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**Folds!**
*(English)*
Zbl 0493.10001

In this series of three papers (see Zbl 0493.10002, Zbl 0493.10003) the authors survey a varied collection of topics which are all related to the so-called paper-folding sequences. Such sequences arise from repeatedly folding a sheet of paper, unfolding it again and considering the sequence of “upward” and “downward” bends. They have a number of highly interesting properties. For example, plane-filling curves can be constructed from them. They can also be used to construct sequences of integers \(u(h)\) satisfying
\[
\sup_{0\leq\theta\leq 2\pi}|\sum_0^{n-1} (1)^{u(h)}e^{ih\theta}|\leq (2+\sqrt 2)\sqrt n,
\]
the lower bound \(\sqrt n\) being trivial. Some alternative ways of generating related sequences are generation by automatons and by symmetry operations. For example, \(\sum g_hX^h\) is algebraic over \(\mathbb F_p[X]\) if and only if the sequence \(g_h\) can be generated by a so-called \(p\)-automaton. Moreover, \(\sum g_hp^{-h}\) is a transcendental number in that case. Furthermore, the continued fraction of the Fredholm series \(g^{-2^h}\) can be given by a sequence generated by symmetry operations.

By generalization of the paperfolding idea, one can construct bizarre, plane-filling curves, which, drawn on a piece of paper yield intricate patterns that arouse ones fantasy. In all, the paper contains much information, is written in an entertaining form and worth wile reading.

By generalization of the paperfolding idea, one can construct bizarre, plane-filling curves, which, drawn on a piece of paper yield intricate patterns that arouse ones fantasy. In all, the paper contains much information, is written in an entertaining form and worth wile reading.

Reviewer: Frits Beukers (Utrecht)

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11B83 | Special sequences and polynomials |

11B85 | Automata sequences |

05B30 | Other designs, configurations |

68Q45 | Formal languages and automata |

11J81 | Transcendence (general theory) |

11A55 | Continued fractions |

### Keywords:

paperfolding sequences; automata; systems of functional equations; dragon curves; dimension of plane curves; Rudin-Shapiro sequences; Fredholm series
Full Text:
DOI

### Online Encyclopedia of Integer Sequences:

The Rudin-Shapiro or Golay-Rudin-Shapiro sequence (coefficients of the Shapiro polynomials).Characteristic function of powers of 2, cf. A000079.

### References:

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[2] | A. Cobham: Uniform tag sequences.Math. Systems Theory 6 (1972), 164–192. Also: On the Hartmanis- Stearns problem for a class of tag machines,Technical Report RC2178 (1968) IBM Research Centre, Yorktown Heights, N.Y., and: On the base dependence of sets of numbers recognisable by finite automata.Math. Systems Theory 3 (1969), 186-192 · Zbl 0253.02029 |

[3] | Philip J. Davis: Visual geometry, computer graphics and theorems of perceived type. Proc. Symposia in Appl. Math 20 (1974), 113–127 (Amer. Math. Soc); see also: Philip J. Davis, Reuben Hersh:The Mathematical Experience (Birkhäuser Verlag, 1980) |

[4] | Chandler Davis, Donald E. Knuth: Number representations and dragon curves I.J. Recreational Math. 3 (1970) 61–81; II 3 (1970), 133-149 |

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[6] | William Feller:An introduction to probability theory and its applications (Wiley, 1950) · Zbl 0039.13201 |

[7] | Martin Gardner: Mathematical games. ä Scientific American 216 (March 1967), 124–125; (April 1967), 118-120; 217 (July 1967) 115. Also in:Mathematical Magic Show (Knopf NY, 1977). |

[8] | M. Mendès France,A. J. van der Poorten: Arithmetic and analytic properties of paperfolding sequences (dedicated to Kurt Mahler),Bull. Austral. Math. Soc. 24(1981), 123–131 · Zbl 0451.10018 |

[9] | M. Mendès France, G. Tenenbaum: Dimension des courbes planes, papiers plies et suites de Rudin-Shapiro.Bull. Soc. Math. France 109 (1981), 207–215 · Zbl 0468.10033 |

[10] | W. Rudin: Some theorems on Fourier coefficients.Proc. Amer. Math. Soc. 10 (1959), 855–859 · Zbl 0091.05706 |

[11] | H. S. Shapiro :External problems for polynomials and power series. Thesis MIT (1951) |

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