Cliff & Danny Stoll
We make Klein bottles and other topological manifolds for the math community. Much fun - mathematicians suggest designs, and we create glass shapes.
Recently, we've been playing with linked glass Klein bottles and other objects.
A half-twist Mobius loop can be mapped onto a Mobius loop with three twists. This works for Klein Bottles as well: By connecting any odd number of Klein Bottles, you'll make a shape that's homeomorphic to a Klein bottle.
What happens if you try make a Mobius Loop with two half twists? Or connect an even number of Klein bottles? Now that's a different story...
Two Klein bottles linked together do not form a Klein Bottle!
This double-Klein bottle is two sided and orientable ... it's homeomorphic to a torus!
It's made from two "standard" glass Klein bottles, cut using a diamond saw, and connected by welding in a glass lathe.
This was commissioned by cosmologist Paul Steinhardt for the cover of his book, "Endless Universe: Beyond the Big Bang"
Three connected Klein Bottles, linked externally to form a single manifold -- this is homeomorphic to a "standard" R3 immersed Klein Bottle.
We built this by welding together three stock Klein bottles. The borosilicate glass has a low coefficient of expansion, so we can apply a torch to one seam without creating expansion cracks. This weighs about 500 grams, and has 3mm thick walls.
Like our other Klein bottles, this is R3 immersed (not embedded), Riemannian, unbounded, affine, Hausdorffian, compact, and nonorientable.