Wow, the title does not make it justice: I only went to look at it in an idle moment after lunch. Much better to transcribe the beginning of the video:
4D Toys is a toy box filled with 4-dimensional toys.
By 4-dimensional I mean that they exist in a world with 4 dimensions of space and 1 dimension of time, instead of 3 dimensions of space and 1 dimension of time.
It turns out that the rules of how objects bounce, slide and roll around can be generalized to higher dimensions, and this unique toy lets you experience what that would look like.
There was an indie game that had 4D puzzles (as in actual 3D + time), and the person/team behind the game developed a framework and map editor to work with those properly...
I cannot recall the name, but it looked awesomely complex!
Was the game Miegakure [0]? It has the same author as this paper/post, and pretty much seems to be the premiere 4D game. The parent link says that the research shown here was developed for Miegakure, so it might be the game you're thinking of.
I'll ask any physicists out there. A 4th spatial dimension probably doesn't exist, right?
Even if we couldn't perceive shapes across a 4th dimension, we would still perceive things moving through the 4th dimension like we see in 4D toys. In reality we don't ever see anything like this (spatially anyway). Is that correct?
String theories require extra spatial dimensions (to total of 10, 11 or 26), but those dimension are so small and/or looping so that we cannot detect them.
Other approach is to treat our 3D space as slice of higher dimensional space.
String theories have a lot of problems (biggest ones is that they are currently unverifiable using experiments and they do not give any meaningful predictions about our 3D view of universe), but many theoretical physicists are working on them.
It is perfectly possible to formulate theories of physics in higher spatial dimensions. In fact, many high-energy theories like string theory require many spatial dimensions for mathematical consistency. It is something we actively look for signatures of in experiments.
What we have found is that our observations are incompatible with 'extended' additional dimensions. Extended here means that you can go a sizeable distance. It's not that the other dimensions would just stop, but they'd be more like Pac Man---these dimensions might have periodic boundary conditions, making them circles. If the circle's radius (or circumference, equivalently) becomes very large, such a 'compactified' dimension starts to have a lot of space in it, much like the familiar 3 dimensions. So, we can put an upper bound.
My most recent recollection. We may yet have missed dimensions as large as 1mm in [diameter, circumference, I forget]. The most sensitive probes with the fewest assumptions about the structure + particle content of the universe tend to be gravitational, and gravity is very hard to measure precisely on very short length scales.
If there were 4 spatial dimensions + 1 time dimension we would end up in an unstable universe.
Ehrenfest (1917/1920) studied the hydrogen in n dimensions and concluded for n> 3 that neither classical atoms nor planetary orbits can be stable, because the inverse square law of electrostatic and gravity becomes an inverse cube law. When n > 3 there are no stable orbits to the two body problem: an incoming light body attracted by a heavy one would either escape to infinity or get sucked into collision.
For n = 3 we get stable elliptic orbits or non-bound parabolic and hyperbolic orbits.
Collision only occurs when the lighter body heads directly towards the heavy body within 2R (R being the heavier body's radius), ie. the impact parameter is zero [2]
Keep in mind that on Earth-scale, movement through the fourth dimension would be pretty easy to notice. There's a lot of stuff on Earth, so seeing objects bounce in and out of view would be a pretty big deal.
But our universe as a whole is almost entirely empty space, with visible objects appearing to make up only about 5 percent of the universe's total volume. Noticing 4D things flit in and out of our 3D universe might be difficult at that scale.
So what about subatomic particle scale? Would it look like particles spontaneously appearing and disappearing? Quantum tunneling through obstacles? (they actually went around in another dimension)
I understand what you're saying but it doesn't make sense to me why this would be a rare occurrence. How could all the varied matter we perceive be so perfectly smooth and inertia free along the 4th dimension that we see no collisions resulting in movement along a 4th dimension?
I forgot most of the details already but as far as I remember, some classes of splines are 4 dimensional constructs that are then projected to 3 dimension to get your 3d spline curve.
I would like to see the 4th dimension projected into 3d and not just a slide, same as you could project the 3d into 2d, by showing further objects smaller and closer objects bigger.
You can absolutely do this (and it would be quite interesting to try with a VR headset, e.g.!); the idea is a special case of the projective transform [0] in 4 dimensions. It may make for a cool demo.
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 dimensional graphics to be able to say if you would be able to do projection directly from 4D->2D (as our displays still are 2D), or if it is better to go 4D->3D->2D.
Bit disappointing that the video spends a lot discussing general 4D stuff and less about the dynamics part which is the actual subject.
Here is video of one of the other visualization systems referred in the paper which if I interpret it correctly does projection with depth of field https://www.youtube.com/watch?v=dT5YCs84jJU
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.
I believe this is the author's GitHub: https://github.com/neverhood311 (I followed a YouTube link from the paper, and the YouTube profile pic matches this one's.)
Well, we get 3D scenes/projections on a 2D screen by being in a 3D world/context and looking at a 2D screen. With that I mean to say that I think (not sure) a 2D image of a 3D projection can only feel 3D if you're in a 3D world.
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.
Makes me wonder if the human brain if born into a 4D world would be able to function, or if fundamentally it is impossible for the brain to process a 4D word.
The thing is, 3D to 2D projections work well because that's how our eyes work in the first place, so we can intuitively understand a projected image on the screen.
When projecting 4D to 2D, weird things start happening. Instead of a ray, each pixel now represents a plane in 4D space. So even while it can be done mathematically, the brain is just overwhelmed by what happens when you e.g. rotate the projection
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 bit like using ed instead of a modern text editor.
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 long way off.
Yeah, I've mostly ignored it so far as well. I did get Dual Contouring working on 4D signed distance fields, but the resulting meshes are kind of janky. My thoughts are to eventually get boolean operations working on arbitrary tetrahedral meshes and do some CSG, or create a Blender-style 4D mesh editor.
Another game idea is The Incredible Machine in 4D, but it would be so hard to play, and even harder to design the puzzles.
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 I missed it, any films actually done that?
One thing Annihilation did for me was really get me to the point mentally where I could start to appreciate just how mindblowing it is to encounter such different life like in the movie. By the time the end of the movie came around, I could really get a grip on how horrifying the "choreographed" scene would be, or encountering a floating mandelbrot set, c'mon. That is pure terror.
And that's not an easy perspective to hold on to in a world that barrages me with so many sci-fi/fantasy movies and ideas.
if you're into anime, check out Seikaisuru Kado.[0]
The antagonist is a being from another dimension who routinely shows off dominion over N dimensions, with the effect looking something similar to what is demonstrated in the siggraph video.
The story and ending are rather weak, but the physics questions posed are interesting. I like any franchise that discusses the teletransportation paradox.
Nothing exactly related but your description made me recall the section in The Watchmen comics where Dr. Manhattan flees to mars and reflects on things, seeing all time together.
_____________________________________________________________________________________________________________________________________________________________________Interstellar kinda touched on that.
In the Hyperion books by Dan Simmons there is a being that can move in higher dimensions and the books will reportedly get a movie. Maybe they will adapt some effects like this.
You should read Spaceland by Rudy Rucker. It’s a hilarious 4D riff on Flatland. A venture capitalist in Silicon Valley gets harassed by a 4D monster called Momo
In the last episodes of TV show Legion, they bring in Monsters from another dimension (time). So it was not exactly 4D in spatial sense as is in this paper.
Awesome! Would be cool to see the extension in non-flat spaces as well, like geometry with a low "speed of light". Clifford algebras can handle that too.
SIGGRAPH 2020 in July is all online because Washington DC is keeping their convention center as a makeshift hospital through the summer. I have not seen revised registration fees for this conference yet and hope they are reduced. I attend approximately every other year and its on the pricey side.
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 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.
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.
Why do these n-dimension demos always have the little 2d man seeing the world as if he could fly away from his 2d plane and see it from above? In that example where narrator says the 2d man would be intrigued about the circle floating, he would actually only see a line moving up and down. The ball crossing the plane would be just a line stretching and shrinking. If he lived in a 2d world, what he could actually see would be /pretty/ limited.
If you rotate a square, then the length will oscillate. A circle will remain fixed (but presumably would have surface detail to clue you in that it's rotating.)
Assuming you have "1d depth vision", ie computing distance to the eye using two eyes set some distance apart, you'd be able to tell them apart rather easily, in addition to shading and movement differences.
"In previous sections I have said that all figures in Flatland present the appearance of a straight line, yet now I am about to explain to my Spaceland critics how we are able to recognize one another by the sense of sight.
That this power exists in any regions and for any classes is the result of Fog.
If Fog were non-existent, all lines would appear equally and indistinguishably clear; and this is actually the case in those unhappy countries in which the atmosphere is perfectly dry and transparent. But wherever there is a rich supply of Fog, objects that are at a distance, say of three feet, are appreciably dimmer than those at a distance of two feet eleven inches; and the result is that by careful and constant experimental observation of comparative dimness and clearness, we are enabled to infer with great exactness the configuration of the object observed.
Suppose I see two individuals approaching whose rank [shape] I wish to ascertain. They are, we will suppose, a Merchant and a Physician, or in other words, an Equilateral Triangle and a Pentagon: how am I to distinguish them?
It will be obvious, to every child in Spaceland who has touched the threshold of Geometrical Studies, that if I can bring my eye so that its glance may bisect an angle of the approaching stranger, my view will lie evenly between his two sides that are next to me, so that I shall contemplate the two impartially, and both will appear of the same size.
Now in the case of the Merchant [Equilateral Triangle], what shall I see? I shall see a straight line, in which the middle point will be very bright because it is nearest to me; but on either side the line will shade away rapidly into dimness, because the sides recede rapidly into the fog; and what appear to me as the Merchant's extremities will be very dim indeed.
On the other hand in the case of the Physician [Regular Pentagon], though I shall here also see a line with a bright center, yet it will shade away less rapidly into dimness, because the sides recede less rapidly into the fog; and what appear to me the Physician's extremities will not be not so dim as the extremities of the Merchant.
The Reader will probably understand from these two instances how -- after a very long training supplemented by constant experience -- it is possible for the well-educated classes among us to discriminate with fair accuracy between the middle and lowest orders [different shapes] by the sense of sight. If my Spaceland Patrons have grasped this general conception, so far as to conceive the possibility of it and not to reject my account as altogether incredible - I shall have attained all I can reasonably expect."
Edwin A. Abbott in Flatland: A romance of many dimensions
(slightly edited)
If you find this interesting, you'll really love the short book "Flatland: A Romance of Many Dimensions." It's a deep dive into what a dimension really is, intuitively, and also satirical commentary about Victorian society.
This is an awesome accomplishment. I remember seeing this work when it first came online years back — pretty mind-blowing, especially some of the transitions between [dimensions] in his flagship game.
Many years ago, in my freshman year of college, I wanted to create a 4d space game, and spent about a year grinding out an OpenGL rendering engine. I taught myself the requisite mathematics and realized that my conception of hyperspace as operative in my game, was fundamentally wrong. So I moved on. Still have the code laying around, although I lost the Linux port I ginned up at one point. Really should polish it and release it. :)
my initial thought was that hyperspace would allow a _shortening_ of the distance traveled from point P0(x,y,z,q) to P1(x,y,z,q). Since I was _really_ into space combat games at the time... that would have been really cool!
However, - and this is obvious when you work through it - when you use Euclidean distance (https://mathworld.wolfram.com/Distance.html - see the General equation), the distance travelled increases, thus obviating a key part of my game, which was to "jump" through hyperspace to shorten the trip. I didn't care to rewrite math enough to make the game work out, and since I'd spent so much time learning math, graphics,and larger-scale program design, I was burnt out. It was a great experience.
One day I might return to the notion, but it'll be around the hyper-dimensionality experience, not as a combat game. I don't like the slice mode of 4D games qua the OP, I far prefer projections from 4D as a way to understand the 4D realm.
The simplest answer is that I was aiming at doing a projection from 4D into 2D. 4 dimensions in Euclidean space have 6 planes. A 4D rotation thus has to have a ton more of sine/cosine calculations than a 2D one. Projections down from the 4th induce this as well. You have to project each point into the space you're visualizing.
I didn't really get into camera / FOV calculations, I kept that fixed, don't remember much about those decisions. It was 2001 when I started the project, and I took a final swing around 2005 to get a final thing.
Not sure of the state of the information today, but at that time, I was unable to find almost anything about the mathematics involved in the 4d->3d work. I was a freshman at a community college, literally printing off articles from Wolfram and interlibrary loaning books on graphics. I had to deduce from the principles of 3d->2d projection what it'd take to do a 4d->3d projection.
I went on to take a minor in mathematics and retain a good deal of that information even now. Given an adequate graphing/windowing context with a PutLine/PutPoint in 3d or 2d, I could cook up a 4D library over a weekend of focus. I just don't know what to do with that library besides goofing off rotating a cube or whatever.
N.b., one of the issues with doing something here is that you really need a 4D CAD tool. Think Blender, but 4D. Otherwise you're laboriously typing in `Point p = new Point(0,0,0,1); Point p2 = new Point(/ugh/);` or some such, maybe in a text file. Haven't looked into the area in, you know, 15 years. Maybe there's such a tool out there today.
That's a complicated question you're asking. It belongs to differential geometry, which I specialised in 15 years ago.
So, you're looking for interesting 4 dimensional manifolds. Problem is, there are many of those. Too many to count, iirc.
Contrast this with the 3 dimensional manifolds, of which there are only eight (and simple compositions of these eight).
Now, unfortunately, the Klein bottle is a 2 dimensional manifold (it's surface is the manifold, and that is locally isomorphic to a R^2). It's interesting because it does not have an embedding into 3d space, unlike most other 2d manifolds, so you need to embed it into R^4. I don't know if there is a list of these.
I can look up more details and give more pointers if there is interest.
Interesting side note: the Klein bottle was originally called "kleinsche Fläche" (Klein surface). When saying this German word with an English voice it'll sound like "Kleinsche Flasche" which translates to "Klein bottle".
100 comments
[ 3.2 ms ] story [ 127 ms ] thread4D Toys is a toy box filled with 4-dimensional toys.
By 4-dimensional I mean that they exist in a world with 4 dimensions of space and 1 dimension of time, instead of 3 dimensions of space and 1 dimension of time.
It turns out that the rules of how objects bounce, slide and roll around can be generalized to higher dimensions, and this unique toy lets you experience what that would look like.
It's easy enough, albeit annoying, to brush away the inflated mediocre. It's so much harder to locate the unseen gems.
Edit: ah, I see - the video is from 2017, and there was a thread back then too: https://news.ycombinator.com/item?id=14471931. I guess we'll make an ugly hybrid of the titles.
I cannot recall the name, but it looked awesomely complex!
[0] - https://miegakure.com/
Wish people wouldn't call "game with time manipulation mechanics" "4d" though.
I do remember there was a game which was like, an rts, but with time manipulation bits.
Ah, but that didn't really feature 3d movement, so probably not what they are thinking of?
I also remember a 4d spaceship shooter game..
Bill Thurstons Three-Dimensional Geometry and Topology is a great read
You can then play around with objects in 3D, 4D, and beyond:
https://enkimute.github.io/ganja.js/examples/coffeeshop.html...
Even if we couldn't perceive shapes across a 4th dimension, we would still perceive things moving through the 4th dimension like we see in 4D toys. In reality we don't ever see anything like this (spatially anyway). Is that correct?
String theories require extra spatial dimensions (to total of 10, 11 or 26), but those dimension are so small and/or looping so that we cannot detect them.
Other approach is to treat our 3D space as slice of higher dimensional space.
https://en.wikipedia.org/wiki/String_theory#Extra_dimensions
String theories have a lot of problems (biggest ones is that they are currently unverifiable using experiments and they do not give any meaningful predictions about our 3D view of universe), but many theoretical physicists are working on them.
Makes me appreciate that relativity and quantum mechanics are verifiable experimentally.
It is perfectly possible to formulate theories of physics in higher spatial dimensions. In fact, many high-energy theories like string theory require many spatial dimensions for mathematical consistency. It is something we actively look for signatures of in experiments.
What we have found is that our observations are incompatible with 'extended' additional dimensions. Extended here means that you can go a sizeable distance. It's not that the other dimensions would just stop, but they'd be more like Pac Man---these dimensions might have periodic boundary conditions, making them circles. If the circle's radius (or circumference, equivalently) becomes very large, such a 'compactified' dimension starts to have a lot of space in it, much like the familiar 3 dimensions. So, we can put an upper bound.
My most recent recollection. We may yet have missed dimensions as large as 1mm in [diameter, circumference, I forget]. The most sensitive probes with the fewest assumptions about the structure + particle content of the universe tend to be gravitational, and gravity is very hard to measure precisely on very short length scales.
edit to add: the latest summary of extra dimensional searches from the Particle Data Group http://pdg.lbl.gov/2019/listings/rpp2019-list-extra-dimensio...
even better, here's the PDG's review, rather than their technical summary: http://pdg.lbl.gov/2019/reviews/rpp2018-rev-extra-dimensions...
Ehrenfest (1917/1920) studied the hydrogen in n dimensions and concluded for n> 3 that neither classical atoms nor planetary orbits can be stable, because the inverse square law of electrostatic and gravity becomes an inverse cube law. When n > 3 there are no stable orbits to the two body problem: an incoming light body attracted by a heavy one would either escape to infinity or get sucked into collision.
For n = 3 we get stable elliptic orbits or non-bound parabolic and hyperbolic orbits.
Collision only occurs when the lighter body heads directly towards the heavy body within 2R (R being the heavier body's radius), ie. the impact parameter is zero [2]
[1] https://doi.org/10.1002/andp.19203660503
[2] https://en.wikipedia.org/wiki/Impact_parameter
edit: grammar
But our universe as a whole is almost entirely empty space, with visible objects appearing to make up only about 5 percent of the universe's total volume. Noticing 4D things flit in and out of our 3D universe might be difficult at that scale.
Nothing (in the classical, non-quantum) setting is solely determined by position coordinates.
Recently there has been super interesting research on using these nurbs as ansatzfunctions e.g. for FEM stuff. They got a lot of applications.
-----
[0] https://en.wikipedia.org/wiki/Homography
edit: [1] https://4dtoys.com
I don't know enough about higher dimensional graphics to be able to say if you would be able to do projection directly from 4D->2D (as our displays still are 2D), or if it is better to go 4D->3D->2D.
Bit disappointing that the video spends a lot discussing general 4D stuff and less about the dynamics part which is the actual subject.
Here is video of one of the other visualization systems referred in the paper which if I interpret it correctly does projection with depth of field https://www.youtube.com/watch?v=dT5YCs84jJU
https://scholarsarchive.byu.edu/etd/6968/
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.
https://en.wikipedia.org/wiki/Anaglyph_3D
When projecting 4D to 2D, weird things start happening. Instead of a ray, each pixel now represents a plane in 4D space. So even while it can be done mathematically, the brain is just overwhelmed by what happens when you e.g. rotate the projection
I imagine that after awhile the human brain would learn to recognize patterns in 4D spatial reasoning.
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 bit like using ed instead of a modern text editor.
Another game idea is The Incredible Machine in 4D, but it would be so hard to play, and even harder to design the puzzles.
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 I missed it, any films actually done that?
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.
One thing Annihilation did for me was really get me to the point mentally where I could start to appreciate just how mindblowing it is to encounter such different life like in the movie. By the time the end of the movie came around, I could really get a grip on how horrifying the "choreographed" scene would be, or encountering a floating mandelbrot set, c'mon. That is pure terror.
And that's not an easy perspective to hold on to in a world that barrages me with so many sci-fi/fantasy movies and ideas.
https://www.youtube.com/watch?v=v27jdRub2GQ
Will definitely check it out.
Thanks @all for suggestions
The antagonist is a being from another dimension who routinely shows off dominion over N dimensions, with the effect looking something similar to what is demonstrated in the siggraph video.
The story and ending are rather weak, but the physics questions posed are interesting. I like any franchise that discusses the teletransportation paradox.
[0]: https://myanimelist.net/anime/32032/Seikaisuru_Kado
Will check SK, thanks
The 4D creature mentioned by parent is amazing. There are more 4D puzzles like a temple moving back through time.
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.
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.
Just like our 2d eyes do in our 3d world.
That this power exists in any regions and for any classes is the result of Fog.
If Fog were non-existent, all lines would appear equally and indistinguishably clear; and this is actually the case in those unhappy countries in which the atmosphere is perfectly dry and transparent. But wherever there is a rich supply of Fog, objects that are at a distance, say of three feet, are appreciably dimmer than those at a distance of two feet eleven inches; and the result is that by careful and constant experimental observation of comparative dimness and clearness, we are enabled to infer with great exactness the configuration of the object observed.
Suppose I see two individuals approaching whose rank [shape] I wish to ascertain. They are, we will suppose, a Merchant and a Physician, or in other words, an Equilateral Triangle and a Pentagon: how am I to distinguish them?
It will be obvious, to every child in Spaceland who has touched the threshold of Geometrical Studies, that if I can bring my eye so that its glance may bisect an angle of the approaching stranger, my view will lie evenly between his two sides that are next to me, so that I shall contemplate the two impartially, and both will appear of the same size.
Now in the case of the Merchant [Equilateral Triangle], what shall I see? I shall see a straight line, in which the middle point will be very bright because it is nearest to me; but on either side the line will shade away rapidly into dimness, because the sides recede rapidly into the fog; and what appear to me as the Merchant's extremities will be very dim indeed.
On the other hand in the case of the Physician [Regular Pentagon], though I shall here also see a line with a bright center, yet it will shade away less rapidly into dimness, because the sides recede less rapidly into the fog; and what appear to me the Physician's extremities will not be not so dim as the extremities of the Merchant.
The Reader will probably understand from these two instances how -- after a very long training supplemented by constant experience -- it is possible for the well-educated classes among us to discriminate with fair accuracy between the middle and lowest orders [different shapes] by the sense of sight. If my Spaceland Patrons have grasped this general conception, so far as to conceive the possibility of it and not to reject my account as altogether incredible - I shall have attained all I can reasonably expect."
Edwin A. Abbott in Flatland: A romance of many dimensions (slightly edited)
https://www.gutenberg.org/files/201/201-h/201-h.htm
https://www.youtube.com/watch?v=N0WjV6MmCyM
Many years ago, in my freshman year of college, I wanted to create a 4d space game, and spent about a year grinding out an OpenGL rendering engine. I taught myself the requisite mathematics and realized that my conception of hyperspace as operative in my game, was fundamentally wrong. So I moved on. Still have the code laying around, although I lost the Linux port I ginned up at one point. Really should polish it and release it. :)
However, - and this is obvious when you work through it - when you use Euclidean distance (https://mathworld.wolfram.com/Distance.html - see the General equation), the distance travelled increases, thus obviating a key part of my game, which was to "jump" through hyperspace to shorten the trip. I didn't care to rewrite math enough to make the game work out, and since I'd spent so much time learning math, graphics,and larger-scale program design, I was burnt out. It was a great experience.
One day I might return to the notion, but it'll be around the hyper-dimensionality experience, not as a combat game. I don't like the slice mode of 4D games qua the OP, I far prefer projections from 4D as a way to understand the 4D realm.
I didn't really get into camera / FOV calculations, I kept that fixed, don't remember much about those decisions. It was 2001 when I started the project, and I took a final swing around 2005 to get a final thing.
Not sure of the state of the information today, but at that time, I was unable to find almost anything about the mathematics involved in the 4d->3d work. I was a freshman at a community college, literally printing off articles from Wolfram and interlibrary loaning books on graphics. I had to deduce from the principles of 3d->2d projection what it'd take to do a 4d->3d projection.
I went on to take a minor in mathematics and retain a good deal of that information even now. Given an adequate graphing/windowing context with a PutLine/PutPoint in 3d or 2d, I could cook up a 4D library over a weekend of focus. I just don't know what to do with that library besides goofing off rotating a cube or whatever.
N.b., one of the issues with doing something here is that you really need a 4D CAD tool. Think Blender, but 4D. Otherwise you're laboriously typing in `Point p = new Point(0,0,0,1); Point p2 = new Point(/ugh/);` or some such, maybe in a text file. Haven't looked into the area in, you know, 15 years. Maybe there's such a tool out there today.
So, you're looking for interesting 4 dimensional manifolds. Problem is, there are many of those. Too many to count, iirc.
Contrast this with the 3 dimensional manifolds, of which there are only eight (and simple compositions of these eight).
Now, unfortunately, the Klein bottle is a 2 dimensional manifold (it's surface is the manifold, and that is locally isomorphic to a R^2). It's interesting because it does not have an embedding into 3d space, unlike most other 2d manifolds, so you need to embed it into R^4. I don't know if there is a list of these.
I can look up more details and give more pointers if there is interest.
Interesting side note: the Klein bottle was originally called "kleinsche Fläche" (Klein surface). When saying this German word with an English voice it'll sound like "Kleinsche Flasche" which translates to "Klein bottle".
If you really want to enjoy 4D Toys in the meantime download his tablet game: http://4dtoys.com