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.