Re: [math-fun] mechanical realization of a 3-torus
Seb, That's a very nice Java application! Just in case you're feeling ambitious, would you be willing to also build a Java program that displays three rotatable rods, U, V, W, of equal length L in the plane, each fixed at one end -- at points P, Q, R, respectively, that form the vertices of an equilateral triangle of side S. The lines connecting the variable ends of U, V, W -- i.e., P + U(x), Q + V(y), R + W(z) -- could be thick and colored red, blue, and green. (Here 0 <= x,y,z < 2pi are the angles of U, V, W.) This would display each point of the configuration space as a distinct colored triangle -- among triangles whose vertices lie on the three circles of radius L about P, Q, R. Since triangles are familiar objects, this might be a palatable way to digest the idea of a configuration space. Finally, it would be interesting to experiment with several values for the ratio S/L -- I'd say especially S/L = 0, 1, 2 -- to see which mechanism looks more interesting or informative. (Of course S/L = 0 means P = Q = R.) If you're feeling *really* ambitious, there could also be an automatic mode in which the user can push U, V, W off to keep moving by itself along some geodesic. Since almost all ideal such geodesics are dense, this will give a "grand tour" of the configuration space T^3. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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Dan Asimov