After all the discussions that began with the question of Moebius's band. I would like to address the question that was raised at the very beginning: Where is the mathematics?
1. One of the most important concepts in the study of geometry and physics is that of *orientation*. In a two dimension world (see the marvelous pamphlet by Abbott: Flatland), a person living on a Moebius band would not be able to tell left from right. In the practical world of chemistry, it is quite important to distinguish *handedness*. The potency of certain drugs is increased by orders of magnitudes if the *handedness* is correct.
To get an intuitive idea of the difference, one may construct a small flag say:
and move it around the usual Euclidean plane, or on the surface of a globe and try to become convinced that it is not possible to turn it into a flag that looks like:
On a Mobius band, this is still the case as long as one does not *go around*. When one does, one comes back in the other form.
2. More advanced, but still accessible at the school mathematics level, is the following assertion:
The *infinite Mobius band* can be used to *coordinatize* the collection of all the straight lines in the plane.
Putting coordinates on an interesting collection of objects is the first step of a bridge between geometry and algebra. One can not do any physics (or problems in the world of finance) without the idea of *coordinates*. There is no fast rule on how one can put coordinates on a collection. Some choices do a better job than others.
We go back to Euclidean geometry of the plane. The basic objects are a) points and b) lines. Descarte introduced a breakthrough idea by introducing coordinates where *points* correspond to *ordered pairs of real numbers*. A Cartesian coordinate frame then consists of an order pair of *arrows* (or vectors) each of length 1 and making an angle of 90 degrees (this is analogous to the little flags we drew).
If we let the starting point (the point of intersection) be (a,b) and the two endpoints be (u,v) and (x,y) respectively, then
Exercise: Show that the 2 x 2 determinant
| u-a x-a | det | | | v-b y-b |
is not zero. Show that we can move this Cartesian coordinate frame into the usual one (0,0), (1,0), (0,1) if and only if the above determinant is +1. In fact, the above determinant must be either +1 or -1. Its absolute value must be the area of the parallelogram (square in this case) where two of the adjacent edges are the given arrows.
One may just define the above determinant as
(u-a)(y-b) - (x-a)(v-b)
and not worry about the general (and difficult) determinant.
By using graph paper, one can move the cartesian frame around and do a few calculations of the above *invariant*. This now makes the qualitative concept of *orientation* quantitative.
With this *warm-up*, we may go on to the more difficult assertion.
With respect to a chose Cartesian coordinate system, we may ask:
How do we describe a line?
There is the usual scheme of writing down the equation
Ax + By = C
Thus, (A, B, C) may be thought of as a *coordinate description* of a line. However, there are two important things to observe:
A) Either A or B must be different from 0. B) kA x + kB y = kC defines the same line when k is not 0.
We may forecast that B) leads to the *real projective plane* if we consider the collection of all triples (A,B,C) distinct from (0,0,0) and identify (A,B,C) with (kA,kB,kC) for k distinct from 0. For any (A,B,C) not (0,0,0), (kA,kB,kC) ranges over the points on the line determined by (0,0,0) and (A,B,C) with the exception of the origin (0,0,0). In this picture, condition A) simply tells us that we must omit the line consisting of all (0,0,C). In other words, the collection of all lines in the plane is identifiable with the "real projective plane with one point removed".
Let us look at a second approach.
To each line L in the plane, we pretend that it is the hull of a *long* ship and that we are the sonar operator on a submarine. Now, the target ship can be located by sending out the sonar beam in a suitable direction and measure the return signal that is bounced back from the target boat represented by L. Thus, we send out the signal in a direction \theta relative to the positive x-axis of our fixed coordinate system (the one on the sonar screen). (\theta is given in radians in mathematics, but in degrees in real life, so the students would have to learn the conversion process and can do a few exercises on ratios and fractions). The reflected signals will be received if and only if the signal beam is perpendicular to L. The time needed for the return signal to be detected can be converted to distance (provided that we know the speed of sound in the given medium--this depends on various parameters, such as temperature, salinity, etc. which can either be read off some standard table or by checking it against some known target).
Note: as sonar operator, we *first* send out the signal in a specific direction and *then* measure the *distance from the origin to L* in the specified direction. If the target boat is directly on us (L passes through the origin), we are in trouble. As a mathematical problem, (\theta, 0) still make perfectly good sense as long as we understand that \theta represents the direction perpendicular to L. Moreover, there is no problem interpreting (\theta, r) for r < 0. This is the line corresponding to (\theta + \pi, -r). Repeating, we see that each line corresponds to the infinite collection of pairs:
(\theta + n*\pi, (-1)^n *r)
We may now coordinatize the collection of all lines in the plane by the collection of ordered pairs of real numbers (\theta, r) where \theta is non-negative but stricly less than \pi, and r is arbitrary and real. In other words, \theta vary over the half open interval [0, \pi}. We next note:
the point (\pi,r) and (0,-r) determine the same line.
Namely, if we *glue the vertical axis with the line x = \pi but with an upside down flip, then we have the assertion:
the collection of lines in the plane is in one to one correspondence with the points of the *infinite Moebius band*
It is infinite because the height is infinite.
We can now do a batch of exercises in the form of making up a dictionary:
Euclidean plane Infinite Moebius band
Line Point Point P ?P
We note that through a point P, we have the *pinwheel* of all lines L passing through P (jargon: the pencil of lines through P). This pinwheel clearly determines P. Each such line L corresponds to a point in the infinite Moebius band. We therefore need to know the coordinate description of the line L.
We begin with the polar coordinate description of the point P in the form (\alpha, R). For the moment, we assume R > 0. Draw a line L through P. Then L corresponds to (\theta, r) satisfying:
R*cos(\theta - \alpha) = r
Thus, the pencil of such lines L through P is a curve in the infinite Moebius band that can be viewed as the *Moebius coordinate description* of the point P, i.e. ?P is the graph of the curve:
(\theta, R*cos(\theta - \alpha)), \theta in [0,\pi)
We note that this works just fine if R = 0. In other words, ?P is a cosine graph with *amplitude* R and *phase angle* \alpha.
The Euclidean statement: two distinct points in the plane determine a line becomes,
two distinct cosine graphs in the coordinatized infinite Moebius plane have one common point of intersection.
In terms of pencils of lines through P and Q, this amounts to to the statement that these two pencils have a common member-- the unique line through P and Q.
Exercise: Translate your favorite statements in the Euclidean plane in terms of the coordinatized infinite Moebius band.
Exercise: Show that the collection of directed lines in the coordinate plane can be coordinatized by the infinite cylinder. Namely, the collection of all pairs (\theta,r), r any real number, \theta in [0,2*\pi) where (\theta,r) is identified by (\theta + 2n*\pi,r) for any integer n.
There is no shortage of exercises at this point.
In particular, the two different ways of coordinatizing the collection of all lines in the plane show that the real project plane with one point removed is the 'same' as the infinite Moebius band. This is totally analogous to the statement that the surface of a round globe with a point removed is the 'same' as the Euclidean plane. To *see* this latter, put the globe on the xy-plane with the south pole resting on the origin and let the missing point be the north pole. Now, take any point P on the globe other than the north pole. The line joining the north pole with the point P in 3-space must pierce the xy-plane in exactly one point P'. This sets up the one-to-one correspondence. (Jargon: this is called a "stereographic projection".)
Exercise: Suppose the sphere has radius 1. a) Determine its equation. b) For a point P on the sphere (i.e. with P = (x,y,z) satifying a), find the coordinates (x',y',0) of P'.
Exercise: Show that a "finite" Moebius band (one of the usual ones constructed by glueing the ends of a thin strip of paper with a single twist) may be identified to the real projective plane with an open disk removed.
Note: One may generalize the Moebius band by making more twists. These are gadgets of interests to biochemists. They are also of interests in the frontier of interaction between mathematics and quantum physics. They will soon be part of science and mathematics in the 21st century. "Dead" mathematics have a tendency of being revived many, many years later. They not not just technical things of interest to topologists.