12 ms·

4D Toys: how objects bounce and roll in 4D – paper accepted to SIGGRAPH 2020

429 points by doppp 6y ago

aabajian 6y ago
If you like this, then checkout the presentation on Geometric Algebra from 2019 SIGGRAPH: https://www.youtube.com/watch?v=tX4H_ctggYo https://www.youtube.com/watch?v=tX4H_ctggYo You can then play around with objects in 3D, 4D, and beyond: https://enkimute.github.io/ganja.js/examples/coffeeshop.html#qcga3d_points_and_more https://enkimute.github.io/ganja.js/examples/coffeeshop.html...
- xixixao 6y ago
  This amazing, thanks for the pointer!
AlbertoGP 6y ago
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.
- dang 6y ago
  Ok, let's take the title from the video instead. Thanks for pointing this out! Sometimes the best work hides behind dry or modest language. 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 https://news.ycombinator.com/item?id=14471931. I guess we'll make an ugly hybrid of the titles.
  - chadmeister 6y ago
    Thank you this is still much better than the useless original title
jfkebwjsbx 6y ago
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!
- thatcherc 6y ago
  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. [0] - https://miegakure.com/ https://miegakure.com/
  - garaetjjte 6y ago
    Sadly, it might be one of games with longest development..
  - drdeca 6y ago
    probably not, because they are describing a 3+1 game where there are substantial time-manipulation puzzles. 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..
BFatts 6y ago
That's really cool!
rcpt 6y ago
3-manifolds have some pretty great descriptions to https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-three-torus/ https://blogs.scientificamerican.com/roots-of-unity/a-few-of... Bill Thurstons Three-Dimensional Geometry and Topology is a great read
jayd16 6y ago
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?
- terramex 6y ago
  Obligatory "I'm not a physycist, but...". 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 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.
  - dehrmann 6y ago
    > String theories...are currently unverifiable using experiments Makes me appreciate that relativity and quantum mechanics are verifiable experimentally.
- evanb 6y ago
  I am a physicist. 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-dimensions.pdf 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.pdf http://pdg.lbl.gov/2019/reviews/rpp2018-rev-extra-dimensions...
guardian5x 6y ago
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.
- LolWolf 6y ago
  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. ----- [0] https://en.wikipedia.org/wiki/Homography https://en.wikipedia.org/wiki/Homography
  - jimmySixDOF 6y ago
    Absolutely this should be in VR and I'm a little bit surprised it isn't tbh edit: [1] https://4dtoys.com https://4dtoys.com
zokier 6y ago
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 https://www.youtube.com/watch?v=dT5YCs84jJU
- zokier 6y ago
  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/ https://scholarsarchive.byu.edu/etd/6968/
  - scintill76 6y ago
    I believe this is the author's GitHub: https://github.com/neverhood311 https://github.com/neverhood311 (I followed a YouTube link from the paper, and the YouTube profile pic matches this one's.)
- elihu 6y ago
  I suppose you could also compromise by drawing a 4-D wireframe overlaid over a traditional render of a 3-D cross section.
- mettamage 6y ago
  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.
echelon 6y ago
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.
- gridlockd 6y ago
  The author is also developing a videogame (Miegakure) that has 4D mechanics. It's a puzzle game though, not a shooter.
- beaumayns 6y ago
  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.
  - gliese1337 6y ago
    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.
    - beaumayns 6y ago
      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.
throwaway_pdp09 6y ago
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?
- deleted 6y ago
  [deleted]
- jsd1982 6y ago
  Futurama had a 4-D "space" white whale in the Bermuda Tetrahedron in S6 E21 "Mobius Dick".
- anigbrowl 6y ago
  Annihilation (2018) Spoiler, obviously: https://www.youtube.com/watch?v=uBsJgceM0KI 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.
  - hombre_fatal 6y ago
    Perfect music for that scene. 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.
    - peter_l_downs 6y ago
      if you like those sounds you may like Burial, of which it immediately reminded me https://www.youtube.com/watch?v=v27jdRub2GQ https://www.youtube.com/watch?v=v27jdRub2GQ
miohtama 6y ago
Has there been any attempts to visualize and project 4D objects to 3D space so that we could "grasp" them better? E.g. using colours, shadows, etc.
nimish 6y ago
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.
peter303 6y ago
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.
dllu 6y ago
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.
- centimeter 6y ago
  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.
  - h-cobordism 6y ago
    > Does such a thing exist at higher dimensions? 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.
dang 6y ago
A related thread from 2017: https://news.ycombinator.com/item?id=14471931 https://news.ycombinator.com/item?id=14471931
EL_Loco 6y ago
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.
- jblow 6y ago
  He would be able to see a circle the same way you can see a sphere. Do you feel like you can only see discs, not spheres?
  - FPGAhacker 6y ago
    How different would a 2d square look projected onto a 1d view, vs a circle?
    - javajosh 6y ago
      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.)
    - jobigoud 6y ago
      If the object is lit at an angle the shading will be completely different, the edge of the square will create a harsh line.
    - shezi 6y ago
      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. Just like our 2d eyes do in our 3d world.
- avhon1 6y ago
  "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)
elil17 6y ago
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. https://www.gutenberg.org/files/201/201-h/201-h.htm https://www.gutenberg.org/files/201/201-h/201-h.htm
gdubs 6y ago
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.
pier25 6y ago
If you liked that video check this other one by Carl Sagan: https://www.youtube.com/watch?v=N0WjV6MmCyM https://www.youtube.com/watch?v=N0WjV6MmCyM
pnathan 6y ago
I love 4D graphics. 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. :)
- beaumayns 6y ago
  I'm curious, in what way was your conception wrong? Sounds like it could still be interesting.
  - pnathan 6y ago
    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 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.
- joekunin 6y ago
  I would love to know more about the kind of math involved in this.
  - pnathan 6y ago
    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.
beefield 6y ago
Wonder if there is somewhere a list or some document of "interesting" 4d shapes? For example klein bottle, which is the only one I know of.
- shezi 6y ago
  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".
deleted 6y ago
[deleted]
slowhadoken 6y ago
Oh yeah this is old. I think this is the same guy making Miegakure @ https://miegakure.com/ https://miegakure.com/
kensai 6y ago
Marc is amazing. Can't wait for his masterpiece game to be finally completed! http://miegakure.com http://miegakure.com If you really want to enjoy 4D Toys in the meantime download his tablet game: http://4dtoys.com http://4dtoys.com
kmbriedis 6y ago
The most surprising aspect of this is that it's a solo paper accepted for a major conference!
jbinney 6y ago
The comparison to children playing with blocks makes me wonder: If you spent long enough playing with these 4D toys, would you develop a deeper intuition for the 4th dimensional shape of the objects?
- kmill 6y ago
  I've played with some programs I made to explore 4D objects, and, yes, you do gain some intuition. Interestingly, I was told one of George Boole's daughters got lessons from a guy on 4D objects by using colored wooden blocks. She really picked it up, and later in life mathematicians would consult her about intersections of 4D polytopes.
- hinkley 6y ago
  If you haven't read Mimsy Were the Borogroves, you should. https://en.wikipedia.org/wiki/Mimsy_Were_the_Borogoves https://en.wikipedia.org/wiki/Mimsy_Were_the_Borogoves
- robocat 6y ago
  That would be a perfect application of VR. I’m sure most of us could learn a visceral understanding of 4D if we could push and twist 4D objects.
- dehrmann 6y ago
  Sort of, but not really. The problem is you don't have sufficient information to know what should happen.
robofanatic 6y ago
This is super interesting but hurting my brain. A person in 3D can see 3D, 2D and 1D. But a person in 2D can only see 1D. How can a person living in 2D world see 2D things like Circles.
- bespokealgo 6y ago
  What we (3D beings) see is a 2D projection of the world around us, 2D beings would indeed see a 1D projection of their 2D world (e.g. we only really see a circle when looking at a sphere). Just as we use depth perception to differentiate 3D objects from 2D ones, 2D beings would also be able to discern a circle from a line in a 2D world.
bawana 6y ago
How does one model 4D gravity? Consider the 3D to 2D analogy. If the 2D person lived on a planet, that would be a circle. Then 2 planets orbiting each other would appear to the 2D person as two line segments oscilating up and down (because the 2D person cannot see the entire circle, just the edge view). Based on the velocity of oscillation and the separation of the line segments, a 2D newton could come up with an analogous law of gravition with an inverse square relation. But if two 3d planets were orbiting transversely through the 2D plane of our inhabitants, the 2D person would see two separate line segments growing and shrinking although the line segments could maintain a constant distance of separation. The velocity of growing and shrinking would be changing even though their separation was constant. Extending this analogy to our universe makes me wonder if our calculation that the universe is expanding is just an illusion. Rather, at astronomical distances, our universe could be traversing 4 dimensions. And what we perceive as an accelerating expansion of the universe is just motion into the 4th dimension. And if we could travel into that dimension, we might be able to get to places that appear far away but are actually not. Though it sounds far fetched, it also might explain the stuff that is postulated to happen at the quantum scale-where quantum fluctuations are postulated to occur. The only thing we would need to be able to do is to 'coordinate' the fluctuations of lots of adjacent voxels of space. And then we could wander through 4D space.
DreamScatter 6y ago
Check out my geometric algebra implementation in Julia language at https://github.com/chakravala/Grassmann.jl https://github.com/chakravala/Grassmann.jl

429 points by doppp 6y ago

If you like this, then checkout the presentation on Geometric Algebra from 2019 SIGGRAPH: https://www.youtube.com/watch?v=tX4H_ctggYo https://www.youtube.com/watch?v=tX4H_ctggYo You can then play around with objects in 3D, 4D, and beyond: https://enkimute.github.io/ganja.js/examples/coffeeshop.html#qcga3d_points_and_more https://enkimute.github.io/ganja.js/examples/coffeeshop.html...

xixixao 6y ago
This amazing, thanks for the pointer!

AlbertoGP 6y ago

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.

dang 6y ago
Ok, let's take the title from the video instead. Thanks for pointing this out! Sometimes the best work hides behind dry or modest language. 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 https://news.ycombinator.com/item?id=14471931. I guess we'll make an ugly hybrid of the titles.
- chadmeister 6y ago
  Thank you this is still much better than the useless original title

jfkebwjsbx 6y ago

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!

thatcherc 6y ago
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. [0] - https://miegakure.com/ https://miegakure.com/
- garaetjjte 6y ago
  Sadly, it might be one of games with longest development..
- drdeca 6y ago
  probably not, because they are describing a 3+1 game where there are substantial time-manipulation puzzles. 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..

BFatts 6y ago

That's really cool!

rcpt 6y ago

3-manifolds have some pretty great descriptions to https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-three-torus/ https://blogs.scientificamerican.com/roots-of-unity/a-few-of... Bill Thurstons Three-Dimensional Geometry and Topology is a great read

jayd16 6y ago

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?

terramex 6y ago
Obligatory "I'm not a physycist, but...". 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 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.
- dehrmann 6y ago
  > String theories...are currently unverifiable using experiments Makes me appreciate that relativity and quantum mechanics are verifiable experimentally.
evanb 6y ago
I am a physicist. 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-dimensions.pdf 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.pdf http://pdg.lbl.gov/2019/reviews/rpp2018-rev-extra-dimensions...

guardian5x 6y ago

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.

LolWolf 6y ago
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. ----- [0] https://en.wikipedia.org/wiki/Homography https://en.wikipedia.org/wiki/Homography
- jimmySixDOF 6y ago
  Absolutely this should be in VR and I'm a little bit surprised it isn't tbh edit: [1] https://4dtoys.com https://4dtoys.com

zokier 6y ago

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 https://www.youtube.com/watch?v=dT5YCs84jJU

zokier 6y ago
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/ https://scholarsarchive.byu.edu/etd/6968/
- scintill76 6y ago
  I believe this is the author's GitHub: https://github.com/neverhood311 https://github.com/neverhood311 (I followed a YouTube link from the paper, and the YouTube profile pic matches this one's.)
elihu 6y ago
I suppose you could also compromise by drawing a 4-D wireframe overlaid over a traditional render of a 3-D cross section.
mettamage 6y ago
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.

echelon 6y ago

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.

gridlockd 6y ago
The author is also developing a videogame (Miegakure) that has 4D mechanics. It's a puzzle game though, not a shooter.
beaumayns 6y ago
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.
- gliese1337 6y ago
  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.
  - beaumayns 6y ago
    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.

throwaway_pdp09 6y ago

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?

deleted 6y ago
[deleted]
jsd1982 6y ago
Futurama had a 4-D "space" white whale in the Bermuda Tetrahedron in S6 E21 "Mobius Dick".
anigbrowl 6y ago
Annihilation (2018) Spoiler, obviously: https://www.youtube.com/watch?v=uBsJgceM0KI 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.
- hombre_fatal 6y ago
  Perfect music for that scene. 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.
  - peter_l_downs 6y ago
    if you like those sounds you may like Burial, of which it immediately reminded me https://www.youtube.com/watch?v=v27jdRub2GQ https://www.youtube.com/watch?v=v27jdRub2GQ

miohtama 6y ago

Has there been any attempts to visualize and project 4D objects to 3D space so that we could "grasp" them better? E.g. using colours, shadows, etc.

nimish 6y ago

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.

peter303 6y ago

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.

dllu 6y ago

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.

centimeter 6y ago
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.
- h-cobordism 6y ago
  > Does such a thing exist at higher dimensions? 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.

dang 6y ago

A related thread from 2017: https://news.ycombinator.com/item?id=14471931 https://news.ycombinator.com/item?id=14471931

EL_Loco 6y ago

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.

jblow 6y ago
He would be able to see a circle the same way you can see a sphere. Do you feel like you can only see discs, not spheres?
- FPGAhacker 6y ago
  How different would a 2d square look projected onto a 1d view, vs a circle?
  - javajosh 6y ago
    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.)
  - jobigoud 6y ago
    If the object is lit at an angle the shading will be completely different, the edge of the square will create a harsh line.
  - shezi 6y ago
    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. Just like our 2d eyes do in our 3d world.
avhon1 6y ago
"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)

elil17 6y ago

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. https://www.gutenberg.org/files/201/201-h/201-h.htm https://www.gutenberg.org/files/201/201-h/201-h.htm

gdubs 6y ago

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.

pier25 6y ago

If you liked that video check this other one by Carl Sagan: https://www.youtube.com/watch?v=N0WjV6MmCyM https://www.youtube.com/watch?v=N0WjV6MmCyM

pnathan 6y ago

I love 4D graphics. 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. :)

beaumayns 6y ago
I'm curious, in what way was your conception wrong? Sounds like it could still be interesting.
- pnathan 6y ago
  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 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.
joekunin 6y ago
I would love to know more about the kind of math involved in this.
- pnathan 6y ago
  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.

beefield 6y ago

Wonder if there is somewhere a list or some document of "interesting" 4d shapes? For example klein bottle, which is the only one I know of.

shezi 6y ago
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".

deleted 6y ago

[deleted]

slowhadoken 6y ago

Oh yeah this is old. I think this is the same guy making Miegakure @ https://miegakure.com/ https://miegakure.com/

kensai 6y ago

Marc is amazing. Can't wait for his masterpiece game to be finally completed! http://miegakure.com http://miegakure.com If you really want to enjoy 4D Toys in the meantime download his tablet game: http://4dtoys.com http://4dtoys.com

kmbriedis 6y ago

The most surprising aspect of this is that it's a solo paper accepted for a major conference!

jbinney 6y ago

The comparison to children playing with blocks makes me wonder: If you spent long enough playing with these 4D toys, would you develop a deeper intuition for the 4th dimensional shape of the objects?

kmill 6y ago
I've played with some programs I made to explore 4D objects, and, yes, you do gain some intuition. Interestingly, I was told one of George Boole's daughters got lessons from a guy on 4D objects by using colored wooden blocks. She really picked it up, and later in life mathematicians would consult her about intersections of 4D polytopes.
hinkley 6y ago
If you haven't read Mimsy Were the Borogroves, you should. https://en.wikipedia.org/wiki/Mimsy_Were_the_Borogoves https://en.wikipedia.org/wiki/Mimsy_Were_the_Borogoves
robocat 6y ago
That would be a perfect application of VR. I’m sure most of us could learn a visceral understanding of 4D if we could push and twist 4D objects.
dehrmann 6y ago
Sort of, but not really. The problem is you don't have sufficient information to know what should happen.

robofanatic 6y ago

This is super interesting but hurting my brain. A person in 3D can see 3D, 2D and 1D. But a person in 2D can only see 1D. How can a person living in 2D world see 2D things like Circles.

bespokealgo 6y ago
What we (3D beings) see is a 2D projection of the world around us, 2D beings would indeed see a 1D projection of their 2D world (e.g. we only really see a circle when looking at a sphere). Just as we use depth perception to differentiate 3D objects from 2D ones, 2D beings would also be able to discern a circle from a line in a 2D world.

bawana 6y ago

How does one model 4D gravity? Consider the 3D to 2D analogy. If the 2D person lived on a planet, that would be a circle. Then 2 planets orbiting each other would appear to the 2D person as two line segments oscilating up and down (because the 2D person cannot see the entire circle, just the edge view). Based on the velocity of oscillation and the separation of the line segments, a 2D newton could come up with an analogous law of gravition with an inverse square relation. But if two 3d planets were orbiting transversely through the 2D plane of our inhabitants, the 2D person would see two separate line segments growing and shrinking although the line segments could maintain a constant distance of separation. The velocity of growing and shrinking would be changing even though their separation was constant. Extending this analogy to our universe makes me wonder if our calculation that the universe is expanding is just an illusion. Rather, at astronomical distances, our universe could be traversing 4 dimensions. And what we perceive as an accelerating expansion of the universe is just motion into the 4th dimension. And if we could travel into that dimension, we might be able to get to places that appear far away but are actually not. Though it sounds far fetched, it also might explain the stuff that is postulated to happen at the quantum scale-where quantum fluctuations are postulated to occur. The only thing we would need to be able to do is to 'coordinate' the fluctuations of lots of adjacent voxels of space. And then we could wander through 4D space.

DreamScatter 6y ago

Check out my geometric algebra implementation in Julia language at https://github.com/chakravala/Grassmann.jl https://github.com/chakravala/Grassmann.jl