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 the whole manifold. To further enjoy this unique and visually delightful shape, we've made adorable tiny glass Klein bottles necklaces! The Klein bottles are handmade from high borosilicate glass and come on your choice of a 24 inch steel chain, adjustable brown leather cord, or adjustable black leather cord. The bottles measure about 1.5in (~37mm) from top to bottom. They function properly, with a tiny opening all the way through the neck and into the bottle. If you have sand from a special place, you could even put it in your pendant…just be aware that this shape has no lid and the contents can fall out if it’s turned the wrong way. Take the necklace off before doing cartwheels and it’ll probably be fine. Alternatively, if you want us to include a little bag of black decorative sand that you can put into your pendant, select one of the style options that end in "+ bag of black sand." Perfect for: Mathematicians Topologists Math teachers and students ...Anyone who likes to think about unique, interesting things!