A Computer Technique for Displaying N-Dimensional Hyperobjects (1967) [pdf](reprints.gravitywaves.com)
reprints.gravitywaves.com
A Computer Technique for Displaying N-Dimensional Hyperobjects (1967) [pdf]
http://reprints.gravitywaves.com/VIP/ViewExtraSpaceDims/Noll-1967_AComputerTechniqueForDisplayingNDimensionalHyperObjects.pdf
28 comments
Maybe it's only possible to project/fake one additional dimension and not two? So if we had a 3D display, holographic or something, then a projected 4-D object might make more sense.
Sounds like an awesome VR project to me
https://store.steampowered.com/app/619210/4D_Toys/
4d VR simulator
Now live inside this app exclusively for a year and see if 3D is weird for you when you come back...
You’re right! Now I have an extra reason to learn math.
shamelessly sharing my UI for viewing 4D objects (works with touch screen or mouse)
https://transdimensional.xyz/
https://transdimensional.xyz/
Cool, Thanks. To me, 4-D rotation is completely “unpredictable” or “contraintuitive”, or both. I wonder, you as a developer of that visualization, did you get an intuition for it? do you know in advance how the projection changes when you apply one or the other rotation?
I did actually! As jcun2148 pointed out, the game of trying to return the hyper cube to the start position is a great way to start building intuition.
To describe what each rotation axis does in words: There are six axes of rotation because there are 6 perpendicular planes in 4D (xy, xz, xw, yz, yw, zw). The 3 of these which only use xyz are just the regular axes of rotation. The other three rotate between a regular axis and the 4th dimension. This has the effect of a kind of toroidal wrapping (you may have animations of this, where the inner cubes wrap around to become the outer cube) oriented in the direction of the other xyz coordinate.
Would be cool if the individual rotation columns responded to the mousewheel.
This is really cool, but maybe I'm too dumb to know what I'm actually doing with it. It would be nice if you could add some texture or some color gradient or some other pattern to make it more obvious what's going on.
Doesn't seem to do anything other than show lots of unmovable static 3D cubes?
Interesting, firefox says "gah, your tab just crashed".
FWIW I'm on Firefox 96.0.3 (64-bit), on Windows 10 64 bit and it works fine.
My brain can't handle it but my browser can :-)
My brain can't handle it but my browser can :-)
this is great
I remember encountering this approach in A. K. Dewdney's Computer Recreations column of the April 1986 issue of Scientific American: "A program for rotating hypercubes induces four-dimensional dementia". It included enough pseudo-code for my young self to get something working in qbasic, even adding support for red-blue cellophane glasses so that the hypercubes could be seen in 3d. Great memories!
Flatland, referenced in this paper, is a wonderful and thought-provoking little story.
EDIT: I removed the link because I was worried about the server I linked to. But you can find the PDF of flatland if you just google "flatland pdf"
EDIT: I removed the link because I was worried about the server I linked to. But you can find the PDF of flatland if you just google "flatland pdf"
Genius! I've seen the animation on youtube a hundred times, but never a write-up. Supposedly, the basis for a non-euclidean sculpting robot ;)
Maybe this blog entry is another good starting point: https://ciechanow.ski/tesseract/ and references therein (at the end).
It has been discussed on HN a while ago.
It has been discussed on HN a while ago.
Could you link the video? Sounds cool
Not OP, is it the 4D Hypercube?
https://www.youtube.com/watch?v=q5Qh2XpoCsY
side note it is a mind f (to me) to do point rotations in graphics case, just trying to make a rotating cube in WebGL wow.
side note it is a mind f (to me) to do point rotations in graphics case, just trying to make a rotating cube in WebGL wow.
I was curious how they did the typesetting of the computer generated graphics back then. Printers weren't really a thing.
It could be just a high contrast photograph. I know that at some point they used to wait for all the vectors to be drawn on a CRT with a "permanent"-resettable phosphor and then take a photo with everything on it, clear the screen, and start drawing the next frame. Fun stuff.
Your thought intrigued me and I wanted to look further.
https://piratefsh.github.io/2019/01/07/computer-art-history-... https://piratefsh.github.io/2019/01/07/computer-art-history-...
So there were plotters, which would need to be photographed, like you were talking about, but there's also this:
> And then they were the electronic or microfilm plotters. It works in a similar way as the mechanical plotter, but instead of a pen, it uses an electron beam and instead of paper, it “draws” on microfilm. The idea is that it plots an electron beam in a cathode ray tube, runs it through a camera lens system, which records these beams onto microfilm. I think of it like how a CRT display works, but it also exposes light onto film. The microfilm can then be viewed via a microfilm reader, printed or even animated."
https://piratefsh.github.io/2019/01/07/computer-art-history-... https://piratefsh.github.io/2019/01/07/computer-art-history-...
So there were plotters, which would need to be photographed, like you were talking about, but there's also this:
> And then they were the electronic or microfilm plotters. It works in a similar way as the mechanical plotter, but instead of a pen, it uses an electron beam and instead of paper, it “draws” on microfilm. The idea is that it plots an electron beam in a cathode ray tube, runs it through a camera lens system, which records these beams onto microfilm. I think of it like how a CRT display works, but it also exposes light onto film. The microfilm can then be viewed via a microfilm reader, printed or even animated."
"At first it was thought that the computer-generated movies of the four-dimensional hyperobjects might result in some "feeling" or insight for the visualization of a fourth spatial dimension. In particular, perhaps some visualization of a solid four-dimensional hyperobjeet would be gained from the distortions in the three-dimensional perspective projection. Unfortunately, this did not happen, and we are still as puzzled as the inhabitants of Flatland in attempting to visualize a higher spatial dimension."
When I wrote my first 3D engine as a kid, I immediately extended it to 4D because I was a nerd and I thought the same thing - maybe I'll be able to train my brain to understand 4D objects.
No. Your brain remains confused.
Perhaps if you did it from birth. Like they did with those kittens that never saw horizontal and vertical lines. This would be a totally unethical experiment to carry out on a human child though :)