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?
The surface of a 4D sphere (a 3-sphere) is itself 3-dimensional (just like the surface of an ordinary 3D ball is 2D).
So when I use the hypersphere in intermediate computations, I’m not actually adding an extra dimension to the world.
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.
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These projections, how do they make sense?
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?
The surface of a 4D sphere (a 3-sphere) is itself 3-dimensional (just like the surface of an ordinary 3D ball is 2D). So when I use the hypersphere in intermediate computations, I’m not actually adding an extra dimension to the world.
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.