Quasicrystals and Geometry brings together for the first time the many strands of contemporary research in quasicrystal geometry and weaves them into a coherent whole. The author describes the historical and scientific context of this work, and carefully explains what has been proved and what is conjectured. This, together with a bibliography of over 250 references, provides a solid background for further study. The discovery in 1984 of crystals with 'forbidden' symmetry posed fascinating and challenging problems in many fields of mathematics, as well as in the solid state sciences. Increasingly, mathematicians and physicists are becoming intrigued by the quasicrystal phenomenon, and the result has been an exponential growth in the literature on the geometry of diffraction patterns, the behaviour of the Fibonacci and other nonperiodic sequences, and the fascinating properties of the Penrose tilings and their many relatives.
Author(s): Senechal M.
Publisher: CUP
Year: 1996
Language: English
Pages: 306
Tags: Физика;Физика твердого тела;
Cover......Page 1
Title Page......Page 6
Copyright Page......Page 7
Contents......Page 8
Preface......Page 12
1.1 Demise of a paradigm......Page 18
1.2 Ancient views on the structure of crystals......Page 24
1.3 From building-blocks to lattices......Page 30
1.4 The ascendance of symmetry......Page 33
1.5 Hilbert's 18th problem......Page 38
1.6 Toward a new definition of `crystal'......Page 44
1.7 Notes......Page 48
2.1 Introduction......Page 51
2.2 Lattices and their duals......Page 54
2.3 Voronoi cells......Page 58
2.4 Symmetry......Page 62
2.4.1 Isometries......Page 63
2.4.2 Symmetry groups......Page 65
2.4.3 The crystallographic restriction......Page 67
2.5 Orbits of Z-modules......Page 69
2.6.1 The projection method......Page 71
2.6.2 Uniform distribution......Page 75
2.6.3 Five-fold symmetry......Page 76
2.7 Lattices with icosahedral point groups......Page 80
2.7.1 The 16 quasicrystal......Page 81
2.7.2 The root lattices......Page 82
2.7.3 D6 and E8 quasicrystals......Page 85
2.7.4 Is the projection formalism necessary?......Page 88
2.8 Notes......Page 89
3.1 Introduction......Page 91
3.2 Diffraction geometry......Page 94
3.2.2 Two-point diffraction......Page 95
3.2.3 N-point diffraction......Page 100
3.3 Fourier transforms and the Wiener diagram......Page 103
3.4 Dirac deltas and Dirac combs......Page 108
3.5 The diffraction condition......Page 112
3.6 Searching for deltas......Page 114
3.6.2 Poisson combs......Page 116
3.7 Notes......Page 118
4.1 One-dimensional crystals......Page 123
4.2 Fibonacci numbers, strings, and sequences......Page 126
4.3 Projected crystals......Page 131
4.3.1 Projected Fibonacci sequences are crystals......Page 137
4.3.2 Generalizations......Page 140
4.4 Substitution sequences......Page 141
4.5 Circle map sequences......Page 145
4.6 Staircases......Page 148
4.7 Notes......Page 149
5.1 Tilings and crystals......Page 152
5.2 Some basic concepts of tiling theory......Page 153
5.3 Atlases......Page 157
5.4 Which shapes are tiles?......Page 161
5.5 Orderliness......Page 162
5.6 Some construction techniques......Page 169
5.6.2 Dualization......Page 170
5.6.3 Substitution......Page 173
5.6.4 Multigrids......Page 179
5.6.5 Projection......Page 184
5.7 Notes......Page 186
6 Penrose filings of the plane......Page 187
6.1 Penrose tiles and matching rules......Page 188
6.2 Up-down generation......Page 200
6.3 Pentagrids and their duals......Page 207
6.4.1 The canonical projection......Page 212
6.4.2 The vertex atlas......Page 213
6.4.3 The role of the shift vector......Page 216
6.5 Other generalizations of the Penrose tilings......Page 218
6.6 The space P......Page 222
6.7 Notes......Page 223
7.1 Matching rules and their classification......Page 226
7.2 The SCD (Schmitt-Conway-Danzer) tile......Page 228
7.3 `Octagonal' tilings......Page 232
7.4.1 Danzer's triangular prototiles: seven-fold symmetry......Page 237
7.5.1 The three-dimensional Penrose tilings......Page 239
7.5.2 The Socolar-Steinhardt tilings and Danzer's aperiodic tetrahedra......Page 241
7.6 Toward a theory of matching rules......Page 242
7.6.1 Matching rules for projected tilings......Page 243
7.6.2 Uncountability is intrinsic......Page 248
7.6.3 Errors are intrinsic......Page 250
7.7 Notes......Page 253
8.1 The atlas......Page 257
8.2.1 Lattice-vacancy tilings......Page 262
8.2.3 Generalized Penrose tilings......Page 263
8.2.5 A pinwheel tiling......Page 265
8.4.1 The tilings......Page 266
8.4.2 `fourier'.c......Page 270
8.4.3 Line arrangements and zonotopal tilings......Page 273
8.4.4 QuasiTiler......Page 277
Appendix I. A mathematical toolbag......Page 278
Appendix II. De Bruijn's generalized functions......Page 286
References......Page 291
Index......Page 302
Plate 1......Page 214
Plate 2......Page 215
Back Cover......Page 306
