The mobius band revisited

Here’s something I only recently realized: the mobius band arises naturally when you consider the space of all lines in the plane.

To begin with, the open mobius band (i.e. the usual mobius band without its boundary) is the same as the space of all lines in the plane. The easiest way to see that is to use the fact the lines in the plane are all of the form a x + b y + c = 0, where a and b can’t simultaneously be zero. And two lines ax+by+c=0 and a^\prime x + b^\prime y + c^\prime=0 are the same if there’s a non zero \lambda for which (a^\prime, b^\prime, c^\prime) = \lambda (a,b,c).

The space of all (a,b,c) with the above equivalence and constraint is RP^2 with one point removed (namely the point (0,0,c) with c non zero). And that’s the open mobius band (S^2 with a pair of antipodal points removed is topologically the same as an open cylinder, and it covers RP^2 without a point, and the corresponding open cylinder covers the open mobius band).

Furthermore, to get the “classic” mobius band (i.e. closed and with a boundary), just take the space of all lines in the plane that intersect the closed unit disc (and if you restrict to the open unit disc, you get the open mobius band).

And, of course, the above spaces are the same as the space of all chords of a circle (where the degenerate chords consisting of one point form the boundary of the closed mobius band). So that’s another way the mobius band comes up naturally.

Anyway, I hope this has been of interest to those of you who already know some mathematics and have only seen the mobius band as a weird pathological example, disconnected from the rest of mathematics, in a topology course or a popular mathematics book. It actually does come up naturally in the context of fairly elementary mathematics. The general equation for all lines in R^2 is something that I first saw in grade 11 or 12, and the mobius band is something I’ve known about for about as long, and yet it wasn’t until a week ago that I realized that there was a strong connection between them.

Here are some extra details for those of of you who can’t get enough of this sort of thing:

If you don’t like using the equation a x + b y + c = 0 to identify lines in the plane with RP^2 - {(0,0,1)}, here’s another approach. Consider the plane z=1 in R^3. A line in that plane is uniquely determined by a plane in R^3 through the origin (0,0,0). The only plane through the origin that doesn’t result in a line in z=1 is the plane z=0. Finally, the planes through the origin are in one to one correspondence with the lines in R^3 through the origin (just consider the line perpendicular to the plane). And the space of lines in R^3 through the origin is precisely RP^2.

Also, if instead of considering lines in the plane you consider directed lines, then that space is S^2 - \{(0,0,1), (0,0,-1)\}, which is the open cylinder. The double covering of the space of directed lines to the space of lines corresponds exactly to a double covering of the open cylinder to the open mobius band.

Finally, to get a parametrization for the space of all line in R^2, use the following parameterization of the space of directed lines:
(\theta, s) \to L, where L is the line parametrized by
t \to s(cos \theta, sin \theta) + t (-sin \theta, cos \theta).

Restricting \theta to the values 0 \leqslant \theta \textless 2\pi gives the homeomorphism from S^1 \times R^1 to the space of directed lines, and the values 0 \leqslant \theta \textless \pi give the identification of the open mobius band with the space of undirected lines. And then restricting s to the values -1 \leqslant s \leqslant 1 gives the identification of the mobius band (with boundary) to the space of lines intersecting the closed unit disc D^2.

Author: Walter Vannini

Hi, I'm Walter Vannini. I'm a computer programmer and I'm based in the San Francisco Bay Area. Before I wrote software, I was a mathematics professor. I think about math, computer science, and related fields all the time, and this blog is one of my outlets. I can be reached via walterv at gbbservices dot com.