From 2D hyperbolic forests to 3D Euclidean entangled thickets
Applied Mathematics, Research School of Physical Sciences,
Australian National University, Canberra, A.C.T. 0200, Australia
2 LPTMCNRS ESA 8089, Université de Cergy Pontoise, 5 Mail G. Lussac, 95031 Cergy-Pontoise, France
Published online: 15 August 2000
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
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2000