From 2D hyperbolic forests to 3D Euclidean entangled thickets |
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Authors: | ST Hyde C Oguey |
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Institution: | (1) Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, A.C.T. 0200, Australia, AU;(2) LPTM (CNRS ESA 8089), Université de Cergy Pontoise, 5 Mail G. Lussac, 95031 Cergy-Pontoise, France, FR |
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Abstract: | A method is developed to construct and analyse a wide class of graphs embedded in Euclidean 3D space, including multiply-connected
and entangled examples. The graphs are derived via embeddings of infinite families of trees (forests) in the hyperbolic plane, and subsequent folding into triply periodic minimal
surfaces, including the P, D, gyroid and H surfaces. Some of these graphs are natural generalisations of bicontinuous topologies to bi-, tri-, quadra- and octa-continuous
forms. Interwoven layer graphs and periodic sets of finite clusters also emerge from the algorithm. Many of the graphs are
chiral. The generated graphs are compared with some organo-metallic molecular crystals with multiple frameworks and molecular
mesophases found in copolymer melts.
Received 10 December 1999 |
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Keywords: | PACS 61 50 Ah Theory of crystal structure crystal symmetry calculations and modeling - 61 25 Hq Macromolecular and polymer solutions polymer melts swelling - 61 30 Cz Theory and models of liquid crystal structure |
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