Book contents
- Frontmatter
- Contents
- Preface
- 1 Stringology
- 2 Number Theory and Algebra
- 3 Numeration Systems
- 4 Finite Automata and Other Models of Computation
- 5 Automatic Sequences
- 6 Uniform Morphisms and Automatic Sequences
- 7 Morphic Sequences
- 8 Frequency of Letters
- 9 Characteristic Words
- 10 Subwords
- 11 Cobham's Theorem
- 12 Formal Power Series
- 13 Automatic Real Numbers
- 14 Multidimensional Automatic Sequences
- 15 Automaticity
- 16 k-Regular Sequences
- 17 Physics
- Appendix Hints, References, and Solutions for Selected Exercises
- Bibliography
- Index
17 - Physics
Published online by Cambridge University Press: 13 October 2009
- Frontmatter
- Contents
- Preface
- 1 Stringology
- 2 Number Theory and Algebra
- 3 Numeration Systems
- 4 Finite Automata and Other Models of Computation
- 5 Automatic Sequences
- 6 Uniform Morphisms and Automatic Sequences
- 7 Morphic Sequences
- 8 Frequency of Letters
- 9 Characteristic Words
- 10 Subwords
- 11 Cobham's Theorem
- 12 Formal Power Series
- 13 Automatic Real Numbers
- 14 Multidimensional Automatic Sequences
- 15 Automaticity
- 16 k-Regular Sequences
- 17 Physics
- Appendix Hints, References, and Solutions for Selected Exercises
- Bibliography
- Index
Summary
Quasicrystals, discovered by the materials scientist Dan Shechtman in April 1982, are materials that are intermediate between crystalline and random structures (glasses). They are formed, for example, in certain alloys of aluminum with other metals, such as copper or manganese.
After their experimental discovery, it was proposed that a certain tiling of the plane, known as the Penrose tiling, could serve as a theoretical model of the structure of these alloys. The Penrose tiling is not periodic, but possesses fivefold symmetry and looks “regular” see Figures 17.1 and 17.2. Figure 17.1 illustrates a partial Penrose tiling of the plane by two kinds of pieces: kites and darts. A local fivefold symmetry can be seen (local invariance under a rotation of 2π/5). In Figure 17.2, some pieces have been glued together to form bow ties that are either long or short. If such an infinite line, or worm, of short and long bow ties can be found in a Penrose tiling, the bow ties are arranged according to the infinite Fibonacci word f = 010010100100101001010 … introduced in Section 7.1, where a short bow tie is replaced by 0 and a long one by 1.
Because the Fibonacci word is associated with the Penrose tiling, theoretical physicists began to study the properties of f.
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- Chapter
- Information
- Automatic SequencesTheory, Applications, Generalizations, pp. 455 - 470Publisher: Cambridge University PressPrint publication year: 2003