could you give details?
Ah yes it is an awful spelling! It's the "hopf fibration"