31–40 of 101 posts
In general, using Lie groups for this sort of thing is great. Things like time-derivatives become very natural in any dimension.
Bivectors are an elegant choice for dealing with rotations be cause they are isomorphic to skew-symmetric matrices, which are the Lie algebra of the special orthogonal group SO(n). The Lie group SO(n) is also known as n-dimensional rotation matrices. In general, using Lie groups for this sort of thing is great. Things like time-derivatives become very natural in any dimension.
Does such a thing exist at higher dimensions? I vaguely recall something about having complex numbers for 2D rotation, quaternions for 3D rotation, and octonions for (I'm guessing) 4D rotation, but I'm curious if the loss of associativity with octonions screws with this relationship somehow.
It would be interesting to see how it would look with the 4D scenes properly projected instead of cross-sectioned. I mean that is how we get 3D scenes on our puny 2D displays; we don't generally do the sort of cross-sections for 3D like they show in the flat-land example except in some specialized applications. Of course cross-section is still valid method of visualization. I don't know enough about higher dimensiona…
Just for reference, I found this 2017 about 4D animation suite "Fourveo", the thesis discusses the different ways to visualize (cross-section, projection via 3d and directly) 4D scenes. Unfortunately the software itself doesn't seem to be available anywhere, and there is scant other information about it. https://scholarsarchive.byu.edu/etd/6968/
This would make for a really fun video game. Think about a 4D shooter in the vein of Portal. I imagine that after awhile the human brain would learn to recognize patterns in 4D spatial reasoning.
I've been working towards making something like a 4D Descent, but I keep getting sidetracked by problems like 4D physics, collision detection, and mainly how to model interesting 4D objects. What Marc's done with Miegakure, from what's publicly visible, is pretty incredible. I have no idea how he's managed to seemingly create a coherent 4D world while only being able to view a slice of it at a time. I guess it's a bi…
It would be interesting to see how it would look with the 4D scenes properly projected instead of cross-sectioned. I mean that is how we get 3D scenes on our puny 2D displays; we don't generally do the sort of cross-sections for 3D like they show in the flat-land example except in some specialized applications. Of course cross-section is still valid method of visualization. I don't know enough about higher dimensiona…
Earlier quoted context omitted.
I've been working towards making something like a 4D Descent, but I keep getting sidetracked by problems like 4D physics, collision detection, and mainly how to model interesting 4D objects. What Marc's done with Miegakure, from what's publicly visible, is pretty incredible. I have no idea how he's managed to seemingly create a coherent 4D world while only being able to view a slice of it at a time. I guess it's a bi…
Ditto. My solution so far has been to simply ignore the problem of modelling interesting objects--the only objects are components of the environment made of simple geometric shapes, and the game challenges are all related to navigation. I've been fiddling off-and-on with procedurally generating some 4D creatures for a sequel game (e.g., by using a genetic algorithm to evolve 4D shapes that can walk), but that's a lon…
Another game idea is The Incredible Machine in 4D, but it would be so hard to play, and even harder to design the puzzles.
Bivectors are an elegant choice for dealing with rotations be cause they are isomorphic to skew-symmetric matrices, which are the Lie algebra of the special orthogonal group SO(n). The Lie group SO(n) is also known as n-dimensional rotation matrices. In general, using Lie groups for this sort of thing is great. Things like time-derivatives become very natural in any dimension.
Correct me if I'm wrong - I'm just trying to make sure I understand the generality of your statement. 3-bivectors are commonly referred to as the "axis angle" representation, and have an obvious embedding as the lie algebra to the lie group of rotation quaternions. [x,y,z] -> 0+xi+yj+zk -exp-> rotator quaternion. Does such a thing exist at higher dimensions? I vaguely recall something about having complex numbers for…
Yes, everything in your first paragraph extends to any number of dimensions (replacing "quaternion" with "rotor").
> I vaguely recall something about having complex numbers for 2D rotation, quaternions for 3D rotation, and octonions for (I'm guessing) 4D rotation
Bivectors and rotors faithfully represent rotations in any number of dimensions. The octonion product can't, because as you said, it's not associative, but rotations obviously have to compose associatively.
It would be interesting to see how it would look with the 4D scenes properly projected instead of cross-sectioned. I mean that is how we get 3D scenes on our puny 2D displays; we don't generally do the sort of cross-sections for 3D like they show in the flat-land example except in some specialized applications. Of course cross-section is still valid method of visualization. I don't know enough about higher dimensiona…
This also means such a thing should be possible for living in a 2D space and then projecting a 2D world on a 1D screen.
If you're going to play with 3D projections of 4D objects, then in order for it to really feel 4D we need to live in a 4D world. In that sense, this is quite similar to living in a 2D world while seeing a 3D projection. That's way more trippy.
To recap:
Feel 3D: 3D context -> 3D projection -> 2D screen
Feel 2D: 2D context -> 2D projection -> 1D screen
Feel 4D: 4D context -> 4D projection -> 3D screen
What we can do for 4D: 3D context -> 4D projection -> 3D screen
Compare issue with 3D: 2D context -> 3D projection -> 2D screen
I hope this makes sense (and I hope I have it right).
Man, this is tough to write about. I'm not sure if I fully understood all the nuances of your comment.
I am genuinely surprised for a long time that I've not seen 4D monsters in films. It has been so obvious to me there's potential there. It would of course be a flat 2D projection of a 3D slice through a 4D creature, so it would look like a smoothly morphing between different creatures, and in and out of existence so we've sort of had morphing for a long while (since Willow, C. 1985) but... there's potential. In case…
Spoiler, obviously: https://www.youtube.com/watch?v=uBsJgceM0KI - if you haven't seen it cannot recommend it highly enough, just wait and watch the whole thing.