Cover

Title

Copyright

Contents

Preface

1 Introduction to geometrical frustration

1.1 From cubism to icosahedrism

1.2 Geometry

1.3 Geometrical frustration

2 Ideal models

2.1 A unified approach to very different materials

2.2 Simple two-dimensional examples

2.3 Metals

2.4 The {3, 3, 5} polytope: an ideal template for amorphous metals

2.5 Covalent tetracoordinated structures

2.6 Frustration in lamellar liquid crystals and amphiphiles

2.7 Lamellar structures in curved spaces

2.8 Frustration and curved space structure for blue phases

2.9 Frustration in polymers

3 Finite structures

3.1 Finite clusters

3.2 Toroidal vesicles with phospholipid membranes

4 Decurving and disclinations

4.1 Disclinations

4.2 Wedge and screw disclinations

4.3 Coordination number, disclination density and Regge calculus

4.4 Conservation laws

4.5 Defects and topology

5 Hierarchical polytopes

5.1 Hierarchical polytopes and symmetry groups

5.2 Hierarchy and scaling

5.3 Matrix formulation of the hierarchical structures

5.4 Disorder and non-commutative defects

5.5 Deflation of the orthoscheme

6 Some physical properties

6.1 Structure factor of polytopes and orientational order

6.2 Specific volume variation in disordered solids: a simple model

6.3 Landau theory of frustrated systems

6.4 Two-level systems

6.5 Frustration-limited domain theory

6.6 Excitation spectrum

7 Periodic structures with large cells

7.1 Frustration and large cell crystals

7.2 Complex structures in metals

7.3 Melting of model structures

7.4 Tetracoordinated structures

7.5 Liquid crystal structures

8 Quasiperiodic order and frustration

8.1 Quasicrystals: the spectacular appearance of quasiperiodic order in solid state physics

8.2 Hierarchical clusters in quasicrystals

8.3 Random tilings

8.4 Random tilings in one dimension

8.5 Two-dimensional tilings

8.6 Three-dimensional rhombohedral tilings

8.7 Glass-like properties in quasicrystals

Al Spaces with constant curvature

A1.1 The three geometries

Al .2 Spherical spaces

Al .3 Two- and three-dimensional cylindrical spaces

A 1.4 Intrinsic curvature

A2 Quaternions and related groups

A2.1 Quaternions

A2.2 Some continuous groups acting on spheres

A2.3 Discrete groups

A3 Hopf fibration

A3.1 Fibrations

A3.2 Hopf fibration

A4 Poly topes and honeycombs

A4.1 Symmetries and orthoscheme tetrahedra

A4.2 Polytopes and honeycombs

A5 Polytope {3, 3, 5}

A5.1 The geometry of polytope {3, 3, 5}

A5.2 Description in terms of toroidal shells

A6 Frank and Kasper coordination polyhedra

A6.1 Frank and Kasper polyhedra

A6.2 Positive and negative disclinations

A7 Quasiperiodic tilings: cut and projection

A7.1 Cut and projection algorithm

A7.2 Codimension 1 approximants

A7.3 Approximants of the octagonal tiling

A7.4 An almost octagonal quasiperiodic tiling: the labyrinth

A8 Differential geometry and parallel transport

A8.1 Manifold and tangent space

A8.2 Geodesic equation

A8.3 Parallel transport and curvature

A9 Icosahedral quasicrystals and the ES lattice

A9.1 Introduction

A9.2 The ES lattice

A9.3 A discrete Hopf fibration on the Gosset polytope

A9.4 Shelling the quasicrystal

A9.5 The 2 d - ld aspect of the shell-by-shell construction of the quasicrystal

A9.6 Quasicrystals of lower dimension

Bibliography

Index