STEVE SWEENEY'S MOBIUS LETTER

Dear Rick,

I thought I should finally get around to writing at least a few words on the Mobius strip problem. It's not going to be more than a few words, however, since I'm just about completely burnt out on this problem and have virtually no interest in it anymore. The source of my interest for so long was that I kept thinking, "If I just do this one little thing, I can get a few more pieces, and that'll be the maximum." I now admit defeat, and am quite apathetic about it. First, let me restate the problem, verbatim:

Consider the torus, a doughnut-shaped solid that is perfectly circular at each perpendicular cross section, and a Mobius strip which has a single 180 degree twist and a uniform curvature throughout its length. Suppose a torus is sliced three times by a knife that each time precisely follows the path of such a Mobius strip. What is the maximum number of pieces that can result if the pieces are never moved from their original positions?

I've discussed the problem with, or at least heard from, about twenty different people. Of these approximately twenty people (including One-in-a-Million members and people who solved the cones/cylinder problem), almost everyone had a different interpretation. A few shared the same numerical answer, but had different (often erroneous) ways of getting it.

The above, as stated, is problematic for several reasons. By far the greatest source of controversy is the interpretation of the Mobius strips. Several people searched for a mathematical definition of "curvature" and argued that "uniform curvature" was impossible. Others, such as myself, were content with an intuitive understanding of "uniform curvature." With this interpretation, the Mobius strip is traced by a line segment as it rotates 180 degrees at a constant rate, with its midpoint tracing out a circle, also at a constant rate. This is clearly what Ron Hoeflin had in mind. Even if we accept this interpretation, however, we still do not arrive at a definitive solution.

Another problem involves the interpretation of Hoeflin's torus. Most people did not have any problem with it, but a couple notable people did. The torus is "perfectly circular at each perpendicular cross section." "Perpendicular to what?" asks Chris Langan. This question was not commonly asked, since most of us have [a] ready-made idea of what a torus is. Chris Langan does point out, however, that this is an unsatisfactory definition. He also points out that nothing stated prevents the circular cross sections from varying in diameter [one guy did give an answer of infinity, but I don't know how he got it]. A satisfactory, albeit long-winded definition would be: "a solid whose surface is formed by rotating a circle about an axis such that the circle and the axis are in the same plane, but such that the axis does not intersect the circle."

One or two people even complained about the "knife." It doesn't bother me much, but they do have a point, I suppose. The "knife" would have to be a line segment, or a piece of the Mobius strip. One guy let the knife be a line and got some interesting results. The same result can be achieved if we just use a really long line segment. In this case the Mobius strip intersects itself, which was also an issue of controversy. Is such a figure a Mobius strip? My argument was that it isn't, because it seems that such a figure is four-sided, whereas a Mobius strip should be one-sided. But it is not even clear that such a figure is four-sided. The self-intersection creates a kind of "fence" in the area of intersection. I think both the one-sided and four-sided views are reasonable. [What do members think?]

A plot of submitted answers would surely be multi-modal. I should note that most of the solutions I have seen contained errors, even given a particular interpretation of the surfaces. Even erroneous solutions, however, raised legitimate issues, some of which are detailed above.

I guess I finally stopped caring about this problem when it became clear that the solution is almost certainly extremely unaesthetic, that is, really ugly. Since diagrams are not possible in the current format of Noesis, I won't submit any. [I previously said we couldn't do diagrams, but Chris Cole says we can. So send in any diagrams y'all have lying around.--Ed.] I probably wouldn't anyway, since it would take too much time. Let me just say that I'm certain that the answer is 29 or greater, and possibly much more (this is under the popular interpretations of the surfaces). This is achieved by letting each Mobius strip be tangent to the surface of the torus twice. I also let each Mobius strip be wide enough so that it is tangent to its own edge at one point. [If we let it intersect itself, we get a lot more pieces.]

But I don't necessarily think I'm onto the solution here. One guy gave an answer in the thirties with an intriguing approach, but I haven't checked it out thoroughly due to lack of interest; i.e. bum-out, This was the only guy who described his surfaces using geometry (he uses cylindrical coordinates). He might have the right idea. Maybe not. But to illustrate a point, he now claims his original answer was too low--now he can get 76 pieces. It just goes on and on.

TANGENTIAL CHALLENGE: If we accept the most popular interpretation of the Mobius strip, but we trace it out with a line (as opposed to a line segment), the set of points of self-intersection trace out something that looks very much like a normal, or Gaussian curve. Can anyone determine whether this curve is normal? What is it?

Future considerations: from our correspondence, it sounds like Ron Hoetlin will scrap this problem entirely, and go back to his original Trial Test "A" version, where the three Mobius strips are confined to the interior of the torus. He will still have to come up with satisfactory definitions of the surfaces, however. My suggestion to him was to state the problem something like this:

Consider a torus, a surface formed by rotating a circle about an axis such that the circle and the axis are in the same plane, but such that the axis does not intersect the circle. An consider a Mobius strip with a single 180 degree twist, formed as follows: take a line segment and let its midpoint trace out a circle at a constant rate, and at the same time, let the line segment rotate 180 degrees, also at a constant rate. Now let three such Mobius strips intersect inside a torus. What is the maximum number of pieces thus formed?

This is the best I could do. I can foresee a potential problem with the rotation of the line segment, but I don't know how to avoid it without adding several more sentences. The bottom line should be that intelligent people share the same interpretation, and I think they would under my definition, while they do not under Ron Hoeflin's definition.

Of course the problem with this wording is that it is very long and will produce a lot of blank stares. Not ideal for an IQ test. So can any members come up with a shorter and more comprehensible wording for this problem? I'm sure Ron Hoeflin would be interested in seeing it.

Note to members: If you have comments on any of the above, please send them to Rick Rosner for publication in Noesis. If you have or are interested in diagrams, you can write me at the address below. I won't draw any more diagrams, but I do have a huge stack of diagrams [two inches thick--Ed.] drawn by myself and others, and I might be willing to copy some.

Take it easy Rick.

Sincerely,
Steve Sweeney