Hexagulation numbers: symmetry breaking on the gyroid surface

Abstract The regular arrangements of identical spheres on the gyroid surface have been investigated using Monte Carlo simulations of hard spheres undergoing the Alder transition. In our preceding study, we showed that these regular structures on saddle-shaped triply periodic minimal surfaces are systematically classified by the hexagulation number H, analogous to the Caspar–Klug triangulation number for icosahedral viruses. In this paper, we focus on the non-centrosymmetric regular structures that emerge when H is a multiple of three on the gyroid surface. In these cases, the space group Ia3¯d of the gyroid is reduced to its non-centrosymmetric subgroup I4¯3d. By considering both inversion structures, we construct an efficient bilayer configuration that forms a truss-like structure. These physically efficient designs may provide inspiration for future applications in soft-matter science, DNA-origami nanotechnology, metamaterials, and even architectural design. This article is part of the theme issue ‘Geometry, materials and the imagination’.