150 N. Rivier and J.-F. Sadoca cylinder. The cylinder is cut and flattened. This elegant geometric construc-
tion cannot be considered as a material unit for two reasons:(a) The regular tetrahedron is not a three-dimensional space filler. Indeed, its
dihedral angle is 2π/(5.1), and that is geometrically frustrated because the
.1 leaves empty space, and the fivefold symmetry is not crystallographic.
Indeed, the perfect, close packing of spheres exists in the positively
curved spaceS^3 ; it is polytope{3, 3, 5}, containing 120 spheres and 600
tetrahedra.
(b) As a one-dimensional structure, it is not periodic, not even quasi-periodic,
that is extensible from a small finite nucleus by substitution rules or
by cut-and-projection through a sequence of approximants. The num-
ber of edges or of spheres per turn of the helix^1 is 2π/cos−^1 (− 2 /3) =
[2, 1 , 2 , 1 , 2 , 1 , 1 , 2 , 1 , 1 , 7 , 6 ,...]≈ 2 .7312, that is neither a rational number,
describing a periodic structure, nor a quadratic irrational, necessary con-
dition for context-free inflation–deflation symmetry.^2
It is possible to resolve the difficulty (b). One can construct a quasicrys-
talline Coxeter helix with a number of spheres per turn equal to 1 +√
3=
[2,1] =2.73205, with exactly the same rational convergents through the
first 112 amino acids and 41 turns. Notably, the periodic helices 30/11 =
[2, 1 , 2 , 1 ,2] = 2. 7272 ...of Fig. 8.2c), and 41/15 = [2, 1 , 2 , 1 , 2 ,1] = 2. 733 ...
are rational convergents of the B–C and quasicrystalline Coxeter helices.
Neither have axis perpendicular to the base of the cylinder; as befits
successive rational convergents, it lies on either side of the axis of the qua-
sicrystalline Coxeter helix, which has an irrational slope in the underlying
triangular lattice. There exists a third helix, with its axis exactly perpendic-
ular to the base of the cylinder. It has 14 amino acids for 5 turns exactly, i.e.
42 /15 = 14/5=[2, 1 ,4] = 2.8 that has the same convergents (principal and
intermediate) through the first 11 amino acids and 4 turns.^3 This is the basic
helix of collagen.(^1) Three types of helical chains of spheres in contact, with different pitch and chi-
rality, can be distinguished in the B–C helix. We refer here to the flattest, right-
handed helix, called Coxeter chain. Each one of the three polypeptide chains
constituting the collagen molecule, is a left-handed helix of intermediate pitch
(Fig. 8.3), also constructible on the B–C helix. TheGlysit on the steepest, right-
handed helix, that is also a fibre in the Hopf fibration of polytope{ 3 , 3 , 5 }.
(^2) A number is represented by its continuous fraction, as q 0 + 1
q 1 +q 2 +^1 ... =
[q 0 ,q 1 ,q 2 ,...]. A quadratic irrational has a periodic continuous fraction expan-
sion, with the period underlined, e.g.
√
3=1+1+^11
2+1+^1 ...
=[1, 1 , 2 ]. Rational
approximants are given by truncation of the sequence [5].
(^3) In the collagen chain (Fig. 8.3), the axis goes throughGly(0),X(7),Y(14) and
Gly(21). The (left-handed) helix with an axis perpendicular to the base is 21/6,
or 7/2 for identical vertices. It has been identified by Okuyama et al. [1].