of 50 copies (some after symmetry) of the pattern which appears on Fig. 5. The load is a force field f orthogonal to the plane, and all sides are clamped. Displacements are measured along the direction orthogonal to the plane, and are thus zero on the boundary. (The case of freely moving edges, of course more realistic with space structures in mind, is similar, but slightly more complicated). Suppose we want to compute the displacements. The idea consists of ‘imbedding' this problem into an artificial, equivariant new one, to which the general method will apply. Let us consider three structures, obtained by mirroring the first one with respect to the axes indicated on Fig. 5. Let us stick them side by side (Fig. 6). Now how will this larger structure behave if: (1) the displacements on opposing sides should be equal (e.g., same displacements in A and A'); (2) the load is determined from the original load /by anti-symmetric prolongation: f(M') is set equal to —/(Af),/(Af") to etc.? Because of this skew-symmetry of the loading, the displacements of the large structure are skew-symmetric. Therefore, they are null on the symmetry axes and on the edges. So, the displacements of the left upper quarter are equal to those of the original structure with the load given in the first place. But now the new problem is equivariant, and the general theory works well. There is a group of invariance, with two kinds of operations: pseudo-translations, which consist in first performing a translation, then putting modules which fall from a side back on the opposite side, and pseudo mirror-symmetries, similarly defined. In other words, the big structure is periodic in both directions (as if it was wrapped around a torus). The new problem can thus be reduced to a set of ‘small' structural problems (50, in this case) on the symmetry-cell (the hashed region in Figs. 5 and 6).
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