My coworker Chris Brewer created a visualization that allows you to experience a 4D sphere (or "hypersphere") by zooming around a 3D space.
Set Viewing Distance to 1000 and choose Map 1.
What appears to be a spherical container is actually the "equator" of the hypersphere. The three spokes are meridians (lines of longitude), and they meet at one of the poles. Mind-blowing.
9:44:27 AM jona: so tell me what this is exactly - it seems that I'm inside a sphere that gets distorted as I move around it
9:45:00 AM BrewerC1: Sure, so do you know what a hypersphere or 4 dimensional sphere is?
9:45:38 AM jona: is that something where x^3 + y^3 + z^3 + a^3 = 5?
9:45:49 AM BrewerC1: Yeah, exactly like that
9:45:52 AM BrewerC1: er except
9:45:55 AM BrewerC1: squared
9:45:57 AM jona: ok
9:46:29 AM jona: how does the x, y, z that I'm moving around relate to that equation's x, y, z, and a?
9:46:58 AM BrewerC1: so in this spaces, you are moving through 4d space, but with the constraint that you are always R^2 units from the origin of the space
9:47:06 AM BrewerC1: which confines you to the surface of the hypersphere
9:47:11 AM jona: ok
9:47:19 AM BrewerC1: So the surface of a sphere is a 2d space
9:47:25 AM BrewerC1: and the surface of a 4d sphere is a 3d space