>>12793
Be care on your path Anon. Love awaits you.
Related to pic 1
As remarked above: What you get is the Klein bottle. Put differently: The result is what you get when you take two Möbius strips (which both have one boundary) and glue both boundaries together (which does not work when embedded in 3d space but works in theory).
Related to pic 2
In the course of experimenting with 3-D models, I developed a method for twisting a toroid of arbitrary cross section. I called these forms Möbioids. Of course, if the cross-section is circular the result will be indistinguishable. What I did was to consider cylinders of non-circular cross-section. However, then the twist angles are quantized. Depending on the particular cross-section and number of twists, you can get forms with one or more surfaces. The figures below show an astroid cross-section with one twist of π/4 (left) and a pentacuspid cross-section with six twists of 2π/5 (right). Each has a single surface. Again, not all twists lead to a single surface. You can find more images and some animations at A New Twist on Möbius.
Sauce:
https://math.stackexchange.com/questions/217513/is-there-a-möbius-torus