9 - Rigidity of polyhedral frames

Summary

The engineering design of space-frames is often facilitated by the use of an idealisation in which the actual structure is replaced conceptually by an assembly of rods and frictionless ball joints, or (as Maxwell put it) a collection of lines and points. If the idealised assembly is rigid when all of the bars or lines are inextensional – as distinct from being a mechanism – then the actual physical structure under consideration can be expected to carry loads applied at its joints primarily by means of tension and compression in its members. The next stage of the engineering calculation for such a structure is to perform a statical analysis of the tensions in the members, to invoke Hooke's law and then to compute the displacements of the assembly. But for the purposes of this appendix, we are concerned only with the question of the rigidity (or otherwise) of idealised frameworks made up from inextensional bars or lines.

This problem is one which attracts the attention of pure mathematicians. (Consideration of elasticity etc. would make the problem ‘applied’.) These workers are inclined to think of the assembly of lines and points as their real structure, and any physical representation of the system by means of (e.g.) rubber connectors and wooden bars, or even structural steelwork, as conceptual idealisations of the reality under consideration. Here, then, we have a complete inversion of the engineer's view that the geometrical array of lines and points is a conceptual idealisation of the physical reality under consideration; the mathematician's is the platonic as opposed to the aristotelian view of nature.