## Fivefold SymmetryFivefold symmetry is common in flowers, fruits, molecules, logos, and buildings, but it is a forbidden symmetry in the world of crystals. A few years ago, the so-called quasicrystals were discovered displaying fivefold symmetry, and it caused a minirevolution in crystallography. There has been increased awareness of fivefold symmetry in all domains of human interest ever since. The present book brings together authors and ideas on a common theme from mathematics, the sciences, design, and anthropology to history, literature, and the arts. Its 29 chapters are an offering by scientists and humanists from 13 countries to a broad readership of professionals and laypersons about fivefold symmetry and the areas that are being bridged by this unique concept. |

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See also http://www.sciencemag.org/cgi/reprint/sci;318/5855/1383a.pdf

### Contents

Fivefold Symmetry in Mathematics Physics Chemistry | 11 |

The Relationship Between Mathematics and Mysticism of | 33 |

800YearOld Pentagonal Tiling From Maragha Iran and | 67 |

Plane Projections of Regular Polytopes with Fivefold Symmetries | 87 |

Fivefold Symmetry in Hyperbolic Crystallography | 129 |

The Pentasnow Gasket and its Fractal Dimension | 141 |

How to Inscribe a Dodecahedron in a Sphere | 167 |

The Discovery of Space Frames with Fivefold Symmetry | 205 |

Notes on Some Pentagonal Mysteries in Egyptian | 307 |

The Icosahedral Design of the Great Pyramid | 333 |

A Mystic History of Fivefold Symmetry in Japan | 361 |

Fivefold Symmetry in the Literature | 395 |

Certain Quinary Aspects of the Hindu Civilization | 423 |

On the Shape of Five in Early Hindu Thought | 445 |

Albrecht Durer and the Regular Pentagon | 465 |

Fivefold Symmetry in the Graphic Art of M C Escher | 489 |

The New Zome Primer | 221 |

A Calvinist Church | 235 |

Pentagon and Decagon Designs in Islamic Art | 263 |

An Islamic Pentagonal Seal from Scientific Manuscripts of | 283 |

Nomothetical Modelling of Spiral Symmetry in Biology | 505 |

Hawaiian Flowers with Fivefold Symmetry | 529 |

### Common terms and phrases

ancient appears approximate architectural axis Beowulf century chaos circle construction corresponding crystallographers crystals cube decagon described diagonal diameter Diirer dimensions divergence angle divine dodecahedron Durer's dynamic edges Egyptian elements equal Euclidean example faces Fibonacci number figures fivefold axes fivefold rotational fivefold symmetry flowers four fourfold fractal geometric golden gnomon golden mean golden ratio golden section golden triangle graphics H.S.M. Coxeter Hargittai Hindu hyperbolic icosahedral icosahedron Islamic iteration Japanese Julia sets lattice length lines M.C. Escher Maragha mathematical mathematicians non-periodic number five Osiris pair pattern Penrose tiling Pentagonal Seal pentagram phyllotaxis plane Platonic solids polygons polyhedra Pyramid quasi-crystals regular hexagon regular pentagon represent rhombi Rig Veda rotational symmetry sequence shape shown in Fig shows side space space-fillings spiral square star decagon structure symbol symmetry group tessellation three-dimensional transformations translation triacontahedron two-dimensional Underweysung Upanishad vectors Vedic vertex vertices York zonohedra