I wrote:
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.
Let me be more concrete about this, and phrase this as a puzzle: What is the topology of the set of triples of unit complex numbers (u,v,w) satisfying u+v+w = some fixed complex number z? Clearly all that matters is the magnitude of z, so we might as well assume z is a real positive number; call it r, and let S_r = {(u,v,w): |u|=|v|=|w|=1, u+v+w=r}. If r > 3, S_r is empty; if r = 3, S_r contains just the one point (1,1,1). But what about r < 3? (Quick: Do you think S_r is connected when r = 2.9, or do you think it has two components?) You may wish to imagine a linkage consisting of three struts of unit length joining points A, B, C, and D; A and D are held fixed (r units apart) while B and C are free to move as long as they stay 1 unit apart from each other and from A and D respectively. (Think of A as 0, B as u, C as u+v, and D as u+v+w.) If you hold A and D fixed, what sort of freedom do you have in moving B and C? (Note that this is related to Dan Asimov's observation that if we just want a mechanical model of the 3-torus, we don't need something as complicated as the mechanism described in my original posting.) Jim