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THEOREM: Rotational symmetries of order greater than six, and also five-fold rotational symmetry, are impossible for a periodic pattern in the plane or in three-dimensional space. The discovery of quasicrystals shattered this fundamental 'law', not by showing it to be logically false but by showing that periodicity was not synonymous with long-range order, if by 'long-range order' we mean whatever order is necessary for a crystal to produce a diffraction pat tern with sharp bright spots. It suggested that we may not know what 'long-range order' means, nor what a 'crystal' is, nor how 'symmetry' should be defined. Since 1984, solid state science has been under going a veritable K uhnian revolution. -M. SENECHAL, Quasicrystals and Geometry Between total order and total disorder He the vast majority of physical structures and processes that we see around us in the natural world. On the whole our mathematics is well developed for describing the totally ordered or totally disordered worlds. But in reality the two are rarely separated and the mathematical tools required to investigate these in-between states in depth are in their infancy.
List of contents
Knotted Tilings.- Solution of the Coincidence Problem in Dimensions d ? 4.- Self-Similar Tilings and Patterns Described by Mappings.- Delone Graphs; Some Species and Local Rules.- What is the Long Range Order in the Kolakoski Sequence?.- Topics in Aperiodicity: Penrose Tiling Growth and Quantum Circuits.- The Diffraction Pattern of Self-Similar Tilings.- Pisot-Cyclotomic Integers for Quasilattices.- Aperiodic Ising Models.- Diffraction by Aperiodic Structures.- Aperiodic Schrödinger Operators.- Symmetry Concepts for Quasicrystals and Non-commutative Crystallography.- Local Rules for Quasiperiodic Tilings.- Almost-Periodic Sequences and Pseudo-Random Sequences.- The Symmetry of Crystals.- Meyer Sets and Their Duals.- Non-crystallographic Root Systems and Quasicrystals.- Remarks on Tiling: Details of a (1 + ? + ?2)-Aperiodic Set.- Aperiodic Tilings, Ergodic Theory, and Rotations.- A Critique of the Projection Method.
