Show HN: Curved Space Shader in Three.js (via 4D sphere projection)
22 bntr 8 5/22/2025, 10:52:44 AM github.com ↗
I made a GLSL shader that bends 3D space using a 4D hypersphere projection.
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.
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.
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... )