Even better, the first comment on that movie says: Daniel Piker <https://vimeo.com/user798992>PLUS <https://vimeo.com/plus>11 years ago You can now order an actual physical 3D print of this from shapeways, here: shapeways.com/model/12988/sudanese_m__bius.html <http://www.shapeways.com/model/12988/sudanese_m__bius.html> It costs only $26.01. And even better than THAT, the image on the Shapeways site is a “digital preview not a photo” that you can rotate as you wish, to inspect it from different angles and move however slowly you need! (Click the 3D button in the upper right corner of the image.) I am humbled that even some great mathematicians on this list find visualizing this surface challenging. I’m ordering one from Shapeways, hoping it will cure my headache. — Mike
On Apr 3, 2020, at 1:43 AM, Scott Kim <scott@scottkim.com> wrote:
Here's a movie of a rotating Sudanese surface. Quite informative. No singularities or going to infinity or self-intersecting...just a remarkably hard to grok surface in ordinary 3-space. https://vimeo.com/2037835
The natural question is if you go in the hole on one side of the circle do you come out the other side? Yes you do...this is a deformation of a loop, after all, so you can link with it.
I once made a polygonal version of this surface, which I find slightly easier to understand. It's based on the observation that the edges of an octahedron are three equatorial squares. To make it, start with a cube that is missing one face. Attach four equilateral triangles to the four open edges of the cube. Fold these triangular flaps alternately up and down, so opposite triangles share a vertex. One of these two vertices will be inside the box, and one will be outside. The edges of these four triangles form the 12 edges of a regular octahedron. Finally fill in one of the two equatorial squares of the octahedron that is NOT edges of the cube with a square panel, and voila, a Möbius strip, with the remaining equatorial square being the sole edge. Well it still hurts my brain, but I do find it a little easier to grasp than the slippery smooth skin of the Sudanese surface.