Aperiodic structures and notions of order and disorder

Abstract

The fabrication of artificial heterostructures is mainly based on substitution systems. We present simple ways to construct double-sided versions of the Fibonacci, Prouhet–Thue–Morse, paperfolding, period doubling and Golay–Rudin–Shapiro sequences. We also construct a generic instance of the two-dimensional Prouhet–Thue–Morse structure and explore its symbolic complexity. The complexity turns out to be polynomial and hence, the entropy goes to zero.

Keywords:

• heterostructures
• double-sided sequences
• 2D Prouhet–Thue–Morse structure
• symbolic complexity

Acknowledgement

We thank Michael Baake from the University of Bielefeld (Germany) and Uwe Grimm from the Open University at Milton Keynes (UK) for fruitful discussions and letting us benefit from parts of their forthcoming book.

Notes

^†Dr Quandt's address after 28 October 2010: School of Physics, University of the Witwatersrand, Wits 2050, South Africa.

Notes

1. Symbolic complexity p_S (w|N) is the number of different words (configurations) w of size N in a given structure S.

2. Topological entropy H is defined as

3. A number (or numeral) system (B|{a ₁, a ₂, … , a _n}) serves to express numbers (in a general sense) expanding in powers of a base B in terms of a set of digits {a ₁, a ₂, … , a _n}. In the present simple example (−2|{0, 1}), we expand the integers in powers of −2. Thus, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6 are written as 1110, 1101, 1100, 1101, 10, 1, 0, 1, 110, 111, 100, 101, 11010, respectively.

4. A canonical paperfolding sequence arises by folding a sheet of paper consistently the same way (from right to left, say). In a generalized paperfolding sequence, the folds follows some different rule or may be even random.

5. The Champernowne sequence arises by successive concatenation of all natural numbers. In its binary representation it is 0,1,10,11,100,101,110,111,1000,1001,1010,1011,1100,1101,1110,1111,10000,10001,10010,… (the commas are inserted only for orientation).

Additional information

Notes on contributors

A. Quandt†

†Dr Quandt's address after 28 October 2010: School of Physics, University of the Witwatersrand, Wits 2050, South Africa.