Ab-initio construction of some crystalline 3D Euclidean networks

We describe a technique for construction of 3D Euclidean (E-3) networks with partially-prescribed rings. The algorithm starts with 2D hyperbolic (H-2) tilings, whose symmetries are commensurate with the intrinsic 2D symmetries of triply periodic minimal surfaces (or infinite periodic minimal surfaces, IPMS). The 2D hyperbolic pattern is then projected from H-2 to E-3, forming 3D nets. Examples of cubic and tetragonal 3-connected nets with up to 288 vertices per unit cell, each linking a pair of 6-rings and a single 8-ring, are derived by projection onto the P, D, Gyroid and I-WP IPMS. A single example of a projection from close-packed trees in H-2 to E-3 (via the D surface) is also shown, that leads to a quartet of interwoven equivalent chiral nets. The configuration describes the channel system of a novel quadracontinuous branched minimal surface that is a chiral foam with four identical, open bubbles. (C) 2003 Editions scientifiques et medicales Elsevier SAS. All rights reserved.