Each sphere is one dimension higher than the last. At every step the same trick lets us see it: shine a light from the north pole N and look at the shadow it casts on flat space one dimension down. The torus is the bridge between S² and S³. Pick a card to open it full size.
The sphere is always one dimension less than its home. S¹ is a curve in the plane, S² a surface in space, and S³ a 3-D "space" that needs four axes to sit in.
The shadow is flat space plus one missing point. A line, a plane, ordinary space: N is the one point with no shadow, and it goes to infinity.
Odd spheres can flow everywhere; even ones can't. S¹ and S³ carry flows with no still point. S² and S⁴ always have one, like the poles of a spinning globe.
A Python companion (python -m sphere_ladder verify) checks the claims on these pages numerically with NumPy. · Research plan · akrishnash.github.io