Paper Trick: Newsletter Challenge (04/26/05)

If you twist the paper and tape the ends together, you create a Mobius Strip which has one side. Cut it and you have two sides. Cut it again, you've got two objects with two sides.

He's Right:

Maybe there's a bored topologist out there who can answer this for me: Starting backwards, if you start with the two strips that are looped and weld their edges together, you get a single strip with a full twist. Weld the edges of this strip together and you get a Moebius strip. So if you then weld the edge of the Moebius strip to itself, do you get a Klein bottle -- an object with one surface but no edges? Seems to me you'd be going from 4 surfaces and 4 edges, to 2 surfaces and 2 edges, to 1 surface and 1 edge, to 1 surface and no edge. Or maybe the object mathematically has no surface (?) and no edge?

I'm having a hard time visualizing it, but a Moebius strip can be welded into a Klein bottle. Ref.

The Science Museum in London has an excellent section on topology. I'll possibly be there within the next few weeks, so I'll see if any of their models illustrate this.

As written in the 5/03 issue of Specs & Techs from GlobalSpec:

The structure is referred to as a "Möebius strip" (or Möebius band), named for A. F. Möebius (1790-1868), a mathematician and astronomer. To construct a Möebius strip, take a strip of paper, twist it 180° and fasten the two ends together. There is only one side to this structure. If you then cut it down the middle, it changes from single-sided to double-sided. And cutting this double-sided structure down the middle results in two double-sided loops. Try it! (Or have your kids try it...)