8 The Structure of Collagen 153Fig. 8.4.The base space of the Hopf fibration of polytope{ 3 , 3 , 5 }is an icosahedron
(a). Each vertex of the icosahedron is only the representative of one fibre. Physically,
the fibre is either a collagen molecule (a triple helix), or (b) the representative of the
amino acidsGly(orX,orY, resp.) of a single PPII chain. Then,a shaded triangular
faceof the icosahedron represents a PPII chain, and the triple helix collagen molecule
is represented by the decorated hexagon in (b). Hydrogen bonds (double lines)are
horizontal bridges between theGlyof one helix and theXof another. Notice the
right-handed chirality of theGlycore and of the triple collagen molecule, opposed
to the left-handed chirality of the PPII helices
in two stages, icosahedron 3^5 to snub cube 3^4 .4, to sigma phase honeycomb
32. 4. 3 .4.
Alternatively, one can represent an entire molecule as a vertex in the base
space. This alternative representation is used hereafter (Fig. 8.5–8.9).
The first step decurves the base space of{ 3 , 3 , 5 }, an icosahedron 3^5 into a
snub cube 3^4. 4 .The snub cube can be decurved one step further, by replacing
a second triangle by a square. One obtains the square–triangle, Archimedean,
z= 5 honeycomb 3^2. 4. 3 .4 (major skeleton of the sigma phase of intermetallic
compounds, a Frank–Kasper phase).^6 Decurving is complete, because the base
space is now flat and infinite, and the fibres are periodic B–C helices 42/15.
(^6) The alternative honeycomb 3 (^3). 42 , in which the two squares are neighbours, forfeits
the isotropy of the original polytope{ 3 , 3 , 5 }.