@@HyperCubist Truly! I’ve seriously watched many many many videos on this topic to try and get the Euclidean horror trip out of my psyche 💯😅 man thank you so much for your content! Foreal man!!! I’m super hype for your next video!
When problem solving, being able to simplify the question is a really useful skill to have which you seem to have done really well here. Simplifying a 4D sphere into a 1d line through a few steps is a really good idea as you can build it back up.
Very interesting video, thank you! There's a math field calld Geometric Algebra or Clifford Algebra which is tightly related to rotations For example if you have two unit vectors, 𝐮 and 𝐯, formula for rotating arbitrary vector 𝐰 in the direction from 𝐮 to 𝐯 would be: 𝐰′ = 𝐮𝐯𝐰𝐯𝐮 Where infix operator is geometric product which is really easy And that would work in any number of dimentions
1:00 If you use the Minkowski hyperboloid model, then you can use rotations to perform translations as you say, but I think Hyperbolica actually uses the Poincaré disc model, and instead uses something known as gyrovectors, which instead of performing 4D rotations to achieve translations, perform 3D Möbious transformations to achieve the same thing.
The Poincaré disc doesn't have as nice properties as the hyperboloid model, though, IMO, because in the hyperboloid model, several chained rotations trivially combine to just one rotation by simply multiplying the matrices together, which is useful when optimizing the rendering of hierarchical scenes in computer graphics (and probably for other things too). On the other hand, while several chained Möbvious transformations also combine to form another Möbvious transformation, the mathematics for combining them is much less straightforward and somewhat tricky to derive.
The developer of Hyperbolica mentions that he uses gyrovectors in his devlog "Rendering Hyperbolic Spaces - Hyperbolica Devlog #3", but it may also be the case that he later switched to using the hyperboloid model. I know that he has changed other things that he also spoke about in his devlogs. In anyway, it's an interesting game concept.
I still have no idea what Im looking at! My mind simply can't understand the hypersphere... Maybe I can understand the hypercube - I have 2 videos on my channel about it - but the hypersphere might forever elude me.
3:47 this isn't really correct, in all 3 cases it is a plane in 2d, you are rotating in the xy plane in 3d, you are also rotating in a plane, but this time the plane can be oriented in an arbitrary direction in 4d, of course you are also rotating in a plane rotations fundamentally happen in planes, thinking about it in terms of an axis of rotation is technically incorrect, and it only works in 3d due to the coincidence that the number of basis vectors (x, y, z) is the same as the number of basis planes (xy, yz, zx) axis rotations are also kinda weird because the direction of rotation around an axis is undefined, instead it has to be defined by convention using the right hand rule, but with bivectors the direction is obvious; for example, rotation in the yz plane brings the vector y to the vector z planes can be used for rotation in any number of dimensions, if you try 5d you will see that this still remains true
Not incorrect, just a different way to look at the same thing, right? Though, granted, it's perhaps more natural to talk about the plane of rotation when talking about different numbers of dimensions.
As kantomega mentioned its just a different look at the same thing. Also I was talking about how its usually implemented and used in games. Namely Unity implements Quaternion.AngleAxis(angle, axis) to do rotations in 3D. Atleast my prefered way of doing 3d rotations.
3:50 NOO!!! All rotations happen inside a plane! in 0D and 1D you can't rotate because there aren't any planes, in 2D there's one plane of rotation so 1 value to describe rotations. in 3D there are 3 planes, in 4D there are 6 and so on with the formula of planes = (d * (d - 1)) / 2. It's a common misconception that rotations happen in d-2 (axis for 3D, point for 2D, plane for 4D) but that's not true. The reason we use planes in 4D for rotations is that you can't get away with using axes. in 3D you can get away with it by saying Y axis instead of XZ plane because that's the only plane perpendicular to the Y axis. but in 4D, the Y axis is perpendicular to the XZ, ZW and XW planes. edit: saw someone else said the same thing. Sure, its technically different ways to say the same thing but its still kind of misleading. like if you use use planes in 4D rotation and points in 2D, then rotating *around* the XY plane in 4D means rotating *in* the ZW plane. anyway I liked your video!
3:47 This is actually pretty misleading. Rotations are a 2d phenomenon, meaning it can only happen within a 2d plane. When you rotate in 2d space, there is only a single point that does not rotate. In 3d, you still fundamentally only rotate along a single plane, but that plane is extruded along the 3rd axis, rotating the whole object with a line being stationary. You can also add another plane to rotate along, but it only ends up rotating the original plane along a non-parallel plane. You can verify this yourself by grabbing an object, imagining a plane slicing through the middle, and arbitrarily rotating the object. You can rotate it however you want, but it will still rotate along the imaginary plane that is itself rotating. However, in 4d, you can create a plane and still have 2 more axes to create yet another plane. If you rotate one of those planes, the other is unaffected. Assuming the first plane is still rotating, there is no reason to assume the 2nd plane can't. Rotating both at the same time (or in sequence to get from one 4d rotation to another) is called a double rotation.
We made it up, so he can say it ideolectally. He just needs to get around 675 million or so other people to agree with him and then his take would be in the majority. There's no "correct" way, or way you're "supposed" to speak, it only matters that you're understood Again lol, there's nothing "natural" about english syntactics in this sense, we just made it all up.
What's the Wii axis? Is it something to do with the gaming console from Nintendo?
loved the discussion of projection of the trig unit circle to unit sphere and unit hypersphere, that "exchange of values" in rotations
THANK YOU FOR THE UPDATE!!!!!!! I’m so happy someone on UA-cam actually cares about what I ponder about before I go to sleep ❤️
4D is a small world, huh?
@@HyperCubist Truly! I’ve seriously watched many many many videos on this topic to try and get the Euclidean horror trip out of my psyche 💯😅 man thank you so much for your content! Foreal man!!! I’m super hype for your next video!
Impressive, it's amazing how we can understand better math using visualisations.
When problem solving, being able to simplify the question is a really useful skill to have which you seem to have done really well here.
Simplifying a 4D sphere into a 1d line through a few steps is a really good idea as you can build it back up.
Very interesting video, thank you!
There's a math field calld Geometric Algebra or Clifford Algebra which is tightly related to rotations
For example if you have two unit vectors, 𝐮 and 𝐯, formula for rotating arbitrary vector 𝐰 in the direction from 𝐮 to 𝐯 would be:
𝐰′ = 𝐮𝐯𝐰𝐯𝐮
Where infix operator is geometric product which is really easy
And that would work in any number of dimentions
did you say uwu in a mathematical context on purpose or am I imagining thins
1:00 If you use the Minkowski hyperboloid model, then you can use rotations to perform translations as you say, but I think Hyperbolica actually uses the Poincaré disc model, and instead uses something known as gyrovectors, which instead of performing 4D rotations to achieve translations, perform 3D Möbious transformations to achieve the same thing.
The Poincaré disc doesn't have as nice properties as the hyperboloid model, though, IMO, because in the hyperboloid model, several chained rotations trivially combine to just one rotation by simply multiplying the matrices together, which is useful when optimizing the rendering of hierarchical scenes in computer graphics (and probably for other things too). On the other hand, while several chained Möbvious transformations also combine to form another Möbvious transformation, the mathematics for combining them is much less straightforward and somewhat tricky to derive.
The developer of Hyperbolica mentions that he uses gyrovectors in his devlog "Rendering Hyperbolic Spaces - Hyperbolica Devlog #3", but it may also be the case that he later switched to using the hyperboloid model. I know that he has changed other things that he also spoke about in his devlogs. In anyway, it's an interesting game concept.
I still have no idea what Im looking at! My mind simply can't understand the hypersphere... Maybe I can understand the hypercube - I have 2 videos on my channel about it - but the hypersphere might forever elude me.
3:47 this isn't really correct, in all 3 cases it is a plane
in 2d, you are rotating in the xy plane
in 3d, you are also rotating in a plane, but this time the plane can be oriented in an arbitrary direction
in 4d, of course you are also rotating in a plane
rotations fundamentally happen in planes, thinking about it in terms of an axis of rotation is technically incorrect, and it only works in 3d due to the coincidence that the number of basis vectors (x, y, z) is the same as the number of basis planes (xy, yz, zx)
axis rotations are also kinda weird because the direction of rotation around an axis is undefined, instead it has to be defined by convention using the right hand rule, but with bivectors the direction is obvious; for example, rotation in the yz plane brings the vector y to the vector z
planes can be used for rotation in any number of dimensions, if you try 5d you will see that this still remains true
Not incorrect, just a different way to look at the same thing, right? Though, granted, it's perhaps more natural to talk about the plane of rotation when talking about different numbers of dimensions.
As kantomega mentioned its just a different look at the same thing. Also I was talking about how its usually implemented and used in games. Namely Unity implements Quaternion.AngleAxis(angle, axis) to do rotations in 3D. Atleast my prefered way of doing 3d rotations.
3:50 NOO!!! All rotations happen inside a plane! in 0D and 1D you can't rotate because there aren't any planes, in 2D there's one plane of rotation so 1 value to describe rotations. in 3D there are 3 planes, in 4D there are 6 and so on with the formula of planes = (d * (d - 1)) / 2. It's a common misconception that rotations happen in d-2 (axis for 3D, point for 2D, plane for 4D) but that's not true. The reason we use planes in 4D for rotations is that you can't get away with using axes. in 3D you can get away with it by saying Y axis instead of XZ plane because that's the only plane perpendicular to the Y axis. but in 4D, the Y axis is perpendicular to the XZ, ZW and XW planes.
edit: saw someone else said the same thing. Sure, its technically different ways to say the same thing but its still kind of misleading. like if you use use planes in 4D rotation and points in 2D, then rotating *around* the XY plane in 4D means rotating *in* the ZW plane. anyway I liked your video!
3:47 This is actually pretty misleading. Rotations are a 2d phenomenon, meaning it can only happen within a 2d plane.
When you rotate in 2d space, there is only a single point that does not rotate. In 3d, you still fundamentally only rotate along a single plane, but that plane is extruded along the 3rd axis, rotating the whole object with a line being stationary. You can also add another plane to rotate along, but it only ends up rotating the original plane along a non-parallel plane. You can verify this yourself by grabbing an object, imagining a plane slicing through the middle, and arbitrarily rotating the object. You can rotate it however you want, but it will still rotate along the imaginary plane that is itself rotating.
However, in 4d, you can create a plane and still have 2 more axes to create yet another plane. If you rotate one of those planes, the other is unaffected. Assuming the first plane is still rotating, there is no reason to assume the 2nd plane can't. Rotating both at the same time (or in sequence to get from one 4d rotation to another) is called a double rotation.
👍
you use 4d rotors with 6 degrees of freedom.
It’s “how it would look” or “what would it look like.”
You aren’t supposed to say “how it would look like.”
We made it up, so he can say it ideolectally. He just needs to get around 675 million or so other people to agree with him and then his take would be in the majority. There's no "correct" way, or way you're "supposed" to speak, it only matters that you're understood Again lol, there's nothing "natural" about english syntactics in this sense, we just made it all up.