From new.math.uiuc.edu/oldnew/calculart/
These figures are all 3-D shadows of surfaces embedded
in spherical (positively curved) space.
The Snail begins as a Moebius band whose boundary is a
planar circle. Next, the circle is drawn together to form
a closed crosscap surface. Fly inside to see the pinchpoints
and their Whitney umbrella neighborhoods. Rotating this
projetive plane in 4 space morphs its 3 shadow from the
crosscap to Steiner's Roman surface.
Next we give the ribbon a second half twist, which
expands to form a Clifford torus. Note the Hopf circles,
every pair link each other once. It's fun to fly inside.
It finishes with Brehm's Knotbox. To the topologist, the
Knotbox is a standard spine for the complement of the
trefoil knot in the 3-sphere. Think of a bicycle inner
tube, but tied in a knot; blow it up until there is no
place left in the universe to expand it into. The rubber
walls will merge into a 2-dimensional shape. It could
be the Knotbox.