On tilings and patterns on hyperbolic surfaces and their relation to structural chemistry.

Hyperbolic Periodic Nodal Surfaces (PNSs) have been investigated with respect to surface modulations. The resulting patterns and tilings are beautiful examples of texture information on hyperbolic objects. The construction principle, based on short Fourier series, allows for a strict symmetry control of the generated pattern by choosing appropriate sets of structure factors for information coding. Furthermore, a tailor-made design of complexity is achieved based on a proper choice of the structure factor moduli. In some cases, the resulting patterns are instantaneously transferred into hyperbolic framework information just by our visual perception. In most cases, such correlations can be systematically worked out by utilizing crystallographic and chemical expertise, as shown in this contribution. The presented approach is much simpler than most attempts to generate three-dimensional frameworks because it is subject to boundary conditions like pseudo-two-dimensional topology, given base topology, and symmetry control.