8 The Structure of Collagen 155Fig. 8.6.(a) The unit cell of the square–triangle 3^2. 4. 3 .4 gap structure, drawn
on a triangular lattice. All edges have equal length, but the “square” is in fact a
rhombus. A vertex is the representative of a collagen molecule. (b) The unit cell
of the square–triangle 3^2. 4. 3 .4 overlap structure, drawn on a triangular lattice. The
“square” is slightly rectangular, and there are two types of equilateral triangles, of
sizes in the ratio 2
√
3 /3=2/√- (c) A smaller 3^2. 4. 3 .4 overlap structure, with the
square of the gap structure as unit cell, drawn on a triangular lattice, rotated by
π/2. It is similar to the original overlap structure (b). The factor of similarity is
1 /
√- Also drawn (grey) is the associated Voronoi tiling
Moreover, the axis of the helix is perpendicular to its basis, and the projection
is orthogonal. The structure remains a fibration at all stages.
The identical fibres are helices winding 1:1 around the tori represented
by the triangles and squares of the base space. They have ten vertices in the
original{ 3 , 3 , 5 }polytope (with the tori covered by the Coxeter helix 30/11,
Fig. 8.2c). At the next two stages, the fibres are longer (14 vertices) but they
still wind 1:1 around the longer tori: The torus represented as a triangle in
base space is covered by the Coxeter helix 42/15. Torus and helix are periodic
in Euclidean space (the axis of the strip making up the torus with opposite
sides identified, is perpendicular to the base).