8 The Structure of Collagen 161q=Aj,m=Cj,n=Aj+2,p=Cj+2. (8.4)
The smallest solution (j=1)isq=1,m=2,n=4,p=7.Itcorresponds
to the square of the gap structure. The intermediate convergentq=2,m=3,
n=7,p= 12, which corresponds to the unit cell of overlap structure, yields
the same approximants for
√
3.
Withq=Aj,m=Cj, the two vectors are nearly orthogonal. The scalar
product− 2 b 1 ·b 2 =(m^2 − 3 q^2 )=1forjodd, =−2forjeven, regardless of
the length of the vectors. Indeed,m^2 − 3 q^2 = 1 is known as Pell’s equation [5].
Pell’s equation has infinitely many solutionsCj/Aj(jodd). For intermediate
convergents (defined, forjeven, asA(1)j =Aj− 1 +Aj− 2 ,Aj=2Aj− 1 +Aj− 2 =
A
(1)
j +Aj−^1 ), Pell’s equation ism
(^2) − 3 q (^2) =−3, i.e.q (^2) −3(m/3) (^2) =1,and
q=A(1)j =Cj− 1 ,m=Cj(1)=3Aj− 1 , yielding the same approximant for
√
3 ≈Cj− 1 /Aj− 1 =3Aj− 1 /Cj− 1 ,thus
√
3 ≈Cj− 1 /Aj− 1 as that forj− 1
odd.
Okuyama et al. [1] mention an average distortion of 1.019 in their crystal
of [Gly−Pro−Pro] 10 , which is consistent with these figures.
It is possible to represent the cross-section of the collagen molecule (triple
helix) as a trefoil of three, close-packed hexagons drawn on the underlying
triangular lattice [3, 10]. Contact between trefoils is through an edge of the
triangular lattice in the overlap, and through a vertex in the gap structure.
The trefoil rotates as it goes along the molecule. In the overlap, the smaller
rhombi have four trefoils filling space without any vacant space. The larger
rhombi have a hexagonal hole between the four trefoils, as do the “squares”.
The distances between trefoil centres are 3 and 2
√
3 edges of the underlying
triangular lattice. When the trefoils rotate, they push each other apart to
reach a single maximum distance in the gap. With one out of every five trefoils
missing, the area occupied remains constant. Further rotation of the trefoils
leads to an overlap structure, rotated from the first, with the smaller rhombi
replaced by larger ones, and vice versa.
8.9 Twist Grain Boundary Overlap–(Gap)–Overlap
Figures 8.8 and 8.9 show how the periodic stack overlap–gap–overlap–...can
be constructed, and that the boundaries between gap and overlap structures
are twist grain boundaries, associated with a coincidence site lattice.
The precise value of the rotation angleθ′ at the twist grain bound-
ary between two overlap structures (separated by a gap) is twiceθ′ =
tan−^1 (
√
3 /6) + tan−^1 (√
3 /9) orθ′= tan−^1 (5√
3 /17) whose numerical value is
26. 996 ◦ 27 ◦. Namely the rotation is 3π/10. The tenfold symmetry is visible
in the diffraction pattern of a single overlap structure. It corresponds to a true
coincidence site lattice.
In practice, we place the vertex at the origin of the rotated structure
anywhere within the Voronoi cell of one of the five vertices of the unit cell of