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Issue		Eur. Phys. J. B Volume 16, Number 4, August II 2000
Page(s)		613 - 630
DOI		10.1007/PL00011063

Eur. Phys. J. B 16, 613-630

From 2D hyperbolic forests to 3D Euclidean entangled thickets

S.T. Hyde¹ - C. Oguey²

¹Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, A.C.T. 0200, Australia
²LPTM, Université de Cergy Pontoise, 5 Mail G. Lussac, 95031 Cergy-Pontoise, France
oguey@ptm.u-cergy.fr

Received 10 December 1999

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.

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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