I think this video has finally convinced me beyond all reasonable doubt, I will never fully understand higher dimensions. But I can pretend to by playing this game when it comes out!
what's really annoying to me is that I fully understand it conceptually, it's just impossible for me to visualize. it feels like I'm so close and yet I just can't do it.
@@thezipcreator Yeah that's pretty much my situation, too. I probably can't say I fully understand everything about it right now, but I can tell that even if/when I do I will still never be able to visualize what's "actually going on." It sounds kind of obvious, but it's that impossibility to visualize it that makes it really seem like I'll never _truly_ be able to understand it, which don't get me wrong is perfectly fine, but it is a little irritating...
Yeah I’m with you. I’ve tried to grasp just non-Euclidean geometry for a good while and.. water off a duck. But it’s great to have stuff like this keeping me humble, at least. Lots of reminders that I don’t know shit, and not everything is within my grasp.
It's not the math or simulation that's difficult, it's trying to force a graphics system that was designed for triangles to work with 4D structures and have it be efficient and fast enough for most hardware. There's a lot of crazy tricks for that, you'll see in the next Devlog.
2:04 - "One of my goals for 4D Golf is that I don't want the navigation and movement to be the challenge of the game. That should be as easy as possible, which is the reason I have so many different visual and control options. The 4D stuff is there to give new golfing challenges that are fun to play." Now that we've actually gotten our hands on the game, I really have to say you succeeded at this with flying colors, because this is the exact conclusion I drew on my own. The 4D stuff never felt like the biggest obstacle, because even when I couldn't fully wrap my head around what was happening, there are still enough different perspectives and ways to visualize things that I could often figure _something_ out anyway. You made a 4D game where the 4D stuff wasn't actually the hard part, and that is _seriously_ impressive.
When I first learned about rotations in 4D it blew my mind. The way we're used to thinking about rotations (i.e. with an axis and an angle) only just so happens to work in 3D. In general rotations are actually defined by a plane rather than an axis. We just happen to think about things in terms of the one dimension that _isn't_ rotating, rather than in terms of the 2 dimensions that _are_.
It's so great because after making that realization, you go back and think about rotations in 2D and realize that it was so obvious all along because it's not like you pick an axis in 2D to rotate around, you pick a plane which just so happens to be the only plane.
There is actually a sense in which an axis is more helpful, and it's with regards to parallel rotations. Rotations don't just need a plane to be fully defined, but also a point on that plane to be rotated around. In normal linear algebra, everything has to pass through the origin, so there's an implied center, but the moment you want to move off the origin, things break. Axes not only have a direction perpendicular to the plane of rotation, but more crucially specify the center of rotation. The trade-off is accepting that "axis" doesn't mean "line." An axis could just as easily be a point in 2D, or, mind-bendingly, a plane in 4D. In 3D projective geometry, axes technically _are_ planes that are just rendered as lines where they cut the projective plane. This is related to why it only takes 4 numbers to define a plane in 3D, but 6 to define a line.
Hmm, does that generalize to 5D+? Or is it rather that you're always rotating about an axis, but what "axis" means depends on the dimension? So in 2D you rotate about a point, in 3D you rotate about a line, in 4D about a plane, etc.?
@@angeldude101 It works because the 'point' that becomes the origin of that plane is the intersection point of the (N-2)-space and the complement plane, but you might as well just specify that point instead of increasing the 'complexity' of the model. But in 3D & 4D a line/plane happens to be 'less complicated' than a plane-point pair (although a plane-point in 3D has the same number of parameters as a line in 3D) and in 3D lets you identify a basis of (eulerian) rotations with a basis of the space, in 2D the plane can always be implied so we still end up specifying just a point in either method. So I'd say for more than 4 dimensions it's more natural to use a plane-point pair, and it's really a choice in lower than 4 dimensions, but to be the most 'general' and consistent to higher dimensions when dealing with lower dimensions I'd still argue for plane-point pairs. Especially since this way we have a natural correspondence between rotations and 2D affine subspaces which promotes a decomposition into things we have a better shot at visualising even in more general settings. But I do see the advantage specifically for 4D that complement planes can describe rotations better than plane-point pairs.
Key is analogy ,reverse engineering and emergence . I'm trying to solve the exhaustive enumeration of order 3 magic hypercubes and find a general formula for the dth case. Problem scales up exponentially past 4d and it becomes a great CS problem. At that level I often have absolutely no idea about the visualization part but while proceeding logically along abstract routes ,all the emergent mathematics gets to be pretty interesting and i even get the pseudo feeling that I am conquering higher dimensions. Discover ur particular higher dimensional niche and who knows, u might end up in awe of ur own creations ,even while ur completely baffled. At one level its like how so many Ai developments are blackbox for Ai researchers themselves. But that doesnt prevent them from doing all the back end coding.
i find the flatland one confusing and it doesn't really help me understand it, but this explanation was really helpful and now i actually understand 4d. Especially the view mode where you can swap y with w, it made me actually understand it a lot
I describe 3D shapes in 4D as "flat but not" and you can go up the dimensional ladder and describe every layer below it as "more flat" than the current layer.
The way I like to think about it is that there are different types of "volume". 2d shapes have area, which is just what volume is but in 2d, but they don't have 3d volume. similarly 3d shapes have volume, but they do not have the 4d equivalent to volume, and so on and so forth.
…now thinking about 3D “flip flap” books, changing various parts of a 3D shape on a page by flipping part of a 4D page😵💫🤯 please let me know if this is how this works…
I have always hated the concept of treating time as a 4th spatial dimension. It helps with the mathematics of Relativity, but time is not like a spatial dimension.
Actually, is a misunderstanding to think that time is the fourth dimension. What we consider the fourth dimension is the product of time times the speed of light: tc, and it has the correct units.
I think it's fine, but it depends on what you're going for. You can very much so visualize time as a fourth spatial dimension, you just have to keep in mind certain oddities. And also acknowledge that 4D doesn't inherently mean time, just as 3D doesn't inherently mean space. Most videos on RU-vid are 3D if you are including the time dimension and even can play around with the video in an odd way as a result. I think it's a neat way to visualize things and is technically accurate, but is overgeneralized
honestly, i think the most satisfying explanation of higher dimensions comes about through math - linear algebra was when i felt like i finally understood how and why higher dimensions work the way they do. not that more intuitive explanations can’t work, but i just never really felt like it made sense through pure spatial intuition, since higher dimensions directly conflict with that intuition.
I've always thought I kind of "got" 4 dimensions from all the other kinds of explanations you mentioned at the start of this video. Then the next six minutes gave me a thousand more things to think about and made me realize that I've only barely scratched the surface (no pun intended) of 4D. I came away from this a lot more confused, and a lot more excited to keep learning about higher dimensions, so thanks for that! I'll probably come back to this a few more times to try to wrap my brain around it a little better.
The thing with swapping the axis for the 4th spacial dimensions with one of the first 3 to get an alternative view is something I have thought about for years, probably a decade at this point. And this is the first time I got a visualization of that swap. Thank you so much. I also have a suggestion: How about having both views simultaneously side by side? One View Port in one angle, and the other rotated by 90° in the fourth dimension. I think it would make it very clear whats going on, especially with your ingenious 4d shadowing system.
One view could be up in the corner like a minimap, being able to swap between them at will Though that'd probably mean you'd need twice the processing power to display both at the same time
I wonder if you could get used to processing each image in each eye? You can kind of do that for sound - you can play a different sound in each ear to compare them - but I don't see many people doing that for visuals besides just normal depth perception. Maybe it could be possible to develop a secondary way to process visual data?
@@circuitgamer7759 not too relevant, but this just reminded me of this color experiment i've seen before, where an object is displayed with different colors to each eye (ex. Red in the left, blue in the right). Intuitively, you expect your brain to merge the two images together and make the object one color (purple) but no, the object is instead seen as both red and blue at the same time. Kinda trippy!
@@preferablygeneric yeah i got this nonfiction book with 3d glasses and i also got that effect with the red and blue side, it looks kind of like when two textures overlap in a video game
Another fantastic video, CP! I like how adamant you are about the exact nature of 4D space; you opt for explanations that are difficult to fully grasp, but are as accurate as possible, in contrast to explanations that are very easy to grasp, but are also inaccurate. I feel like I have a much better understanding of 4D space after watching your videos.
omg yes PLEASE keep doing in-depth 4d videos. you're totally right that there's not much on YT beyond "let's imagine a flatlander", and there's so much more to learn-even if it does make my poor little 3d brain hurt
It takes imagination to understand 4D, and it takes massive effort and intelligence to get out the flatland explanation, especially when it's interactive and comprehensible. And I think you're the first one I ever seen. Love the gravity analogy and it lightens me up. Bravo brother.
@@why_oh_elle they can take snapshots of the 3d project passing through and then compile them to get a loose understanding of what the 3d object looks like in 2d
You can't intuitively understand it, but you can understand the mathematical theory behind it and then apply that theory meticulously to an idealized graphical simulation. You can never manifest or interact with a 4d object in 4 dimensions, you can never ever see it or picture it in your head in its entirety, but you can maybe memorize what a sequence of projections along a 3d plane of some 4d object looked like, so you can know the object again when you see a very similar set of projections. This would enable you to, with a lot of training, play simple games like 4d golf. But you will never be able to "rotate a 4d cow in your head." And the flatlanders would never be able to rotate a 3d cow in their head like we can, either. (Sorry to the aphantasia folks out there, but hey the good news is that you have a bunch of comrades in the mental cow rotation problem among all the flatlanders now!)
I would love to watch an almost uncut video or live stream of you creating and testing a level. I feel just being able to see the level design kind of gradually come together would really help me and others to get their head around the weird behaviour of the geometry in this world. Would that be something you might be interested in sharing with us CodeParade? Also, in the next devlog, I think it would be helpful to give a quick refresher on what the orientation indicators in the top right are actually showing. Thanks for the very interesting video as always!
I remember when you announced this. I love to see the updates. One thing I really like is that this explanation evolved naturally after working in the space. It shows that you have made progress on the project. Or to me at least. Keep up the fantastic work
all the videos I've watched about the 4D have prepared me for this video. the more info one gets the harder it becomes to find new information. it all becomes repetitive. that's why i loved your explanation of how 3D objects are flat in 4d and have a 4D top and bottom. that concept WAS new to me
4D space has always fascinated me and was the single biggest thing that got me to love math in the first place. Thank you so much for the visualizations, can't wait for the final game ❤
Thank you so much for this. When you explained the double rotation at the end, I finally managed to visualize the 3D flattened result of rotating a 4D object. The closest I've ever gotten to having an intuition about the next dimension.
Multiplayer for this would be amazing. Even if it's nothing more than being able to see a friend's ball with no collision. Just because I feel like the confusion of trying to see how 2 points in 4d space relate would make for a really interesting way for players to talk about the game with each other.
This is really cool. I've been toying with a 4d game and playing out in my head how things would work. This video confirmed a lot of my 'preconceptions' and I love the 'space station' view where there is no gravity and you float-walk through it. That's exactly the same path I was going down!
this video gave me that mystical feeling again. I used to really have a firm grasp on 4D objects and 4D space, but this really shook everything up. Thanks! I already knew that 3d objects would feel flat, but the way you showed the 3d ground, and then went "inside" of it was so cool. I forgot what its like to not fully get the fourth dimension, thank you for filling that niche on youtube like you were saying.
its AMAZING how you're really pushing unity to do something I never thought it could handle.. seriously, awesome work on this project so far and THANK YOU for these awesome explanation videos that really help get a better understanding/idea of something as complex as 4D
Nice to see a poster who understands these additional things. It was good to finally hear somebody mention solid Klein bottles (which I refer to as Klein Strips in recognition of their higher dimensional equivalence to the lower dimensional Möbius Strip). I will certainly be interested to buy and play your 4D game to observe your 4D implementation.👏
Before this video, my understanding of 4D golf was that it was essentially 3d space and a slider between a number of level layouts, but with rotations and things making everything so much more complicated. Knowing about volume mode has finally confirmed for me that I will buy and enjoy this game. Even with height complicating things, that has suddenly made 4D space so much more intuitive for me!
Your amount of knowledge about these things is unbelievable! I love your videos and explanatory videos and how you try to simplify them for my dumb brain. Love ya (My brain is still like a boiling kettle after watching this video)
Holy cow dude. i kinda want you create a course that explains 4D more - this kinda blew me away. Look, i'm not much a golf person. But damn i'd pay you whatever the golf game costs if you can explain more of this 4D stuff to me. I'm generally interested in exotic physics and have seen quite some videos about the "flat lands" explanation of 4D and always had trouble really grasping it. Your video material really is awesome and you at least seem to know the fourth D deeper. That was amazing!
Well, for biology-related stuff the # of dimensions matters a whole lot, especially in regards to the scaling laws they follow. 3D life uses powers of 4 because it runs in 4D. Yes, that isn’t a contradiction. Fractals are weird. (The book Scale explains)
Thanks for making this! I agree that there is not much out there in terms of more in depth 4D education. I think things like double rotation could probably afford more screen time, even getting explained in multiple ways. It’s such a difficult concept to understand that I don’t think it would hurt to spend more time on this kind of video. But thanks again! I really love trying to wrap my head around 4D and it’s really great to see you unpacking things that unique to higher dimensional space.
Allow me to try. Rotation is a phenomena which occurs in a plane, not around an axis. This is why things can rotate in 2D space, despite there being no z-axis to rotate about. You can think of rotation vectors in 3D as pseudo-vectors, AKA not actually vectors at all but can be described with 3 numbers in 3D space. A more mathematical description would involve an anti-symmetric matrix which describes rotation. So the yx component of the matrix describes the same rotation as the xy component, but reversed. In 3D, the planes of rotation are xy, yz, and zx, which all share at least one axis with each other. In 4D, you can have wx and yz planes of rotation, which share no axes. Hence the double rotation.
YES!!! 4D!!! Your video explained so well some new concepts of higher dimensions that were just so fascinating. Something I would really like to see is how to model 4D objects… It’s just something I really would appreciate to see!
FINALLY someone on RU-vid says something other than “flatland” when talking about higher dimensions. The big problem with infotainment is that once you’ve seen all the big subjects, the only way to find out more is to either go back to college or dive headfirst into scientific papers and hope you understand it. A little variety now and then is very nice.
I remember in middle and high school, my math and physics teachers were always like “What? No. It’s impossible to think spatially in 4D” when I would share my wacky ideas. Thank you for not only having wackier ideas, but truly implementing them! It’s all about going back and building better wings, rather than giving up when your first pair melted.
RE: Klein bottles, I had it explained to me that the familiar 3D Klein bottle was only a "shadow" of a true 4D Klein bottle. This was somewhat intuitive to me as I could see the shadow of a cube unfolding into a net and back again, and still see it as a cube, and that to the proverbial flatlander it would look really weird because parts of the shape would appear to warp and deform to them, but in reality its just going sideways into the third dimension. I believe this was Matt Parker's lecture at the RI which I'm remembering.
Wow. This was the first time watching a video about the fourth dimension that I actually started to grasp the idea. Reminds me of my closed eye visuals during trips when I see the grids here
You gave us a taste of what double rotations look like projected into 3d but I wonder what would the visual difference of the 3d projection look like between two hypercubes double-rotating around one plane in the same direction but in opposite directions for the other plane? I assume it wouldn’t be just a mirror image/time reversed copy since that would probably be what happens if you rotate both planes opposite to the other hypercube. I feel like understanding that would probably allow me to conceptualize double rotations better
This is sooo cool! It’s such a brilliant idea to experiment with graphics engines in higher spatial dimensions. But then actually being able formulate how this would work and then create it.. that takes some serious brain!! Very impressed!
I hate that people continuously say you cant visualize 4D, its clearly logically sound and completely analogous to extending our 2D vision to 3D geometry. In the same way you know that two objects with the same position in the visual field dont have to intersect, you can do the same thing within 3D. I really love your project for this since it genuinely helps me tune my intuition
It depends on what you mean by “visualize.” There’s a difference between developing an intuition for how these things work and actually forming a mental picture of a concrete 4D object in its ‘native’ environment. Our brains only work with the world we have. If you are able to visualize a concrete 4D object in all of its glory and with every direction perpendicular to each other then you should be able to create that in the real world, which isn’t the case. You can’t conjure up a 4D object in the mind’s eye, you can only map the mechanics of 4D onto 3D space. It’s like how you can figure out how color theory would work with 4 primary colors, but that doesn’t mean you can actually ‘see’ the fourth primary color. It’s just not possible.
I think your switching of the y and w axis actually got me just that little bit closer to understanding the fourth dimension. I still can't fully visualize it, but I hope that getting to actually interact with it in this game will change that because I think your system for people to see and view the fourth dimension is very smart and way better than any other visualization I've seen, at least when it comes to actually being in a 4D space.
It makes me think how much we rely on things we dont even think about. Its only when in extremely unusual circumstances that our assumption are shown to be just that. It kinda reminds me of when you see people wearing upsidedown glasses Btw this is such a fantastic thing you are doing. I wish you luck with it and am looking forward to your next video 👍😊
Agree man is insane, but i will say at lest you can have the computer do the math 😄thats whats great about games. But that said, even knowing the right formals for 4D is pretty wild
This looks amazing, I played a few 4D “games” without really grasping what was happening but this really helps to understand this concept better.. BTW I just played hyperbolica and It was really inspiring. I’m a game developer as well, I made a game called Broken Reality and I found a few parallelisms between them, I think you might enjoy it, keep up the great work.
A while ago I thought about the logistics of making a 4 dimensional golf game but I didn't really go anywhere with it, so it is really cool to see this project so far ahead. Can't wait to hit a hypersphere with a 4 dimensional club!
Considering how you can switch between the two views, a miniature of the other view would be helpful. Just so that you can always see how your shot will behave horizontally and vertically.
Damn I was very surprised to see that wireframe view at the beginning. I thought this stuff was rendered with SDF's again, like those 4d demos on shadertoy
Curious. The game seems to follow the balls changing perspective through the 4th demention. But from a 3d persons locked perspective, what does it look like when the ball goes through that last "generated"(7:05) opening in the video. Would they be seeing the ball travel through the wall immediately or does it dissapear then shortly after reappear on the other side of the wall
So Awesome! Im so glad you could give something new to the topic. Amazing you can even make this. But its interesting as once you say it it made alot of sens but i just never head anyone mention these things before
Thank you! You've helped me visualize a (proper) 4-space mobius strip. The feature of having a single surface (3-surface, ofc) was preserved. Much like an R3 strip, seems that it requires a rotation through 4pi radians to end up back on the same "side" (side in the intuitive sense; obviously it's all the same side). Anyway, that was mind-blowing. Irritatingly, I had to use time as my fourth spatial dimension...so in order to test the notion of "going around the mobius strip/volume and ending up on the other 'side'", I had to parametrize in R5 (as the 4pi radian rotation had to occur in a plane involving the time dimension). My head hurts. I may have also used a bit of a dirty simplifying trick (all twist/rotation happens in a single short interval) to avoid having to deal with continuous twisting of the volume...but it's still technically accurate, and this is topology, so nobody cares unless it self-intersects. Double-rotation is also very weird. I've always visualized it as...well, it's pretty much impossible to verbalize, but it's a sort of...inversion...? It's a bit like turning something inside-out in 3+1 dimensions. In fact, this lines up well with the note you made about chirality of R3 volumes under rotation in a plane with k-hat as one of its bases.
Will you try making a 3D texture for your props? That can negate texture flickering and strange strips on the ground (also can make gameplay more intuitive)
That's really cool. Mathematician William Thurston claimed he could visualize 4D space, and he made a convincing case of it by publishing profound results on the topic. Your work shows that it's actually possible, and that you don't have to be a Fields medalist to do it
Yea the time/space analogy I get it . being able to tell left from right in 2D space - ok Having two 4D faces of a 3D object kinda get it... I think. Ok klein Bottle and double spin messed me up a bit. But how on earth am I meant to understand 4D golf? Also WAIT ! What about hyperbolic space in 4D? Are there just more qubes at each corner?
Awesome video! Here's a youtube video title I will share to you all. The title is: "Is there a dimension between third and fourth?" Feel free to share it with other. :-)
When I started trying to render 4D rotations, I was a little confused about them. There are 4 axes, but by analogy, a hypersphere would rotate around a plane instead of a lines somehow. Anyway, as I was working out methods for this, I realized that there are 6 planes of rotation in 4D instead of the 4 that I imagined. I wrote them out and I realized, just as you showed here, that xy and zw could rotate entirely independently - that there were 3 ways you could pair two planes together in 4D and do this double rotation. Thanks for all of your great videos! 😁
What I've always been curious to know; is there even a theoretical situation where someone/something could see in 4D the way we see in 3D? Or would that require what we call light at anothe rlevel?
I don't think so? especially since _we don't actually see in 3d._ We see in *2D,* and then use time, our other senses and our experience of being 3D creatures as interpreting what objects are 2D and what are 3D. For example, I sometimes get thrown off by what 'seems' like a pillar in the distance, only to come closer and realize it was just a 2D cutout. Optical illusions that make street art look 3D is another example. We're just living a movie where the screen our brain shows us looks 2D, it's just that we _know_ it's 3D and can tell it apart from 'fake' 2D (like animations.) to see in 4D, we'd need to have a brain that can just look at an object and immediately see all sides of it. I don't think 4D creatures can see inside 3D things. It's not like we look at squares and see every 1D line in it- it'll take an infinitesimal number of slices and our brains skip computing all that and just go to the shape. To 'see' every object of a 3D object at once and truly achieve 4D vision, we'd need something like a black hole's ability to bend light around it such that the light from the _back_ of a black hole comes back round to the front to be visible- and this is where our brains melt, lol
big thanks for this vid, I really like this stuff, and a clear explanation is appreciated, and yeah there really isnt too many explanations that are clear on the subject, and your game is a really cool display.
