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I'm curious how this would look if the 4D space was projected onto the 3D space instead of taking a cross-section, much like we already project 3D space onto 2D

I'm curious how this would look if the 4D space was projected onto the 3D space instead of taking a cross-section, much like we already project 3D space onto 2D space (your display), to create "3d" graphics.

Rotten194 9y ago
Isn't that what the traditional 3d visualization of a hypercube is? e.g. https://en.m.wikipedia.org/wiki/Hypercube#/media/File%3AHypercube.svg https://en.m.wikipedia.org/wiki/Hypercube#/media/File%3AHype...
- chmod775 9y ago
  Yes. I'd love to play around with that in a sandbox.
- BearGoesChirp 9y ago
  Isn't that a 2d projection of the 3d projection of 4d space? It gets weird because our eyes see almost in 2d, and even with VR, there is a 2d feed to each eye that gets combined to add slight 3d depth perception. So really we would be projecting a 4d world onto two 2d plains that our visual cortex would try to merge to add slight depth perception.
jobigoud 9y ago
There are existing projects for this, for example [1]. I haven't tried OP demo yet (but I've spent a fair amount of time thinking about 4D and VR and implemented my own viewers), and I'm very convinced that the cross section approach is much much better if you want to deeply understand what is happening. The perspective projection folds the fourth dimension into our hyperplane and you can't untangle it. To paste something I wrote here [2], regarding the difficulty of groking the tesseract from the projected figure, I think it’s almost impossible to realize that there are points inside the tesseract that are not inside any of the eight cubes visible in the [projected] image, or that if you trace a diagonal between the corners of the tesseract you do not pass through any of the eight cubes. There is much more space inside the hypercube than meets the eye in the perspective projection. [1]: https://www.youtube.com/watch?v=BFmDLUUhZjE https://www.youtube.com/watch?v=BFmDLUUhZjE [2]: http://1å.fr/blog/down-the-4d-rabbit-hole-navigation/ http://xn--1-2fa.fr/blog/down-the-4d-rabbit-hole-navigation/
indescions_2017 9y ago
One application of these projections would be a point light source in the 4-th Dimension that would be obscured by solid 4-D toys and subsequently cast 4-D shadow "voxel maps" into the 3-D cube space. Having shadows and radiosity illumination would absolutely help with the "disappearing" cubes problem during game play. But it represents a substantial calculation for real-time physics and rendering engines.

Rotten194 9y ago

Isn't that what the traditional 3d visualization of a hypercube is? e.g. https://en.m.wikipedia.org/wiki/Hypercube#/media/File%3AHypercube.svg https://en.m.wikipedia.org/wiki/Hypercube#/media/File%3AHype...

chmod775 9y ago
Yes. I'd love to play around with that in a sandbox.
BearGoesChirp 9y ago
Isn't that a 2d projection of the 3d projection of 4d space? It gets weird because our eyes see almost in 2d, and even with VR, there is a 2d feed to each eye that gets combined to add slight 3d depth perception. So really we would be projecting a 4d world onto two 2d plains that our visual cortex would try to merge to add slight depth perception.

jobigoud 9y ago

There are existing projects for this, for example [1]. I haven't tried OP demo yet (but I've spent a fair amount of time thinking about 4D and VR and implemented my own viewers), and I'm very convinced that the cross section approach is much much better if you want to deeply understand what is happening. The perspective projection folds the fourth dimension into our hyperplane and you can't untangle it. To paste something I wrote here [2], regarding the difficulty of groking the tesseract from the projected figure, I think it’s almost impossible to realize that there are points inside the tesseract that are not inside any of the eight cubes visible in the [projected] image, or that if you trace a diagonal between the corners of the tesseract you do not pass through any of the eight cubes. There is much more space inside the hypercube than meets the eye in the perspective projection. [1]: https://www.youtube.com/watch?v=BFmDLUUhZjE https://www.youtube.com/watch?v=BFmDLUUhZjE [2]: http://1å.fr/blog/down-the-4d-rabbit-hole-navigation/ http://xn--1-2fa.fr/blog/down-the-4d-rabbit-hole-navigation/

indescions_2017 9y ago

One application of these projections would be a point light source in the 4-th Dimension that would be obscured by solid 4-D toys and subsequently cast 4-D shadow "voxel maps" into the 3-D cube space. Having shadows and radiosity illumination would absolutely help with the "disappearing" cubes problem during game play. But it represents a substantial calculation for real-time physics and rendering engines.