Stuart Ramsden

Three-dimensional Euclidean nets from two-dimensional hyperbolic tilings: kaleidoscopic examples.

We present a method for geometric construction of periodic three-dimensional Euclidean nets by projecting two-dimensional hyperbolic tilings onto a family of triply periodic minimal surfaces (TPMSs). Our techniques extend the combinatorial tiling theory of Dress, Huson & Delgado-Friedrichs to enumerate simple reticulations of these TPMSs. We include a taxonomy of all networks arising from kaleidoscopic hyperbolic tilings with up to two distinct tile types (and their duals, with two distinct vertices), mapped to three related TPMSs, namely Schwarzs primitive (P) and diamond (D) surfaces, and Schoens gyroid (G).

Polycontinuous geometries for inverse lipid phases with more than two aqueous network domains.

Inverse bicontinuous cubic phases with two aqueous network domains separated by a smooth bilayer are firmly established as equilibrium phases in lipid/water systems. The purpose of this article is to highlight the generalisations of these bicontinuous geometries to polycontinuous geometries, which could be realised as lipid mesophases with three or more network-like aqueous domains separated by a branched bilayer. An analysis of structural homogeneity in terms of bilayer width variations reveals that ordered polycontinuous geometries are likely candidates for lipid mesophase structures, with similar chain packing characteristics to the inverse micellar phases (that once were believed not to exist due to high packing frustration). The average molecular shape required by global geometry to form these multi-network phases is quantified by the surfactant shape parameter, v/(al); we find that it adopts values close to those of the known lipid phases. We specifically analyse the 3etc(187 193) structure of hexagonal symmetry P6(3) /mcm with three aqueous domains, the 3dia(24 220) structure of cubic symmetry I43d composed of three distorted diamond networks, the cubic chiral 4srs(24 208) with cubic symmetry P4232 and the achiral 4srs(5 133) structure of symmetry P42/nbc, each consisting of four intergrown undistorted copies of the srs net (the same net as in the QII(G) gyroid phase). Structural homogeneity is analysed by a medial surface approach assuming that the headgroup interfaces are constant mean curvature surfaces. To facilitate future experimental identification, we provide simulated SAXS scattering patterns that, for the 4srs(24 208) and 3dia(24 220) structures, bear remarkable similarity to those of bicontinuous QII(G)-gyroid and QII(D)-diamond phases, with comparable lattice parameters and only a single peak that cannot be indexed to the well-established structures. While polycontinuous lipid phases have, to date, not been reported, the likelihood of their formation is further indicated by the reported observation of a solid tricontinuous mesoporous silicate structure, termed IBN-9, which formed in the presence of surfactants [Han et al., Nat. Chem., 2009, 1, 123].

Meditation on an Engraving of Fricke and Klein (The Modular Group and Geometrical Chemistry)

A non-technical account of the links between two-dimensional (2D) hyperbolic and three-dimensional (3D) euclidean symmetric patterns is presented, with a number of examples from both spaces. A simple working hypothesis is used throughout the survey: simple, highly symmetric patterns traced in hyperbolic space lead to chemically relevant structures in euclidean space. The prime examples in the former space are derived from Felix Kleins engraving of the modular group structure within the hyperbolic plane; these include various tilings, networks and trees. Disc packings are also derived. The euclidean examples are relevant to condensed atomic and molecular materials in solid-state chemistry and soft-matter structural science. They include extended nets of relevance to covalent frameworks, simple (lattice) sphere packings, and interpenetrating extended frameworks (related to novel coordination polymers). Limited discussion of the projection process from 2D hyperbolic to 3D euclidean space via mapping onto triply periodic minimal surfaces is presented.

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.  2003 Editions scientifiques et medicales Elsevier SAS. All rights reserved.

EPINET: euclidean patterns in non-euclidean tilings

We present a method for generating 3D euclidean periodic networks from 2D hyperbolic tilings. We utilize triply-periodic minimal surfaces (TPMS) as a mathematical scaffold to guide this process. These surfaces have an intrinsic hyperbolic geometry as well as an underlying set of discrete hyperbolic symmetries, allowing decoration with tilings of matching symmetry. Hyperbolic tilings of a given symmetry can be enumerated (up to finite complexity), projected onto the surface, and used to derive a labeled quotient graph of nodes and edges, representing a 3D periodic net. Although we have visualized intermediate geometries for this process, it is performed entirely as a topological computation using only neighborhood relationships and labels. The final net geometry is derived from the labeled graph, using Systre [Delgado-Friedrichs 2006], enabling uniqueness and 3D space-group identification. We have published the results of these computations at the EPINET website [Ramsden et al. 2005] for an initial sample of tilings, resulting in an online database of many thousands of 3D nets, each with their own webpage, which can be data mined via geometric and topological properties. Tiling combinatorics can provide a rich palette of both geometric and image data, only a fraction of which has been currently explored.

Trading spaces: building three-dimensional nets from two-dimensional tilings

We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic (S2), Euclidean (E2) and hyperbolic (H2) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those spaces into three-dimensional Euclidean space lead to topologically and geometrically complex patterns, including multiple interwoven nets and tangled nets that are otherwise difficult to generate ab initio in three dimensions.