The Smooth Analog
The 3-Sphere is a smoothed out surface of a hypercube:
3{\rm-sphere}
=
\partial(4{\rm-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