Klein Bottle Hats, Moebius Scarves

It is markedly reassuring to learn that there are people in this very world of ours who are inventive enough to be rewarded for their playfulness. Case in point, Cliff Stoll, creator and merchandiser of the Klein Bottle Hat and Matching Moebius Scarf and the Drinking Mug Klein Bottle.

A Klein Bottle is a three-dimensional Moebius Strip, which, according to Wolfram Research, is "a closed nonorientable surface of Euler characteristic 0 (Dodson and Parker 1997, p. 125) that has no inside or outside. It can be constructed by gluing both pairs of opposite edges of a rectangle together giving one pair a half-twist, but can be physically realized only in four dimensions, since it must pass through itself without the presence of a hole. Its topology is equivalent to a pair of cross-caps with coinciding boundaries (Francis and Weeks 1999). It can be cut in half along its length to make two Moebius strips (Dodson and Parker 1997, p. 88), but can also be cut into a single Moebius strip. "

Once we realize that the Klein Bottle is in fact not realizable given the threeness of our dimensionality, we can but only laugh in bewilderment at Mr. Stoll's Klein Bottle products.

A Moebius Strip, on the other hand is in fact quite realizable in three-dimensional space, even though, topologically speaking, it is a one-dimensional object. It's all a bit of a mathematical guffaw, though, because it is, reality speaking, not really one-dimensional at all, at all. And when you cut in in half you get something different than when you cut it in three.