Klein Bottle Necklace

The Klein bottle is a fascinating topological shape first described in 1882 by German mathematician Felix Klein. It's what's called in mathematics a non-orientable, one-sided surface with no boundary.  An orientable surface is one that can't be transformed into its mirror image by moving a shape along the surface. This means that both the Möbius strip and the Klein bottle are non-orientable.  To determine if something is one-sided, imagine walking across the surface. If you can reach every point on the shape by walking, then it is one-sided. The Möbius strip is also one-sided. But unlike the Möbius strip, the Klein bottle has no boundary -- just as a sphere has no boundary. Interestingly, just as with a Möbius strip, if you cut the Klein bottle in half along its length you can make the topological equivalent of two Möbius strips. The formal definition of the Klein bottle is a two-dimensional manifold on which one cannot define a normal vector at each point that varies continuously over