Moiré, Euler and self-similarity – the lattice parameters of twisted hexagonal crystals

A real-space approach for the calculation of moiré lattice parameters superstructures formed by a set rotated hexagonal 2D crystals such as graphene or transition-metal dichalcogenides is presented. Apparent lattices continuously form all rotation angles, and their parameter to good approximation follows hyperbolical angle dependence. Moiré crystals, i.e. decorated with basis, require more crucial assessment commensurabilities lead discrete solutions non-continuous dependence moiré-crystal parameter. In particular, this critically depends on angle, continuous variation can apparently erratic changes The highly complex pattern, which reflects number-theoretical relations between formation crystal. analysis also provides insight into special case 30° constituting lattices, dodecagonal quasicrystalline structure forms.