Jim, I made a simple mockup of the idea by printing linkages onto a laser-printable transparency, cutting them out, punching some holes, and pinning it together with small nuts and bolts on a scrap of cardboard. Here are images with two different choices for which pair of sums is inside the unit circle: http://www.georgehart.com/MoMath/3-torus-A.jpg http://www.georgehart.com/MoMath/3-torus-B.jpg The links with solid lines are your u,v,w directions and the ones with dashed lines are just to maintain parallelism. In this form, there are some interlocking issues that make it difficult to quickly move to any desired configuration. Links will hang up on bolts instead of passing by them. I'm not sure how to get around that. As to your puzzle, my feeling from playing with it is that there is just one connected component to the space when one of the sums (red dots) is fixed with magnitude almost 3. But this prototype is not at all rigid, so it is hard to be confident. As to developing this idea into a possible exhibit for the Museum of Mathematics, I'd be interested to continue discussing that. George http://momath.org/ http://georgehart.com/ On 9/10/2010 5:05 PM, James Propp wrote:

I have an idea for a physical mathematical demo, and I'm wondering if anyone has any thoughts about it (or any interest in actually building it).

Spitzer's "random flight theorem" says that a sum of n independent random unit vectors in the plane has a probability of exactly 1/(n+1) of having magnitude less than 1. In particular, the chance that the sum of 3 independent random unit vectors has magnitude less than 1 is exactly 1/4. It so happens that this particular fact can be deduced from the following geometrical fact, which may or may not be new: If u, v, and w are unit vectors, then exactly two of the eight vectors +u+v+w, +u+v-w, +u-v+w, +u-v-w, -u+v+w, -u+v-w, -u-v+w, -u-v-w have magnitude less than 1 (except in the case when some of these vectors have magnitude exactly 1).

There's a nice way to physically model this "two out of eight" fact with a linkage consisting of bars of length 2, so that eight points move jointly so as to trace out the configuration space {(+u+v+w, +u+v-w, +u-v+w, +u-v-w, -u+v+w, -u+v-w, -u-v+w, -u-v-w): |u|=|v|=|w|=1}. Now imagine that the bars are made of a transparent material like plexiglass, and that the model has a backing that shows a disk of radius 1 centered at 0. As you move around in configuration space, there are always exactly two points that lie inside the disk, except for the borderline case where four (or more) points lie on the boundary of the disk.

I'm wondering if this might make a good object for the Museum of Mathematics, or maybe just a fun object to own and to display in my office. I think I know what happens to the configuration space if you pin down one of the eight points, but it'd be cool to really feel what happens.

Note that one could describe this linkage simply as a mechanical realization of a 3-torus (hence the title of this post).

Jim

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