[A person offering a second person a Klein bottle. The second person rejects the Klein bottle. They have a plate of regular three dimensional shapes in front of them. They have picked up a rectangular prism on their fork. In the upper left corner is the text: “No! A responsible adult says no to non-orientable shapes”]
You haven’t lived till you comprehended this one! If you peel this entire surface into two surfaces like 2ply toilet paper, this is also a halfway point for one way of turning a sphere inside out, depending on which surface peels which way!
I think that’s called a Boy’s Surface. There is a different animation of the same object in the Wikipedia article. It’s a disk with a mobius strip glued to its edge, but most articles get too mathsy too quickly for me to understand, so that’s all the information I can provide :3
The magic happens at the center-point of the surface where the three self-intersections meet. When you ply the surface apart, a tiny cube forms at the triple-point and begins to grow.
Morin’s surface is slightly less complex than splitting the boy’s surface apart, in that sphere eversion halfway model, a trapezoid forms instead of a cube. Inverting a trapezoid in this way is the minimum complexity required to turn a sphere inside out.
Videos I enjoy:
Outside in , which uses a technique different than those above (there’s also a parody out there where the narrators get snarky at eachother)
The optiverse , which uses Morin’s surface mentioned above, but is as ‘smooth’ as mathematically possible.
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