162 N. Rivier and J.-F. Sadoc
the overlap structure. The gap structure, rotated around one of its vertices
by 27◦, has vertices in four out of five Voronoi cells of the original overlap
structure (as indicated schematically in Fig. 8.5b). The next overlap, rotated
by a further 27◦, has its vertices inside all five Voronoi cells of the original
structure. The vertex superposition is only topological; the molecule at the
origin of the structures is only approximatively straight.
The operation is repeated, starting now with the rotated overlap structure.
The primitive cell is translated to have its basis in the same orientation. But
the position of the smaller and larger rhombi is now exchanged, as was sug-
gested in the trefoil model of Bouligand. This is another effect of the rhombic
distortion of the perfect square–triangle structure. Accordingly, the rotation
angle of the TGB is now twice tan−^1 (
√
3 /7)+tan−^1 (√
3 /5) = tan−^1 (3√
3 /8)≈
33 ◦=π/ 3 − 3 π/20, the complementary – to 2π/3–of3π/10. Hence, the appar-
ent tenfold rotation, superposed on the sixfold triangular symmetry, noticeable
in the diffraction patterns. The apparent contradiction between microfibrils
with pentagonal cross-section [11] and quasi-hexagonal alignment of triple
helices [12] is hereby resolved: the collagen fibril exhibits both rotations sym-
metries (Figs. 8.6, 8.8 and 8.9), enforced by the orthorhombic distortion of the
square–triangle pattern on a triangular lattice.
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