But another method is to pass through a immersion of the projective plane, a non-orientable closed compact surface which the sphere double covers. So this man named Boy came up with a smooth immersion of the projective plane when Hilbert assigned the homework to show it can’t be done. Here are a few pictures of my attempts to make a model of it.
You might notice the 3-fold symmetry. You might compare that with the 4-fold symmetry of Morin’s surface.
This is an immersion of the projective plane. The singularity in the center is the standard triple transverse intersection.
It’s the six corners that cause problems. Topologically, they’re cones on a figure eight curve. And they’re not amenable to smoothing.
But if you take these corners in pairs and pull them together….
…you can then undo the badness with a little roll…
At this point every edge can be smoothed. These pictures don’t show it off as well as I would like. Oh well.
One last one for Scott: Boy’s Coffee Cup.