The Smooth Analog

The $3$-sphere is a smoothed out surface of a hypercube:

 $3\text{-sphere}$ $=$ $\partial(4\text{-ball})$ $\approx$ $\partial({\rm hypercube})$ $=$ $\partial(\square\times\square)$ $=$ $(\box\times\square)\cup(\square\times\box)$ $\approx$ $(\circle\times\disc)\cup(\disc\times\circle)$ $=$ Solid Torus $+$ Solid Torus