13 ms·

4D toys

592 points by wenderen 8y ago

anxtyinmgmt 8y ago
This is the best visualization of 4 dimensions I have ever seen.
eerikkivistik 8y ago
What tools were used to build this I wonder?
- MrEldritch 8y ago
  Part of it is that the developer's spent the last decade building a game engine and toolchain for 4D content while developing his 4D puzzle game Miegakure (http://miegakure.com/ http://miegakure.com/)
  - eerikkivistik 8y ago
    That looks incredible. Do you know if the developer has published any papers or code on the subject?
thelastidiot 8y ago
Can't wait to see the 5D version. But seriously, what's the use of that? I don't get it.
- netgusto 8y ago
  It's meant to give an intuition about what a 4th space dimension might feel like, thru self experimentation.
- andybak 8y ago
  > But seriously, what's the use of that? I don't get it. About the same as the use of novels, music, paintings - stuff like that...
- JofArnold 8y ago
  I’m guessing from the word “toy” in the name it’s intended as entertainment. I certainly think it’s fascinating and my mind boggles as to how the developer thinks of these shapes (especially for the “full” third person game)
netgusto 8y ago
There's also Miegakure, another beautiful game based on 4D puzzles. http://miegakure.com/ http://miegakure.com/
- shinryuu 8y ago
  I've been waiting for that game for more than 5 years now I think.. It'll probably stay in development for another 2 years or so.
  - MrEldritch 8y ago
    It's getting really close, though. Puzzles, mechanics, dialogue, effects, all finished. At this point, according to the dev, all that's really left for him to do is "small things like fix collision bugs, and I will keep placing props in levels and program the occasional cool 4D thing. We still need a bunch of 3D modelling done." I'd give a 50% shot it lands on Steam before the end of 2019. (35% it doesn't make 2019 but lands before the end of 2020, remainder that it lands later or never releases.)
    - GuB-42 8y ago
      90% done. It means we are halfway there :p My gut feeling is that we are not there yet, and that 4D toys is an attempt by the author to monetize his development tools in order to be able to complete the main project. I hope it turns out well, Miegakure is definitely in my watch list.
- edanm 8y ago
  Just to make it clear - the original site and your site were created by the same person. The 4d toys are an offshoot of work they did to create Miegakure.
_Codemonkeyism 8y ago
"Goundbreaking 4D+Time Physics Engine that uses new mathematics created for this project." New mathematics no less!
anotheryou 8y ago
Can someone help me out here?: I can represent 3D quite comfortably on 2D monitors, can there be an intuitive mapping of 4D to a 3D VR view? I know 3D mapped to 2D suffers from occlusions and heavily relies on clues like perspective, shadow etc. But given enough time even a less intuitive 4D view could become intuitive with time, too. edit: found this: https://youtu.be/S-yRYmdsnGs?t=252 https://youtu.be/S-yRYmdsnGs?t=252 even better: https://youtu.be/dy_MUfBuq2I https://youtu.be/dy_MUfBuq2I (turn on subtitles)
- chiasmic 8y ago
  I am by no means an expert, but I have thought about trying to visualise the fourth dimension. The issue is that we don’t really see 3D. While in 2D we can view every pixel simultaneously, when we view in 3D we have the issue of aspect and concealed voxels. You can’t visualise simultaneously all voxels of a cube, as most are under others or otherwise obscured. This therefore limits how much we can use 3D to mimic 4D. Perhaps if we were to spend more time with semitransparent objects our brains could learn to visualise 3D better and then maybe 4D.
  - sarreph 8y ago
    What an interesting idea! Have you ever produced anything for a representation of 4D with translucent 3D? I'd be very interested to see it...
    - gitgud 8y ago
      I'm not sure that could practically work. The problem, is that translucency would allow only a partial view of concealed voxels. As more voxels cover something, the translucent effect compounds. Additionally, most voxels would appear different with different view perspectives. Due to more or less voxels covering them. The problem is that in 4D all voxels are visible to the viewer. So viewing a 4D apple would allow you to see the apple from all possible view points simultaneously, including interior views. To me it just doesn't seem possible to replicate this concept in 3D VR.
- Udik 8y ago
rzwitserloot 8y ago
Here's a thought I had watching the bit with 2D man: Actually, the 3D view that 2D man does not understand but which we do understand is... still 2D. My screen is flat. You can use a 2D viewport to render a 3D scene in a way that is natural and easy to understand for us humans: A human watching the 2D scene can very quickly surmise from a glance at the viewport: Which objects are in the scene, and where are they located, in _ALL_ 3 dimensions? This raises the question: Can you render a 4D scene onto a 3D viewport such that us humans are pretty good at understanding where every object in the scene is, in all 4 dimensions? I assume the answer is 'yeah, you can do that'. I wonder what that would look like. It's complicated of course; our eyeballs are involved and they kinda work in 2D and not 3D; where we humans can casually glance at a 2D viewport rendering a 3D scene for a second and know what's happening, we'd have to walk around the 3D screen rendering a 4D scene in order to even see everything.
- anotheryou 8y ago
  I asked the same question a few seconds before you :) found this: https://youtu.be/S-yRYmdsnGs?t=252 https://youtu.be/S-yRYmdsnGs?t=252 and even better, this: https://youtu.be/dy_MUfBuq2I https://youtu.be/dy_MUfBuq2I
  - dm319 8y ago
    Yes, and I think this just shows how a 3d representation on 2d gives our brain the _illusion_ of 3d. This doesn't work for 4d, because our brain doesn't have an internal understanding of 4d objects.
    - anotheryou 8y ago
      I think once we have it interactively our brain will adapt.
      - ianai 8y ago
        It’s t least worth a shot.
abhiminator 8y ago
This is incredibly cool. I feel this would be super useful for teaching children (like my nephew who's 11-years-old) to think about the fourth dimension as a component of the fabric of space instead of time as it's usually understood.
bhouston 8y ago
I would have through the he would have done a projection from 4d space to 2d space similar to how 3d graphics does a projection from 3d to 2d. Similar to http://christopheremoore.net/4d-renderer/ http://christopheremoore.net/4d-renderer/ It I guess in that case it would be difficult to interact with it in VR. I guess this slicing technique works but it would be a bit weird.
MAXPOOL 8y ago
"To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it." – Geoffrey Hinton, A geometrical view of perceptrons, https://www.coursera.org/lecture/neural-networks/a-geometrical-view-of-perceptrons-6-min-sPEhK https://www.coursera.org/lecture/neural-networks/a-geometric... * but remember that going from 13D to 14-D creates as much extra complexity as going from 2-D to 3-D
- forgotpwd16 8y ago
  There is a joke that goes like this: A mathematician and an engineer attend a lecture on the 11-dimensional M-theory. At the end of the lecture the mathematician delighted says how much he enjoyed it. The engineer puzzled asks him how anything even makes sense. The mathematician replies "It's easy. Visualize an N-dimensional space, then let N go to 11."
kranner 8y ago
This passage from Death's End by Cixin Liu really gave me pause to stop and wonder about what the experience of seeing extra dimensions might be like (here translated to English by Ken Liu): -- A person looking back upon the three-dimensional world from four-dimensional space for the first time realized this right away: He had never seen the world while he was in it. If the three-dimensional world were likened to a picture, all he had seen before was just a narrow view from the side: a line. Only from four-dimensional space could he see the picture as a whole. He would describe it this way: Nothing blocked whatever was placed behind it. Even the interiors of sealed spaces were laid open. This seemed a simple change, but when the world was displayed this way, the visual effect was utterly stunning. When all barriers and concealments were stripped away, and everything was exposed, the amount of information entering the viewer’s eyes was hundreds of millions times greater than when he was in three-dimensional space. The brain could not even process so much information right away. In Morovich and Guan’s eyes, Blue Space was a magnificent, immense painting that had just been unrolled. They could see all the way to the stern, and all the way to the bow; they could see the inside of every cabin and every sealed container in the ship; they could see the liquid flowing through the maze of tubes, and the fiery ball of fusion in the reactor at the stern.... Of course, the rules of perspective remained in operation, and objects far away appeared indistinct, but everything was visible. Given this description, those who had never experienced four-dimensional space might get the wrong impression that they were seeing everything “through” the hull. But no, they were not seeing “through” anything. Everything was laid out in the open, just like when we look at a circle drawn on a piece of paper, we can see the inside of the circle without looking “through” anything. This kind of openness extended to every level, and the hardest part was describing how it applied to solid objects. One could see the interior of solids, such as the bulkheads or a piece of metal or a rock—one could see all the cross sections at once! Morovich and Guan were drowning in a sea of information—all the details of the universe were gathered around them and fighting for their attention in vivid colors. Morovich and Guan had to learn to deal with an entirely novel visual phenomenon: unlimited details. In three-dimensional space, the human visual system dealt with limited details. No matter how complicated the environment or the object, the visible elements were limited. Given enough time, it was always possible to take in most of the details one by one. But when one viewed the three-dimensional world from four-dimensional space, all concealed and hidden details were revealed simultaneously, since three-dimensional objects were laid open at every level. Take a sealed container as an example: One could see not only what was inside, but also the interiors of the objects inside. This boundless disclosure and exposure led to the unlimited details on display. Everything in the ship lay exposed before Morovich and Guan, but even when observing some specific object, such as a cup or a pen, they saw infinite details, and the information received by their visual systems was incalculable. Even a lifetime would not be enough to take in the shape of any one of these objects in four-dimensional space. When an object was revealed at all levels in four-dimensional space, it created in the viewer a vertigo-inducing sensation of depth, like a set of Russian nesting dolls that went on without end. Bounded in a nutshell but counting oneself a king of infinite space was no longer merely a metaphor.
yuvalr1 8y ago
This is amazing and looks really fun! I remember reading a short science fiction story once, "Mimsy Were the Borogoves"[1], where small children were playing with such 4D toys, which were found to be educative in unpredictable ways. It really makes me want to try it out. I wonder if it's really the same without VR. [1] https://en.wikipedia.org/wiki/Mimsy_Were_the_Borogoves https://en.wikipedia.org/wiki/Mimsy_Were_the_Borogoves (watch out, there are spoilers here!)
- undershirt 8y ago
  the bottom of the website includes an illustrated frame and excerpt from that very story
  - yuvalr1 8y ago
    Wow, you're right! Didn't see that, Thanks! :)
xtiansimon 8y ago
How curious. What would a haptic VR experience be like? Does a 4D object still exert mass in the other 3Ds? Could you _intuit_ the other 3Ds even if they were not visible?
- marctenbosch 8y ago
  The VR version has haptics. It's kinda interesting. I think there is a little bit of intuition to be gained.
throwawaymath 8y ago
This is an old video [1], but it remains the clearest visual explanation of multiple dimensions that I've ever seen. I really can't recommend it highly enough. I think anyone trying to visualize a 4D object will get something out of watching it. I played with the 4D toys app after it showed up on /r/math a while ago. I like it and I think it's useful. My only complaint would be that it's a little too open ended. While it's nice to provide a simulated tactile experience of four dimensions, I think the app should provide a bit more visual intuition. That's one of the things I like about this video. ________________ 1. https://www.youtube.com/watch?v=90olwwLdEYg https://www.youtube.com/watch?v=90olwwLdEYg
- shawn 8y ago
  Reminds me of https://youtu.be/K4JhruinbWc?t=110 https://youtu.be/K4JhruinbWc?t=110
  - throwawaymath 8y ago
    That was really great, thanks for sharing!
rekshaw 8y ago
The video ends saying that trying to fit a 4D cube in a 4D hole is like a child playing with toy blocks. Hypothetically, if a child was raised with regular VR simulations that allowed the child to manipulate 4D objects, would their brain "learn" it and in a way unlock the 4th dimension?
- freehunter 8y ago
  I was wondering this too as I watched the video. As an adult I know how physics works and I know that what I'm seeing is not normal based on decades of experience. If you showed this to a 1 year old and let them play with 4D virtual toys for a few years, would they more intuitively understand 4D objects? Or would it still remain foreign due to the fact that they can't see things moving around the fourth dimension?
  - vhold 8y ago
    As I played around with 4D toys, I did start to gain a sense of things that were "Behind" or "In Front" of me in 4D space, that is I could start to feel their presence after they had been pushed out of sight, if I knew in what direction they were moving in the 4th dimension. If I didn't know that I could only feel that they were near. It helps that there is a little visualization that is just a line-per-object showing where you are, and where all the objects are intersecting the 4th dimension, that you also use to move "back and forth." One thing I found myself doing was grabbing objects at one of their edges in the 4th dimension, by moving myself to near their boundary, and then using them like brooms. It's really easy to understand with the case of a hypersphere, since at its edge it's just a smaller sphere than at the middle. So you grab that small sphere at the edge, and push in the direction towards its middle in the 4th dimension, and it will act like a bowling ball. You won't see the stuff you are pushing around because the sphere is "ahead" of you, unless they roll around the sphere, then you'll pass them. Once you reach the edge of the 4th dimension, all the stuff you kept pushing will be there. Predicting how 3D intersections change as objects rotate about in the 4th dimension still seems like chaos to me though, except in the case of hyperspheres, which basically don't change as they rotate, but I only played around for about an hour or so. The only way I found to rotate objects in the 4th dimension was to have them collide with each other, or the walls and floor, which makes it kind of hard to carefully experiment with their rotations.
beaumayns 8y ago
I'd love to know the math behind the physics engine, particularly angular momentum. Planar rotations are weird in 4d.
DC-3 8y ago
Using the 4th Dimension, the two interlocking rings can be separated. However, what would be the equivalent puzzle for the 3rd dimension? What 2D system requires the 3rd dimension to be separable? Is there even an equivalent? Clearly a circle within a circle is one such system, but it corresponds to a ball wholly contained by another ball in 3D, not a pair of interlocking rings.
- zwkrt 8y ago
  It is not possible for two ring shapes to interlock in 2D so there is no directly isomorphic trick.
- simias 8y ago
  You can't have interlocking rings in 2D without having some kind of intersection. You have a similar problem with moebius rings and klein bottles: moebius rings are 2D objects but you have to represent them in a 3D world if you don't want them to be self-intersecting. If you attempt to represent a moebius ring in 2D it'll necessarily self-intersect. Klein bottles are the same thing but with an added dimensions: any representation of a klein bottle in 3D makes it look like it's going through itself, even though in 4D it wouldn't: http://s3files.core77.com/blog/images/2013/06/klein-bottle-02.jpg http://s3files.core77.com/blog/images/2013/06/klein-bottle-0... It's also true if you remove a dimension: a 1D moebius strip would simply be a circle, but if you try to draw it in 1D you end up with a segment where both halves of the circle are overlapped. So every time we have a N-dimensional object that can only be properly represented in N+1 dimensions. That's also the same reason you can't solve the problem of connected three objects to three other objects on a 2D planes without intersecting: A B C X Y Z You can't distribute a, b and c to x, y and z on a 2D plane without intersection. Topology is fun.
- lifeformed 8y ago
  It would be a ring with a stick through it. In a 2D slice down the middle, it would look like a donut with a circle (the slice of the stick) in the hole. The circle appears to be stuck inside it, with no way out in 2D. In 3D, you just pull the stick out of the ring.
chubot 8y ago
Wow, cool to see this here. I've been working on reproducing some of it in numpy/matplotlib for the past 2 weeks! I liked their 3D-sliced 120-cell (equivalent of the dodecahedron), and it wasn't too difficult to reproduce. Step 1: Use Schlafli generator from here [1]. Schlafli numbers are a compact description of regular polytopes, and there is a recursive algorithm to generate vertices, edges, faces, etc. from them. The base case of the recursion is dimension 1, so you make 4 calls to get to dimension 4. Step 2: Intersect the edges of the polytope with a hyperplane (a 3D subset of 4D). Step 3: You get a set of 3D points out of step 2. Draw the convex hull of them, which gives you triangles. Step 4: Render the triangles somehow. I used matplotlib's 3d facilities (mplot3d), and we are working on raytracing them. Step 5: Animate over different hyperplanes. Take the min and max in the w plane and that will give you non-empty slices. Now you can "see" the 4D polytope using time as the 4th dimension. I sure he is doing something more advanced (4D collision detection), but this is all we needed to reproduce something that looks kinda cool. [1] https://github.com/aruth2/schlafli https://github.com/aruth2/schlafli
gertgoeman 8y ago
Kinda reminds me of "Flatland": https://en.wikipedia.org/wiki/Flatland https://en.wikipedia.org/wiki/Flatland
- albertgoeswoof 8y ago
  This book is incredible, it’s hard to believe it’s 134 years old.
csomar 8y ago
Can this explain some of the quantum weirdness? Like some of the things quantum entanglement have is due to a forth dimension we are not aware of. Might have to do with general relativity and it is the 4th dimension in smaller scales. Now where is my noble price?
dang 8y ago
Discussed last year: https://news.ycombinator.com/item?id=14471931 https://news.ycombinator.com/item?id=14471931
tehsauce 8y ago
If anyone wants to play with a 4D toy right now in their web browser, here is an interactive tesseract (4d cube) that I built. http://transdimensional.xyz http://transdimensional.xyz It doesn't have physics but features a novel interface that I designed to help build an intuition about different rotations. (There are six axes of rotation in 4 dimensions)
JoeDaDude 8y ago
Some might remember this excellent treatment of higher dimensions [1] in which you first manipulate shapes to solve a puzzle in 2D, then 3D, and finally 4D. YYou develop a keen sense of 4D objects: [1] http://harmen.vanderwal.eu/hypercube/ http://harmen.vanderwal.eu/hypercube/
beefield 8y ago
Agreeably playing with 4d objects is mindboggling enough, but still I would like to somehow visualize myself moving around in 4d space. I am not quite sure how a 4d room would look like and what kind of doors to other 4d rooms would be like, bit for sure navigating around a 4d house could be fun...

4D toys

592 points by wenderen 8y ago

anxtyinmgmt 8y ago

This is the best visualization of 4 dimensions I have ever seen.

eerikkivistik 8y ago

What tools were used to build this I wonder?

MrEldritch 8y ago
Part of it is that the developer's spent the last decade building a game engine and toolchain for 4D content while developing his 4D puzzle game Miegakure (http://miegakure.com/ http://miegakure.com/)
- eerikkivistik 8y ago
  That looks incredible. Do you know if the developer has published any papers or code on the subject?

thelastidiot 8y ago

Can't wait to see the 5D version. But seriously, what's the use of that? I don't get it.

netgusto 8y ago
It's meant to give an intuition about what a 4th space dimension might feel like, thru self experimentation.
andybak 8y ago
> But seriously, what's the use of that? I don't get it. About the same as the use of novels, music, paintings - stuff like that...
JofArnold 8y ago
I’m guessing from the word “toy” in the name it’s intended as entertainment. I certainly think it’s fascinating and my mind boggles as to how the developer thinks of these shapes (especially for the “full” third person game)

netgusto 8y ago

There's also Miegakure, another beautiful game based on 4D puzzles. http://miegakure.com/ http://miegakure.com/

shinryuu 8y ago
I've been waiting for that game for more than 5 years now I think.. It'll probably stay in development for another 2 years or so.
- MrEldritch 8y ago
  It's getting really close, though. Puzzles, mechanics, dialogue, effects, all finished. At this point, according to the dev, all that's really left for him to do is "small things like fix collision bugs, and I will keep placing props in levels and program the occasional cool 4D thing. We still need a bunch of 3D modelling done." I'd give a 50% shot it lands on Steam before the end of 2019. (35% it doesn't make 2019 but lands before the end of 2020, remainder that it lands later or never releases.)
  - GuB-42 8y ago
    90% done. It means we are halfway there :p My gut feeling is that we are not there yet, and that 4D toys is an attempt by the author to monetize his development tools in order to be able to complete the main project. I hope it turns out well, Miegakure is definitely in my watch list.
edanm 8y ago
Just to make it clear - the original site and your site were created by the same person. The 4d toys are an offshoot of work they did to create Miegakure.

_Codemonkeyism 8y ago

"Goundbreaking 4D+Time Physics Engine that uses new mathematics created for this project." New mathematics no less!

anotheryou 8y ago

Can someone help me out here?: I can represent 3D quite comfortably on 2D monitors, can there be an intuitive mapping of 4D to a 3D VR view? I know 3D mapped to 2D suffers from occlusions and heavily relies on clues like perspective, shadow etc. But given enough time even a less intuitive 4D view could become intuitive with time, too. edit: found this: https://youtu.be/S-yRYmdsnGs?t=252 https://youtu.be/S-yRYmdsnGs?t=252 even better: https://youtu.be/dy_MUfBuq2I https://youtu.be/dy_MUfBuq2I (turn on subtitles)

chiasmic 8y ago
I am by no means an expert, but I have thought about trying to visualise the fourth dimension. The issue is that we don’t really see 3D. While in 2D we can view every pixel simultaneously, when we view in 3D we have the issue of aspect and concealed voxels. You can’t visualise simultaneously all voxels of a cube, as most are under others or otherwise obscured. This therefore limits how much we can use 3D to mimic 4D. Perhaps if we were to spend more time with semitransparent objects our brains could learn to visualise 3D better and then maybe 4D.
- sarreph 8y ago
  What an interesting idea! Have you ever produced anything for a representation of 4D with translucent 3D? I'd be very interested to see it...
  - gitgud 8y ago
    I'm not sure that could practically work. The problem, is that translucency would allow only a partial view of concealed voxels. As more voxels cover something, the translucent effect compounds. Additionally, most voxels would appear different with different view perspectives. Due to more or less voxels covering them. The problem is that in 4D all voxels are visible to the viewer. So viewing a 4D apple would allow you to see the apple from all possible view points simultaneously, including interior views. To me it just doesn't seem possible to replicate this concept in 3D VR.
Udik 8y ago

rzwitserloot 8y ago

Here's a thought I had watching the bit with 2D man: Actually, the 3D view that 2D man does not understand but which we do understand is... still 2D. My screen is flat. You can use a 2D viewport to render a 3D scene in a way that is natural and easy to understand for us humans: A human watching the 2D scene can very quickly surmise from a glance at the viewport: Which objects are in the scene, and where are they located, in _ALL_ 3 dimensions? This raises the question: Can you render a 4D scene onto a 3D viewport such that us humans are pretty good at understanding where every object in the scene is, in all 4 dimensions? I assume the answer is 'yeah, you can do that'. I wonder what that would look like. It's complicated of course; our eyeballs are involved and they kinda work in 2D and not 3D; where we humans can casually glance at a 2D viewport rendering a 3D scene for a second and know what's happening, we'd have to walk around the 3D screen rendering a 4D scene in order to even see everything.

anotheryou 8y ago
I asked the same question a few seconds before you :) found this: https://youtu.be/S-yRYmdsnGs?t=252 https://youtu.be/S-yRYmdsnGs?t=252 and even better, this: https://youtu.be/dy_MUfBuq2I https://youtu.be/dy_MUfBuq2I
- dm319 8y ago
  Yes, and I think this just shows how a 3d representation on 2d gives our brain the _illusion_ of 3d. This doesn't work for 4d, because our brain doesn't have an internal understanding of 4d objects.
  - anotheryou 8y ago
    I think once we have it interactively our brain will adapt.
    - ianai 8y ago
      It’s t least worth a shot.

abhiminator 8y ago

This is incredibly cool. I feel this would be super useful for teaching children (like my nephew who's 11-years-old) to think about the fourth dimension as a component of the fabric of space instead of time as it's usually understood.

bhouston 8y ago

I would have through the he would have done a projection from 4d space to 2d space similar to how 3d graphics does a projection from 3d to 2d. Similar to http://christopheremoore.net/4d-renderer/ http://christopheremoore.net/4d-renderer/ It I guess in that case it would be difficult to interact with it in VR. I guess this slicing technique works but it would be a bit weird.

MAXPOOL 8y ago

"To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it." – Geoffrey Hinton, A geometrical view of perceptrons, https://www.coursera.org/lecture/neural-networks/a-geometrical-view-of-perceptrons-6-min-sPEhK https://www.coursera.org/lecture/neural-networks/a-geometric... * but remember that going from 13D to 14-D creates as much extra complexity as going from 2-D to 3-D

forgotpwd16 8y ago
There is a joke that goes like this: A mathematician and an engineer attend a lecture on the 11-dimensional M-theory. At the end of the lecture the mathematician delighted says how much he enjoyed it. The engineer puzzled asks him how anything even makes sense. The mathematician replies "It's easy. Visualize an N-dimensional space, then let N go to 11."

kranner 8y ago

This passage from Death's End by Cixin Liu really gave me pause to stop and wonder about what the experience of seeing extra dimensions might be like (here translated to English by Ken Liu): -- A person looking back upon the three-dimensional world from four-dimensional space for the first time realized this right away: He had never seen the world while he was in it. If the three-dimensional world were likened to a picture, all he had seen before was just a narrow view from the side: a line. Only from four-dimensional space could he see the picture as a whole. He would describe it this way: Nothing blocked whatever was placed behind it. Even the interiors of sealed spaces were laid open. This seemed a simple change, but when the world was displayed this way, the visual effect was utterly stunning. When all barriers and concealments were stripped away, and everything was exposed, the amount of information entering the viewer’s eyes was hundreds of millions times greater than when he was in three-dimensional space. The brain could not even process so much information right away. In Morovich and Guan’s eyes, Blue Space was a magnificent, immense painting that had just been unrolled. They could see all the way to the stern, and all the way to the bow; they could see the inside of every cabin and every sealed container in the ship; they could see the liquid flowing through the maze of tubes, and the fiery ball of fusion in the reactor at the stern.... Of course, the rules of perspective remained in operation, and objects far away appeared indistinct, but everything was visible. Given this description, those who had never experienced four-dimensional space might get the wrong impression that they were seeing everything “through” the hull. But no, they were not seeing “through” anything. Everything was laid out in the open, just like when we look at a circle drawn on a piece of paper, we can see the inside of the circle without looking “through” anything. This kind of openness extended to every level, and the hardest part was describing how it applied to solid objects. One could see the interior of solids, such as the bulkheads or a piece of metal or a rock—one could see all the cross sections at once! Morovich and Guan were drowning in a sea of information—all the details of the universe were gathered around them and fighting for their attention in vivid colors. Morovich and Guan had to learn to deal with an entirely novel visual phenomenon: unlimited details. In three-dimensional space, the human visual system dealt with limited details. No matter how complicated the environment or the object, the visible elements were limited. Given enough time, it was always possible to take in most of the details one by one. But when one viewed the three-dimensional world from four-dimensional space, all concealed and hidden details were revealed simultaneously, since three-dimensional objects were laid open at every level. Take a sealed container as an example: One could see not only what was inside, but also the interiors of the objects inside. This boundless disclosure and exposure led to the unlimited details on display. Everything in the ship lay exposed before Morovich and Guan, but even when observing some specific object, such as a cup or a pen, they saw infinite details, and the information received by their visual systems was incalculable. Even a lifetime would not be enough to take in the shape of any one of these objects in four-dimensional space. When an object was revealed at all levels in four-dimensional space, it created in the viewer a vertigo-inducing sensation of depth, like a set of Russian nesting dolls that went on without end. Bounded in a nutshell but counting oneself a king of infinite space was no longer merely a metaphor.

yuvalr1 8y ago

This is amazing and looks really fun! I remember reading a short science fiction story once, "Mimsy Were the Borogoves"[1], where small children were playing with such 4D toys, which were found to be educative in unpredictable ways. It really makes me want to try it out. I wonder if it's really the same without VR. [1] https://en.wikipedia.org/wiki/Mimsy_Were_the_Borogoves https://en.wikipedia.org/wiki/Mimsy_Were_the_Borogoves (watch out, there are spoilers here!)

undershirt 8y ago
the bottom of the website includes an illustrated frame and excerpt from that very story
- yuvalr1 8y ago
  Wow, you're right! Didn't see that, Thanks! :)

xtiansimon 8y ago

How curious. What would a haptic VR experience be like? Does a 4D object still exert mass in the other 3Ds? Could you _intuit_ the other 3Ds even if they were not visible?

marctenbosch 8y ago
The VR version has haptics. It's kinda interesting. I think there is a little bit of intuition to be gained.

throwawaymath 8y ago

This is an old video [1], but it remains the clearest visual explanation of multiple dimensions that I've ever seen. I really can't recommend it highly enough. I think anyone trying to visualize a 4D object will get something out of watching it. I played with the 4D toys app after it showed up on /r/math a while ago. I like it and I think it's useful. My only complaint would be that it's a little too open ended. While it's nice to provide a simulated tactile experience of four dimensions, I think the app should provide a bit more visual intuition. That's one of the things I like about this video. ________________ 1. https://www.youtube.com/watch?v=90olwwLdEYg https://www.youtube.com/watch?v=90olwwLdEYg

shawn 8y ago
Reminds me of https://youtu.be/K4JhruinbWc?t=110 https://youtu.be/K4JhruinbWc?t=110
- throwawaymath 8y ago
  That was really great, thanks for sharing!

rekshaw 8y ago

The video ends saying that trying to fit a 4D cube in a 4D hole is like a child playing with toy blocks. Hypothetically, if a child was raised with regular VR simulations that allowed the child to manipulate 4D objects, would their brain "learn" it and in a way unlock the 4th dimension?

freehunter 8y ago
I was wondering this too as I watched the video. As an adult I know how physics works and I know that what I'm seeing is not normal based on decades of experience. If you showed this to a 1 year old and let them play with 4D virtual toys for a few years, would they more intuitively understand 4D objects? Or would it still remain foreign due to the fact that they can't see things moving around the fourth dimension?
- vhold 8y ago
  As I played around with 4D toys, I did start to gain a sense of things that were "Behind" or "In Front" of me in 4D space, that is I could start to feel their presence after they had been pushed out of sight, if I knew in what direction they were moving in the 4th dimension. If I didn't know that I could only feel that they were near. It helps that there is a little visualization that is just a line-per-object showing where you are, and where all the objects are intersecting the 4th dimension, that you also use to move "back and forth." One thing I found myself doing was grabbing objects at one of their edges in the 4th dimension, by moving myself to near their boundary, and then using them like brooms. It's really easy to understand with the case of a hypersphere, since at its edge it's just a smaller sphere than at the middle. So you grab that small sphere at the edge, and push in the direction towards its middle in the 4th dimension, and it will act like a bowling ball. You won't see the stuff you are pushing around because the sphere is "ahead" of you, unless they roll around the sphere, then you'll pass them. Once you reach the edge of the 4th dimension, all the stuff you kept pushing will be there. Predicting how 3D intersections change as objects rotate about in the 4th dimension still seems like chaos to me though, except in the case of hyperspheres, which basically don't change as they rotate, but I only played around for about an hour or so. The only way I found to rotate objects in the 4th dimension was to have them collide with each other, or the walls and floor, which makes it kind of hard to carefully experiment with their rotations.

beaumayns 8y ago

I'd love to know the math behind the physics engine, particularly angular momentum. Planar rotations are weird in 4d.

DC-3 8y ago

Using the 4th Dimension, the two interlocking rings can be separated. However, what would be the equivalent puzzle for the 3rd dimension? What 2D system requires the 3rd dimension to be separable? Is there even an equivalent? Clearly a circle within a circle is one such system, but it corresponds to a ball wholly contained by another ball in 3D, not a pair of interlocking rings.

zwkrt 8y ago
It is not possible for two ring shapes to interlock in 2D so there is no directly isomorphic trick.
simias 8y ago
You can't have interlocking rings in 2D without having some kind of intersection. You have a similar problem with moebius rings and klein bottles: moebius rings are 2D objects but you have to represent them in a 3D world if you don't want them to be self-intersecting. If you attempt to represent a moebius ring in 2D it'll necessarily self-intersect. Klein bottles are the same thing but with an added dimensions: any representation of a klein bottle in 3D makes it look like it's going through itself, even though in 4D it wouldn't: http://s3files.core77.com/blog/images/2013/06/klein-bottle-02.jpg http://s3files.core77.com/blog/images/2013/06/klein-bottle-0... It's also true if you remove a dimension: a 1D moebius strip would simply be a circle, but if you try to draw it in 1D you end up with a segment where both halves of the circle are overlapped. So every time we have a N-dimensional object that can only be properly represented in N+1 dimensions. That's also the same reason you can't solve the problem of connected three objects to three other objects on a 2D planes without intersecting: A B C X Y Z You can't distribute a, b and c to x, y and z on a 2D plane without intersection. Topology is fun.
lifeformed 8y ago
It would be a ring with a stick through it. In a 2D slice down the middle, it would look like a donut with a circle (the slice of the stick) in the hole. The circle appears to be stuck inside it, with no way out in 2D. In 3D, you just pull the stick out of the ring.

chubot 8y ago

Wow, cool to see this here. I've been working on reproducing some of it in numpy/matplotlib for the past 2 weeks! I liked their 3D-sliced 120-cell (equivalent of the dodecahedron), and it wasn't too difficult to reproduce. Step 1: Use Schlafli generator from here [1]. Schlafli numbers are a compact description of regular polytopes, and there is a recursive algorithm to generate vertices, edges, faces, etc. from them. The base case of the recursion is dimension 1, so you make 4 calls to get to dimension 4. Step 2: Intersect the edges of the polytope with a hyperplane (a 3D subset of 4D). Step 3: You get a set of 3D points out of step 2. Draw the convex hull of them, which gives you triangles. Step 4: Render the triangles somehow. I used matplotlib's 3d facilities (mplot3d), and we are working on raytracing them. Step 5: Animate over different hyperplanes. Take the min and max in the w plane and that will give you non-empty slices. Now you can "see" the 4D polytope using time as the 4th dimension. I sure he is doing something more advanced (4D collision detection), but this is all we needed to reproduce something that looks kinda cool. [1] https://github.com/aruth2/schlafli https://github.com/aruth2/schlafli

gertgoeman 8y ago

Kinda reminds me of "Flatland": https://en.wikipedia.org/wiki/Flatland https://en.wikipedia.org/wiki/Flatland

albertgoeswoof 8y ago
This book is incredible, it’s hard to believe it’s 134 years old.

csomar 8y ago

Can this explain some of the quantum weirdness? Like some of the things quantum entanglement have is due to a forth dimension we are not aware of. Might have to do with general relativity and it is the 4th dimension in smaller scales. Now where is my noble price?

dang 8y ago

Discussed last year: https://news.ycombinator.com/item?id=14471931 https://news.ycombinator.com/item?id=14471931

tehsauce 8y ago

If anyone wants to play with a 4D toy right now in their web browser, here is an interactive tesseract (4d cube) that I built. http://transdimensional.xyz http://transdimensional.xyz It doesn't have physics but features a novel interface that I designed to help build an intuition about different rotations. (There are six axes of rotation in 4 dimensions)

JoeDaDude 8y ago

Some might remember this excellent treatment of higher dimensions [1] in which you first manipulate shapes to solve a puzzle in 2D, then 3D, and finally 4D. YYou develop a keen sense of 4D objects: [1] http://harmen.vanderwal.eu/hypercube/ http://harmen.vanderwal.eu/hypercube/

beefield 8y ago

Agreeably playing with 4d objects is mindboggling enough, but still I would like to somehow visualize myself moving around in 4d space. I am not quite sure how a 4d room would look like and what kind of doors to other 4d rooms would be like, bit for sure navigating around a 4d house could be fun...