A reader asked me, in response to yesterday’s post, why I failed to make any mention of the Mobius Strip. Addressing that topic seemed like a good way to close the week’s blogging.
Imagine that you take a long thin strip of newspaper. Hold it at the top with your thumb and index finger, and let the bottom dangle loosely. Now grab the bottom and give it a half-twist. Take the narrow side at the bottom, and bring it up so the short edge meets up with the corresponding short edge where you are holding the newspaper. Tape these ends together. The result is a Mobius Strip. Click here for pictures and more information.
The long, thin rectangle of newspaper has the property that if you pick a point at random somewhere between the two long edges, the region near that point looks like a flat surface. That property is maintained after we make the twist and loop the paper back on itself. You might think, therefore, that the Mobious Strip is a two-dimensional manifold just as surely as the spheres, tori and Klein bottles from yesterday’s post.
The trouble is that the Mobius strip has an additional property lacking in the surfaces we considered yesterday. You see, if you choose a point residing on one of the two long edges of the paper, then the local terrain no longer looks like an ordinary, flat surface. Say you are a small bug crawling on the left edge of the paper, starting from the dangling end and moving up toward the end being held. If you look to your right you see an ordinary, flat, two-dimensional surface. But if you look to the left you see only open space. There is no more surface in that direction, and if you move to your left you will fall off the surface altogether.
You might even say this long edge forms a boundary between the surface itself on the one hand and the space in which the surface resides on the other. For this reason, we refer to the Mobious Strip as a two-dimensional manifold with boundary. By contrast, the surfaces considered by the Poincare conjecture are assumed not to have a boundary.
But perhaps we shouldn’t give up. The Mobius Strip has a boundary, and therefore is not a full fledged manifold. But if we could somehow get rid of that boundary edge, perhaps we would be able to produce a new two-dimensional mainfold.
So let us take two Mobius Strips and stitch them together along their boundary edges. The result would be a bona fide two-dimensional manifold, without boundary. If what I said yesterday is correct, then this surface must be equivalent either to a sphere, a torus with some number of holes, or some collection of Klein bottles stitched together.
Our two Mobius strips have a big hole in the middle, so they are not equivlaent to the sphere.
It is less obvious that the two strips are not equivalent to a torus, but indeed they are not. The twist in the Mobius strip gives it a property called “nonorientability.” Without giving a precise definition of what that means, suffice it to say that the torus is “orientable.”
It turns out that two Mobius Strips stitched together along their boundary edge are equivalent to a Klein bottle. Indeed, Felix Klein first defined his bottle in precisely this way. Just as the Mobius strip is said to be a surface with only one side, the stitched together Mobius Strips form a single-sided bottle. Its inside is simultaneously its outside.
Of course, the Mobius Strip possess a seemiingly endless supply of amusing topological properties, but I think that’s enough for now. Time to start the weekend!