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

Hyperbolic Periodic Nodal Surfaces (PNSs) have been investigated with respe ct to surface modulations. The resulting patterns and tilings are beautiful examples of texture information on hyperbolic objects. The construction pr inciple, based on short Fourier series, allows for a strict symmetry contro l of the generated pattern by choosing appropriate sets of structure factor s for information coding. Furthermore, a tailor-made design of complexity i s achieved based on a proper choice of the structure factor moduli. In some cases, the resulting patterns are instantaneously transferred into hyperbo lic framework information just by our visual perception. In most cases, suc h correlations can be systematically worked out by utilizing crystallograph ic and chemical expertise, as shown in this contribution. The presented app roach is much simpler than most attempts to generate three-dimensional fram eworks because it is subject to boundary conditions like pseudo-two-dimensi onal topology, given base topology, and symmetry control.