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.

[1]  Reinhard Nesper,et al.  Nodal surfaces of Fourier series: Fundamental invariants of structured matter , 1991 .

[2]  Stephen T. Hyde,et al.  The Language of Shape: The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology , 1996 .

[3]  J. Cahn,et al.  Metallic Phase with Long-Range Orientational Order and No Translational Symmetry , 1984 .

[4]  R. Nesper,et al.  Die natürliche Anpassung von chemischen Strukturen an gekrümmte Flächen , 1987 .

[5]  Reinhard Nesper,et al.  How Nature Adapts Chemical Structures to Curved Surfaces , 1987 .

[6]  Adams,et al.  Predicted new low energy forms of carbon. , 1992, Physical review letters.

[7]  Alan L. Mackay,et al.  Crystallography and the penrose pattern , 1982 .

[8]  A. Mackay,et al.  From C60 to negatively curved graphite , 1997 .

[9]  Jacek Klinowski,et al.  Systematic enumeration of crystalline networks , 1999, Nature.

[10]  Johannes C. C. Nitsche,et al.  Stationary partitioning of convex bodies , 1985 .

[11]  S. Andersson,et al.  A systematic net description of saddle polyhedra and periodic minimal surfaces , 1984 .

[12]  Stephen T. Hyde,et al.  Continuous transformations of cubic minimal surfaces , 1999 .

[13]  R. Penrose.  Pentaplexity A Class of Non-Periodic Tilings of the Plane , 1979 .

[14]  T. Hahn.  International tables for crystallography , 2002 .

[15]  S. Andersson,et al.  Crystal structure of the synthetic zeolite N, NaAlSiO4· 1.35 H2O , 1982 .

[16]  G Gompper,et al.  Systematic approach to bicontinuous cubic phases in ternary amphiphilic systems. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  G. Langlet,et al.  International Tables for Crystallography , 2002 .

[18]  Nesper,et al.  Elucidation of simple pathways for reconstructive phase transitions using periodic equi-surface (PES) descriptors. The silica phase system. I. Quartz-tridymite , 2000, Acta crystallographica. Section A, Foundations of crystallography.