On 10/26/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
... But I'm tempted instead to investigate 3 spokes on each side of the hub, 2 spiralling one way and 1 in opposition.
A feeble waste of (line-)space. A more plausible construction has 3 spokes spiralling one way (tangentially to the hub flanges) on one side, with 3 spiralling the other way on the other side. This configuration is independent, and I simply cannot visualise how it might be deformed by compressing any combination of spokes: which of course proves nothing. Which in turn is scarcely surprising, because: Theorem: 6 spokes in tension is NOT sufficient to mount rigidly a hub in a rim. For regarded as a 6-ram parallel robot, in order to oppose every applied screw, the ram axes must be linearly independent: therefore every ram may act independently of the others. Then some screw S is opposed by some ram R alone shortening (spoke in tension); and S^(-1) is opposed by R alone lengthening (spoke in compression). QED However 7 rams in tension are (necessary and) sufficient for a robot in general: for example, the Stewart platform with all 6 legs equally extended is opposed by a single extra ram, mounted vertically upward from the platform centre. This presumably constituted the other half of Coxeter's argument, and serves to demolish Edmondson's. It seems improbable that constraining the rams (spokes) to pass through given curves (the circular rim and flanges) could increase this number; but I don't at this point actually have a copperfastened 7-spoke configuration to propose. Fred Lunnon