were not ‘half smaller', in a sense easy to grasp, than the original one (5). For if one knows the components of Eu or Ou on one of the sides of the structure, the other ones are given by the relations S Eu=Eu or S Ou^ — Ou. The size of both linear systems (9) can thus be reduced by a factor of about one half. This results in a net gain, because solving a linear system is a task whose cost increases more rapidly than the number of equations. In geometric terms, the space V of all displacements has been expressed as the ‘direct sum' of two complementary subspaces (let us call them EV and OV), corresponding to even and odd displacements respectively. The main point (expressed by (8)) is that both are globally invariant by the action of A: reaction forces due to an even (odd) displacement also are even (odd). So what we have obtained is a ‘blockdiagonalization' of A (Fig. 4). 2.2 Symmetry in general The theory (whose detailed exposition is beyond the scope of this paper, cf. [3] consists in a generalisation of what has just been described, as follows. The ‘symmetry' of the structure is described by giving a list of all geometrical operations (translations, rotations, plane symmetries) which leave it unchanged as a whole, in all respects (geometrical and mechanical). (Here, we had two such operations, identity and plane
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