From 2D hyperbolic forests to 3D Euclidean entangled thickets

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 innite families of trees (forests) in the hyperbolic plane, and subsequent folding into triply periodic minimal surfaces, including the P;D ,g yroid andH 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 nite 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.