Tetracontinuous net P4232

The most well known spatially ordered networks are mesh networks in 2D, like a New York streetscape from above, or the hexagonal borders in a bees nest.

Others networks occur in 3D, such as the paths through subway stations or the labyrinthine interconnections of tunnels in a cave systems.

Such networks occur on many scales. In this movie a tetracontinuous set of networks is shown in a unique atomically resolved hypothetical liquid crystalline MOF (metal organic framework) stucture. These are attracting attention for their relatively high hydrogen storage potential - a potential enabling technology in a solar hydrogen economy.

Multicontinuous networks are independently branching networks that occupy the same space as other networks, though never crossing paths. In 3D, these multicontinuous networks often fall onto sets of opposite-handed chiral Y* graphs in some liquid crystals such as metal carboxylates (see Luzzati's work in the early to late 60's and my recent paper on Liquid crystal MOFs). These bicontinuous networks have metal ions located on two sets of branching rods. In hyothetical structures they can fall on more than two sets of networks. For example we can predict 4 sets of mutually interpenetrating Y* graphs, as in tetracontinuous networks that are shown in the movie here.

Such multicontinuous structures are not only fundamental to the study of potentially interesting new liquid crystals, or porous gas storage media but also relevant to city networks, communication networks and quite likely at very large scales in the universe.

Fortunately for those interested in the deep geometry and topology of these interpenetrating networks, there is a tool available for mapping these networks - hyperbolic plane or H2.

H2 (like Escher's circular works of shrinking geometric tile patterns) is a very useful mapping tool to find your way around. The usefulness of graphs plotted on the hyperbolic plane is already useful today in forward and backward citations mapping. In H2, the interpenetrating nets are knowns as forests of trees or thickets. The symmetry with which these tile space is given by hyperbolic group theory or hyperbolic crystallography (see the work of Hyde et al over the last decade).

Enjoy the video and thanks for your interest.

Category:

Science & Technology

Tags:

• hyperbolic
• networks
• bicontinuous
• tetracontinuous
• communication networks
• transport systems

[Cuagau1]

thanks jazzcazjazz

[jazzcazjazz]

nice job