This is an old video I made showing how to make a Slinky look like a Klein bottle. It’s the easiest way of making a Klein bottle that I know of!
A Klein bottle is a non-orientable closed 2D manifold. That means that it’s a 2D surface, with no edges, on which you can’t define a consistent sense of direction. This is most clearly demonstrated by the fact that the closed shape doesn’t have an “inside” and an “outside” – you can trace a line along the surface from what looks like the inside to what looks like the outside.
A real Klein bottle can’t exist in 3d space because it would have to intersect itself. By imagining a smooth surface stretched over a Slinky, we can more easily get a sense of what a Klein bottle looks like.
Original idea thanks to Edmund Harris (maxwelldemon.com)