NEW GEOMETRIES FOR
NEW MATERIALS
Contents
Introduction
References
Links
Introduction Exciting and farreaching developments are taking place in the materials sciences. The discovery by Shechtmann in 1982 of an underlying paradoxical fivefold symmetry in an AlMn alloy was followed by the discovery of a wide and growing variety of 'quasicrystalline' alloys, and by the devising of structural models to account for them. The discovery of Buckminsterfullerene C60 and related carbon molecules was followed by the synthesis of nanotubes and the investigation of their various configurations. One may also mention the geometrical ideas developed to understand defects and grain boundaries in crystalline and quasicrystalline materials, the network structures of zeolites and related materials, and liquid crystals. A consequence of this progress is that concepts that once belonged exclusively to the province of 'pure' mathematics  tiling patterns and coverings in Euclidean space, the geometry of higher dimensions, nonEuclidean geometries, topology  have entered the real world. Alan Mackay has for a long time argued for the need to extend the scope of classical mathematical crystallography. This is now taking place. Tilings, coverings, clusters and quasicrystals The study of tilings of the plane with regular polygons has a long history. Kepler introduced the systematic approach to classification but a remarkable intuitive insight is apparent much earlier in the work of Islamic architects. Penrose's aperiodic tiling patterns and Amman's threedimensional generalisation became the basis for our present understanding of quasicrystals. Gummelt's discovery of an aperiodic covering of the plane by overlapping decorated regular decagons led to the recent elucidation of the structure of the decagonal phase of AlNiCo (Steinhardt et al. 1998). We have been able to demonstrate that a decagonal 'quasi unit cell' also exists for other kinds of decagonal phases, and to explore in some detail the analogy between (periodic) trigonal and hexagonal structures and decagonal phases (Lord & Ranganathan 2000). Our search for an analogous quasi unit cell for icosahedral phases led to a pattern of overlapping rhombic triacontahedral clusters, which provides a plausible model for the socalled T2 phase (Lord, Ranganathan & Kulkarni 2000). Hexagonal crystalline phases such as lambda, mu and others have very large unit cells and are apparently 'approximants' of the recently discovered 'hexagonal' quasicrystals. Kreiner and Franzen (1995) have suggested that a basic structural unit for these phases consists of three icosahedra, each sharing a vertex (atom) with two others. This can be extended to a tetrahedron of four icosahedra. Other feasible subunits can be built from slightly distorted icosahedra, which may overlap in various ways, giving rise to FrankKasperlike structures (Kreiner & Schäpers 1997). Much exploratory work is needed, to investigate possibilities and to relate unit cell dimensions of 'approximants' to the average bond lengths. A minimal surface has zero mean curvature and hence negative Gaussian curvature. The classification of triply periodic minimal surfaces (TPMS) is based on topology and symmetry. Following the methods of Koch and Fischer and employing the software 'Surface Evolver', we have found several new types of surface with noncubic symmetry groups (Lord, Colloids & Surfaces 1997) and 14 new cubic types (Lord & Mackay, in preparation). Tiling patterns on triply periodic surfaces correspond to threedimensional networks. Because the minimal surfaces have negative (Gaussian) curvature, these are related to tilings of the hyperbolic plane. A sphere cannot be covered by a 3connected net of hexagons  the positive curvature requires there to be 12 pentagons, as in the structure of the fullerenes. Similarly, the negatively curved TPMS can be tiled by hexagons only if heptagons and octagons also occur. These are the simplest cases; much remains to be explored. 4connected three dimensional nets occur in the structures of silicates, aluminosilicates and zeolites. Three dimensional nets all of whose edge lengths are equal are obviously related to packings of equal spheres  a topic of obvious importance in understanding the structure of many minerals. A method of obtaining a complete topological classification of all possible 3D nets has been reported recently (Friedrichs et al., Nature 1999). Our own approach is based on space group symmetries rather than topology (so that the problem of deciding if a particular net can be realised in ordinary Euclidean space does not arise). The method is based on a tabulation of generators and relations for each of the space groups, analogous to the data given by Coxeter and Moser for the 17 plane symmetry groups. This information allows any net to be assigned a symbol that encodes the net, and from which it can be constructed. A net symbol also allows certain characteristics of a net (connectivities and polyhedral circuits) to be computed algebraically. This work is still in progress. A 3D net can be regarded as the set of edges and vertices of a space filling by polyhedra  which in general do not have flat faces, they are 'saddle polyhedra'. 3D nets and saddle polyhedra are related to TPMS. This is a large subject, offering much scope for future research.

References  
Lord, E A, Mackay, A L & Ranganathan, S (2006) New Geometries for New Materials. Cambridge University Press. 
Coxeter, H S M & Moser, W O J (1957) Generators and Relations for Discrete Groups. Springer. Friedrichs, O D, Dress, A W M, Huson, D H, Klinowski, J & Mackay, A L (1999) Systematic enumeration of crystalline networks. Nature 400, 644647. Gummelt, P (1995) Penrose tilings as coverings of congruent decagons. Geom. Ded. 62, 117. Kreiner, G & Franzen, H F (1995) A new concept and its applcation to quasicrystals of the iAlMnSi family and closely related crystalline structures. Jnl. of Alloys and Compounds 221, 1536. Kreiner, G & Schäpers, M (1997) A new description of Samson's Cd3Cu4 and a model of icosahedral iCdCu. Jnl. of Alloys and Compounds 259, 83114. Lord, E A (1991) Quasicrystals and Penrose patterns. Curr. Sci. 61, 313319. Lord, E A (1997) Triplyperiodic balance surfaces. Colloids & Surfaces A 129130, 279295. NB: The PDF files are copyrighted material. They are made available here for private study only. Lord, E A & Ranganathan, S (2001) The Gummelt decagon as a quasi unit cell. Acta Cryst.A57, 531539. Lord, E A, Ranganathan, S & Kulkarni, U D (2000) Tilings, coverings, clusters and quasicrystals. Curr. Sci. 78, 6472. Lord, E A, Ranganathan, S & Kulkarni, U D (2001) Quasicrystals: tiling versus clustering. Phil.Mag. A 81, 26452651. Lord, E A & Ranganathan S (2001) Sphere packing, helices and the polytope {3,3,5}. EPJ D 15, 335343. Lord, E A & Mackay, A L (2003) Periodic minimal surfaces of cubic symmetry. Curr. Sci. 85, 346362. Lord, E A & Ranganathan S (2004) The gammabrass structure and the BoerdijkCoxeter helix. J. NonCrystalline Solids 334&335, 121125. Mackay, A. L., Finney, J. L. & Gotoh, K. (1977) The Closest Packing of Equal Spheres on a Spherical Surface. Acta Cryst. A33, 98100. Mackay, A (1962). A dense noncrystalloraphic packing of equal spheres. Acta Cryst.15, 916918. Mackay, A (1975). Generalized crystallography. Izvj. Jugosl. Centr. Krist.
(Zagreb) 10, 1536. Mackay, A. L. (1985) Periodic Minimal Surfaces. Nature 314, 604606. Mackay, A (1995) Flexicrystallography. Curr. Sci. 69, 151161. Gandy, J. F., Bardhan, S. & Mackay, A. L. (2001) Nodal surface approximations to the P, G, D and IWP triply periodic minimal surfaces. Chem. Phys. Lett. 336, 187195 Steinhardt P J, Jeong, HC, Saitoh, K, Tanaka, M, Abe, E & Tsai, A P (1998) Experimental verification of the quasiunitcell model of quasicrystal structure. Nature 396, 5557. Miracle, D. B., Lord, E. A. & Ranganathan, S. (2006) Candidate atomic cluster configurations in metallic glass structures. Materials Transactions 47, 17371742.
The project 'New Geometries for New Materials' was sponsored by the Defense, Research and Development Organisation, Ministry of Defense, Government of India (project reference DRDO/MMT/SRG/526). E A Lord 