Pictures of the 3-sphere, or should I say the 4-ball? It’s a 4-dimensional circle.
Even though these drawings of it look completely sweet, I have a hard time parsing them logically. They’re stereographic projections of the hypersphere. All they’re trying to show is the shell of {4-D points that sum to 1}. That’s lists of length 4, containing numbers, whose items add up to 100%. Some members of the shell are
∙ 10% — 30% — 30% — 30%
∙ 60% — 20% — 15% — 5%
∙ 0% — 80% — 0% — 20%
∙ 13% — 47% — 17% — 23%
∙ 47% — 17% — 23% — 13%
∙ 17% — 23% — 13% — 47%
∙ 0% — 100% — 0% — 0%
∙ 5% — 5% — 5% — 85%
The hypersphere is just made up of 4-lists like that.
The 3-sphere was the object of the Poincaré Conjecture (which is no longer a conjecture). Deformations of this shell – this set of lists – are the only simply-connected 3-manifolds. Any other 3-manifold which doesn’t look holey or disjoint must be just some version of the hypersphere.