How can I untwist a flexible strip attached at both sides

Answer 1

This problem seems to be referencing a Mobius Strip. First of all, a Mobius Strip is a 2-dimensional surface. We construct it by taking a 2 dimensional surface with 4 edges and twisting it in the third dimension. We then fuse two of it's edges and get a Mobius Strip, a 2D surface with 2 edges.

Visualization Tip:

(You can actually do this with arts and crafts.) If you use scissors to cut along the middle of the mobius strip, it remains a mobius strip but twice as long and skinnier.

Note: This cut is different from unfusing it because if we unfuse it we have a 2D surface with now 4 edges, a Mobius strip has 2 edges.

So we are not allowed to make cuts that increase our number of edges, but we are allowed to fuse edges.

To undo it, we have to take the surface and self-intersect the Mobius Strip into the fourth dimension. We then fuse the remaining edges. This 2D manifold is known as a Klein Bottle. This is tricky to understand much less visualize but a Klein bottle has 0 edges with 2 sides (inside and outside which are one and the same, just like how you can be on both sides of the Mobius Strip at the same time).

Imagine we were in the second dimension walking along the Mobius strip. You keep walking forward and end up where you started. The Klein Bottle is similar except now we are 3D entities walking along it.

After we turn it into a Klein Bottle, we undo the self intersection in the 4th dimension, which brings us back to having 2 edges and gets rid of that pesky Mobius twist.