8 The Structure of Collagen 151The geometrical frustration (a) is resolved in (positively) curved space [6].
Hopf’s fibration ofS^3 extends to its discrete scaffolding, polytope{ 3 , 3 , 5 }[7].
There are 12 fibres of ten vertices each. They can be divided into four co-axial
tori of three intertwined fibres. All fibres are identical, by definition, and also
equidistant (parallel in curved space). One torus flattened in Euclidean space
is represented in Fig. 8.2c. It is covered by helix 30/11. One notices the three
fibres, the steepest helices winding 1:1 around the torus. Note that edges of
the triangular network are only equal in curved space. In Euclidean space,
there is a slight distortion of the distance between different fibres. The base
space is the icosahedron 3^5 (Fig. 8.4). Each vertex is the representative (a
projective map) of one fibre.^4
Incidentally, the torus covered by helix 42/15 can also be represented by
a cylinder, with its axis perpendicular to the base. The three fibres winding
1:1 around the torus are longer: They have 14 vertices each. These fibres are
identical to the four fibres in the torus covered by a disclinated Coxeter helix
1+41/15, based on a square rather than on a triangle [4]. This enables us
to extend the Hopf fibration of polytope{ 3 , 3 , 5 }to a decurved (flattened)
polytope of 24×14 = 336 vertices, with fibres of 14 vertices instead of ten.
And then, without any further lengthening of the fibres, to the Euclidean
honeycomb with the sigma phase honeycomb 3^2. 4. 3 .4 as base space.
8.3 The Collagen Molecule
The collagen molecule, with its three, intertwined polypeptide chains, can be
represented in curved space on the Hopf fibration of polytope{ 3 , 3 , 5 }.The
base space is an icosahedron.
A single collagen chain (polyproline PPII helix structure) forms a left-
handed helix [Gly−X−Y] 5 , (15/4), drawn in Fig. 8.3 on a torus of the
fibration, i.e. on the right-handed B–C helix 30/11, with twice the vertical
spacing between vertices. Half of the vertices on the torus, constituting a
second left-handed helix parallel to the first, are empty. The fibres (of the
original Hopf fibration, with all vertices identified) are the three, steepest
right-handed helices. One contains fiveGlyseparated by five empty vertices.
The other two, five amino acidsXorY, respectively, separated by five empty
vertices. The torus is mapped on the base space as a triangle (shaded in
Fig. 8.4b).
The collagen molecule is represented as a hexagon on base space, with a
triangular core ofGlyinside (Fig. 8.4b). Within the triangular core, the two
hydrogen side groups ofGlyform a close-packed Coxeter helix 30/11 (with
the 30Hattached to the 3× 5 Glyfrom the three chains). The projection of
the horizontal, interchain, hydrogen bonds are represented as double lines in
Fig. 8.4b.
(^4) We have to distinguish between base space of a non-trivial fibration and base of
a cylinder (a circle).