158 N. Rivier and J.-F. Sadoc
Let us now obtain the metric of the two structures. The inflation multiplier
1+
√
3 must be present both in the longitudinal and in the transverse struc-
tures, which is expected to exhibit several rational approximants of
√
- This
Triangular Lattices 8.5 Transverse Structures (gap, overlap) on Two Orthogonal
lattices. The superposition pattern exhibits “wheels” at rational approximants
of 1 +
√
3, located on (two) square–triangle pattern(s). The wheels are centred
on two copies of the gap structure, which is the main coincidence pattern of
the two perpendicular, triangular lattices.
8.6 The Gap Structure..........................................
The gap structure is shown in Fig. 8.6a. All edges are equal. But the “square”
has a slight rhombic distortion, and the unit cell (4 vertices, 4 triangles, 2
squares, with an area of 112 elementary triangles) is slightly rectangular. The
distortion is equal to 4
√
3 /7=1.010. (see Table 8.1). The gap structure
is metrically regular, but it has no mirror symmetry, and no alignments of
points.
8.7 The Overlap Structure.......................................
The overlap structure is shown in Fig. 8.6b. It is built on rectangular “squares”
with orthogonal sides on the 12 symmetry directions of the underlying tri-
angular lattice, and with 5/4 the vertex density of the gap structure. The
arithmetic construction of the gap and overlap structures is detailed in the
next section.
Table 8.1.The table below lists the geometric manifestations of the various ap-
proximantsCj/Ajto
√
3 in the overlap and gap structures. A rectangular structure
is denoted as [...]; it has one approximant. Without bracket, it refers to an equila-
teral rhombus, with two approximants (a) and (b). The unit cell of the small overlap
structure is the square of the gap structure. The first column gives the value ofj.The
last column gives the (orthorhombic or triclinic) distortion, sup(
√
3/app., app./√
3)
For reference:
j=4
√
3 ≈ 19 /11 in B.C. helix.
j=5
√
3 ≈ 41 /15 in inflated B.C. helix.j
√
3 overlap small lap over-gap distortion
12 / 1 [square] [square] unit cell(a)
unit cell(a) unit cell(a) square(a) 1. 155
25 / 3 [mirror unit cell] [mirror unit cell] 1. 04
37 / 4 unit cell(b) unit cell(b) square(b) 1. 01
[unit cell]