During the past decade, interest has grown tremendously in the design and synthesis of crystalline materials constructed from molecular clusters linked by extended groups of atoms. Most notable are metal-organic frameworks (MOFs), in which polyatomic inorganic metal-containing clusters are joined by polytopic linkers. (Although these materials are sometimes referred to as coordination polymers, we prefer to differentiate them, because MOFs are based on strong linkages that yield robust frameworks.) The realization that MOFs could be designed and synthesized in a rational way from molecular building blocks led to the emergence of a discipline that we call reticular chemistry. MOFs can be represented as a special kind of graph called a periodic net. Such descriptions date back to the earliest crystallographic studies but have become much more common recently because thousands of new structures and hundreds of underlying nets have been reported. In the simplest cases (e.g., the structure of diamond), the atoms in the crystal become the vertices of the net, and bonds are the links (edges) that connect them. In the case of MOFs, polyatomic groups act as the vertices and edges of the net. Because of the explosive growth in this area, a need has arisen for a universal system of nomenclature, classification, identification, and retrieval of these topological structures. We have developed a system of symbols for the identification of three periodic nets of interest, and this system is now in wide use. In this Account, we explain the underlying methodology of assigning symbols and describe the Reticular Chemistry Structure Resource (RCSR), in which about 1600 such nets are collected and illustrated in a database that can be searched by symbol, name, keywords, and attributes. The resource also contains searchable data for polyhedra and layers. The database entries come from systematic enumerations or from known chemical compounds or both. In the latter case, references to occurrences are provided. We describe some crystallographic, topological, and other attributes of nets and explain how they are reported in the database. We also describe how the database can be used as a tool for the design and structural analysis of new materials. Associated with each net is a natural tiling, which is a natural partition of space into space-filling tiles. The database allows export of data that can be used to analyze and illustrate such tilings.

We present a method for geometric construction of periodic three-dimensional Euclidean nets by projecting two-dimensional hyperbolic tilings onto a family of triply periodic minimal surfaces (TPMSs). Our techniques extend the combinatorial tiling theory of Dress, Huson & Delgado-Friedrichs to enumerate simple reticulations of these TPMSs. We include a taxonomy of all networks arising from kaleidoscopic hyperbolic tilings with up to two distinct tile types (and their duals, with two distinct vertices), mapped to three related TPMSs, namely Schwarz's primitive (P) and diamond (D) surfaces, and Schoen's gyroid (G).

Popular summary:Parametrically excited waves are a ubiquitous phenomenon observed in a variety of physical contexts. They span from Faraday waves on the water surface to spin waves in magnetics, electrostatic waves in plasma and second sound waves in liquid helium. Parametrically excited Faraday waves on the surface of vertically vibrated liquids quickly become nonlinear. In dissipative liquids, or in granular media, these nonlinear waves form regular lattices of oscillating solitons (oscillons), resembling in some aspects 2D crystals. If the vertical acceleration is increased, the oscillons do not solely grow in amplitude, their horizontal mobility is also greatly enhanced, and ultimately the lattice melts and becomes disordered. Until recently, the physics of these self-organized waves and their transition to disorder have been studied almost exclusively based on the analysis of the wave motion rather than the motion of their constitutive components, whether they are solid grains or fluid particles.It has recently been discovered that the fluid motion on a liquid surface perturbed by Faraday waves reproduces in detail the statistics of two-dimensional turbulence. This unexpected discovery shifts the current paradigm of order to disorder transition in this system: instead of considering complex wave fields, or wave turbulence, it is conceivable that the 2D Navier-Stokes turbulence, generated by Faraday waves, feedbacks on the wave crystal and disorders it in a statistically predictable fashion. To date, the very mechanism behind the turbulence generation in such waves remains unknown. A better understanding of this phenomenon is important for a wide spectrum of physics applications involving parametric waves.In this paper, we visualize 3D trajectories of floating tracers and reveal that the fluid particles motion injects 2D vortices into the horizontal flow. This is an unexpected and new paradigm for vorticity creation in a 2D flow. The horizontal energy is then spread over the broad range of scales by the turbulent inverse energy cascade. Two-dimensional turbulence destroys the geometrical order of the underlying lattice. The crystal order, however, can be restored by increasing * Nicolas.Francois@anu.edu.au viscous dissipation in the fluid which hinders vorticity creation and thus the development of turbulence. Abstract:We study the generation of 2D turbulence in Faraday waves by investigating the creation of spatially periodic vortices in this system. Measurements which couple a diffusing light imaging technique and particle tracking algorithms allow the simultaneous observation of the threedimensional fluid motion and of the temporal changes in the wave field topography.Quasi-standing waves are found to coexist with a spatially extended fluid transport. More specifically, the destruction of regular patterns of oscillons coincides with the emergence of a complex fluid motion whose statistics are similar to that of two-dimensional turbulence. We reveal that a lattice of oscillons generates vorticity at the osc...

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