**Figure 5**
Structural variations associated with D and G twins. (*a*) Possible stacking orders of catenoidal necks. Each catenoidal neck is spanned by two equilateral triangles having a common threefold axis, where the triangles are parallel or anti-parallel if the arrangements of necks in the two adjacent layers are the same (as in the H surfaces) or different (as in the D surface), respectively. For D twins, necks with parallel triangles exist only at the twin boundary. (*b*) The D surface (1st row) and an H surface (2nd row) shown along with their smallest unit cells and labyrinth networks. Reference unit-cell constants are taken from the cubic D surface, such that and *c*/*a* = (2/3)^{1/2} where *a*_{D} is the cubic lattice constant. (*c*) The G surface (left) and the first (middle) and second (right) simplest polysynthetic G twins. The reference unit-cell constants are taken from the cubic G surface, such that , *a*/*c* = (1/3)^{1/2} and *b*/*c* = (3/8)^{1/2} where *a*_{G} is the cubic lattice constant. In (*b*) and (*c*) all the surfaces are oriented with sides being coloured blue and brown. The dashed green lines indicate the candidate positions of twin boundaries (*i.e.* {111} + 0.5 and {211} + 0.5, respectively), while the green plates indicate actual reflection planes. The periods along orthogonal axes are shown with double-sided arrows and their lengths. |