The idea:
1. Project a model onto a 4D sphere
2. Rotate the sphere
3. Project the model back to 3D
Curious what you think.
The idea:
1. Project a model onto a 4D sphere
2. Rotate the sphere
3. Project the model back to 3D
Curious what you think.
Is 4D sphere the upper limit on this method, or can you project say 3D scene onto 5D sphere? (e.g a 1D line onto a 3D sphere analog)
You could project from 5D down to 3D, but the dimensional mismatch breaks the bijection - you'd lose information or overlap points. However, a 4D → 5D → 4D projection would preserve structure, though it gets harder to visualize.
I chose 3D ↔ 4D specifically because curved 3D space is much more intuitive and has direct physical meaning - it corresponds to positively curved space (see e.g. https://en.wikipedia.org/wiki/Shape_of_the_universe#Universe... )
I can project a 3D item onto a 2D plane, but only observe it because I'm outside of that 2D plane. This is like expecting the 2D plane to see itself and deduce 3D-dimensionality from what it sees. Like a stickman. It would only be able to raycast from its eye in a circle. It could do so from multiple points on the plane, but still, how would it know that it is looking at the projection of a sphere?
What this transformation does give me is a way to imagine a closed, finite 3D space, where any path you follow eventually loops back to where you started (like a stickman walking on the surface of a globe). Whether or not that space “really” needs a 4th spatial dimension is less important than the intuition it gives: this curved embedding helps us visualize what a positively curved 3D universe might feel like from the inside.
I once tried a cross-eye 4D view: https://github.com/bntre/40-js
I've heard of true 3D bump mapping being done in fragment shaders (not just lighting), but I can't really imagine how more radical geometric distortion could be implemented there.
This is the same math as this old program called Jenn3d[0] which I played around with almost twenty years ago. (Amazingly the site is still online!) The crazies who built it also built it to play Go in 3d projective space. I was never able to play Go with it, but I've been in to projective geometries since.
OP - if you want to try something else cool with 4d to 3d projective geometries, here's an idea I ran across working with 3d to 2d.
I make a tool for generating continuous groupings of repetitive objects in architectural computation. [1] When faced with trying to view the inside of lattices containing sets of solids which tile space continuously, I tried a few different methods (one unsuccessful but cool looking one here [2])
So when I created the sphere upon which to project the objects in the lattice, rather than just project the edges I made concentric spherical section planes and projected the intersection of those with the objects. [3] By using objects parallel to the projection plane to cut sections I was able to generate spacings between the final generated section lines that mapped how oblique the surface being cut was from the ray projecting from the centerpoint of the sphere to its surface.
Sorry OP, that's a long description. TL;DR - instead of projecting 3d mesh edges to a 4d sphere then back down to 3d space, what if you tried describing the meshes as the intersection of their 3d geometry with 4d hyperspheres parallel to the projection hypersphere? It would look more abstract, but I bet it would look cool as heck, especially navigating in 3d projective space!
[0] https://jenn3d.org/ [1] https://www.food4rhino.com/en/app/horta [2] https://vimeo.com/698774461 [3] https://vimeo.com/698774461
p.s. Also, if any actual geometers are reading this - I'd love to co-author a math paper that more rigorously considers what I explored / demonstrated with the drawings above. I have a whole set of them methodically stepping through the process, and could generate more at will. I also have a paper about it I can send on request (or if you can hunt down the Design Communication Association Conference Proceedings 2022).
Sourabhsss1•3h ago