## The Golden Ratio and Fibonacci NumbersIn this invaluable book, the basic mathematical properties of the golden ratio and its occurrence in the dimensions of two- and three-dimensional figures with fivefold symmetry are discussed. In addition, the generation of the Fibonacci series and generalized Fibonacci series and their relationship to the golden ratio are presented. These concepts are applied to algorithms for searching and function minimization. The Fibonacci sequence is viewed as a one-dimensional aperiodic, lattice and these ideas are extended to two- and three-dimensional Penrose tilings and the concept of incommensurate projections. The structural properties of aperiodic crystals and the growth of certain biological organisms are described in terms of Fibonacci sequences. Contents: Basic Properties of the Golden Ratio; Geometric Problems in Two Dimensions; Geometric Problems in Three Dimensions; Fibonacci Numbers; Lucas Numbers and Generalized Fibonacci Numbers; Continued Fractions and Rational Approximants; Generalized Fibonacci Representation Theorems; Optimal Spacing and Search Algorithms; Commensurate and Incommensurate Projections; Penrose Tilings; Quasicrystallography; Biological Applications; Construction of the Regular Pentagon; The First 100 Fibonacci and Lucas Numbers; Relationships Involving the Golden Ratio and Generalized Fibonacci Numbers. Readership: Applied mathematicians. |

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### Contents

7 | |

GEOMETRIC PROBLEMS IN TWO DIMENSIONS | 15 |

GEOMETRIC PROBLEMS IN THREE DIMENSIONS | 23 |

FIBONACCI NUMBERS | 35 |

LUCAS NUMBERS AND GENERALIZED | 51 |

CONTINUED FRACTIONS AND RATIONAL | 63 |

GENERALIZED FIBONACCI REPRESENTATION THEOREMS | 71 |

OPTIMAL SPACING AND SEARCH ALGORITHMS | 79 |

COMMENSURATE AND INCOMMENSURATE | 87 |

PENROSE TILINGS | 97 |

QUASICRYSTALLOGRAPHY | 111 |

BIOLOGICAL APPLICATIONS | 123 |

CONSTRUCTION OF THE REGULAR PENTAGON | 137 |

RELATIONSHIPS INVOLVING THE GOLDEN RATIO | 143 |

153 | |

### Common terms and phrases

additive sequence adult rabbit pairs algorithm approaches the golden arrangement baby rabbits calculation cell clockwise clockwise and counterclockwise constructed counterclockwise spirals cube cut and projection cut angle cut line deflation dimensional Fibonacci lattice dimensional Penrose tiling edge length equation example exhibit fivefold symmetry expressed Fibonacci and Lucas Fibonacci numbers Fibonacci representation Fibonacci sequence function geometric geometric sequence given by Eq given in Table golden gnomon golden ratio golden rectangle golden triangles growth angle icosahedron illustrated in Fig indefinitely irrational number long and short Lucas numbers Lucas sequence matching rules mathematical method number of adult octahedron pattern periodic array Platonic solids points properties quasicrystalline quasicrystals quasiperiodic tiling ratio of successive rational approximant recursion relation regular pentagon rescaling rhombohedron rhombuses rotational symmetry seed values short line segments shown in Fig shows small stellated dodecahedron structure successive Fibonacci numbers symmetry characteristics tangent theorem three dimensions tile shapes Tribonacci vertex vertices yield

### Popular passages

Page 11 - the ratio of the lengths of the two segments is the same as the ratio of the length of the longer segment to the

Page 1 - has been called the golden mean, the golden section, the golden cut, the divine proportion, the Fibonacci number and the mean of Phidias

Page 1 - K (the ratio of the circumference to the diameter of a circle) and e

Page 5 - the ratio of the circumference, c, to the diameter, d, of a circle;

Page 2 - the longer segment is the same as the ratio of the length of the longer segment to the length of the

Page 1 - An irrational number is one which cannot be expressed as a ratio of