On 10/28/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
... 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 ...
Obviously the hub can screw in both directions along its axis: in fact, with the hub in position, it is impossible actually to tighten the strings (tension the spokes) at all!
... 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.
For 7-ram sufficiency, the platform configuration (hub location relative to rim) must be given: if it varied generally, 12 rams (in opposing pairs say) would be required for rigidity. This evidently explains Edmondson's confusion --- if it leads only to Fuller's geodesic domes being over-engineered, maybe that's no bad thing!
... but I don't at this point actually have a copperfastened 7-spoke configuration to propose.
My earlier proposal for 4 spokes in opposed pairs spiralling on one side, 3 spokes radial on the other side, still looks convincing to me. To prove it would involve partitioning line space into 7 subsets, corresponding to the 7 bases comprising 6 from 7 spokes (appropriately signed), such that always some component with respect to the corresponding basis is positive (denoting tension). Well, that's perfectly feasible, I suppose ... Fred Lunnon