The Golden Ratio: The Story of Phi, the World's Most Astonishing Number - Mario Livio (2003)

CREDITS

The author and publisher gratefully acknowledge permission to reprint the following copyrighted material:

ARTWORK:

Figs.1, 2, 3, 7, 9, 10, 11, 12, 14a, 14b, 18, 20a, 20b, 20c, 20d, 20e, 21, 24, 25a, 25b, 26, 27, 29, 30, 33a, 33b, 35, 37, 40, 41, 42, 44a, 44b, 49, 57a, 57b, 58, 61, 62, 63, 64, 86, 89, 91, 97a, 97b, 97c, 101a, 101b, 102a, 102b, 103a, 103b, 105, 106a, 106b, 107, 112, 114, 123, 124, and the diagrams in Appendix 2, Appendix 3, and Appendix 4 by Jeffrey L. Ward

Fig. 4: The Bailey-Matthews Shell Museum

Fig. 5: Chester Dale Collection, Photograph © 2002 Board of Trustees, National Gallery of Art, Washington, D.C. © 2002 Salvador Dali, Gala-Salvador Dali Foundation/Artists Rights Society (ARS), New York

Fig. 6: Reprinted with permission from John D. Barrow, Pi In the Sky (Oxford: Oxford University Press, 1992).

Fig. 13: © Copyright The British Museum, London.

Fig. 17: Hirmer Fotoarchiv

Fig. 19: Reprinted with permission from Robert Dixon, Mathographics (Mineola: Dover Publications, 1987).

Figs. 22 & 23, bottom: Reprinted with permission from H. E. Huntley, The Divine Proportion (Mineola: Dover Publications, 1970).

Fig. 23, top: Alison Frantz Photographic Collection, American School of Classical Studies at Athens

Fig. 28: Reprinted with permission from Trudi Hammel Garland, Fascinating Fibonaccis Mystery and Magic in Numbers © 1987 by Dale Seymour Publications, an imprint of Pearson Learning, a division of Pearson Education, Inc.

Figs. 31–32: Reprinted with permission from Trudi Hammel Garland, Fascinating Fibonaccis Mystery and Magic in Numbers © 1987 by Dale Seymour Publications, an imprint of Pearson Learning, a division of Pearson Education, Inc.

Fig. 34: Reprinted with permission from J. Brandmüller, “Five fold symmetry in mathematics, physics, chemistry, biology and beyond,” in I. Hargitta, ed. Five Fold Symmetry (Singapore: World Scientific, 1992).

Fig. 36: Reprinted with permission from N. Rivier et al., J. Physique, 45, 49 (1984).

Fig. 38: The Royal Collection © 2002, Her Majesty Queen Elizabeth II

Fig. 39: Reprinted with permission from Edward B. Edwards, Pattern and Design with Dynamic Symmetry (Mineola: Dover Publications, 1967).

Fig. 43: Credit NASA and the Hubble Heritage Team.

Figs. 46, 45, 47, 50: Alinari/Art Resource, NY

Fig. 47: Perspective lines, reprinted with permission from Laura Geatti, Michelle Emmer Editor, The Visual Mind: Art and Mathematics (Cambridge: the MIT Press,1993).

Fig. 52: Property of the Ambrosian Library. All rights reserved. Reproduction is forbidden.

Fig. 53: Scala/Art Resource, NY

Figs. 55, 56: The Metropolitan Museum of Art, Dick Fund, 1943

Fig. 57: Reprinted with permission from David Wells, The Penguin Book of Curious and Interesting Mathematics (London: The Penguin Group, 1997), copyright © David Wells, 1997.

Figs. 68–69: Kindly provided by the Institute for Astronomy, University of Vienna. Figs. 70, 71, 72: Alinari/Art Resource, NY

Fig. 72: National Gallery, London

Fig. 73: Alinari/Art Resource, NY

Fig. 75: Scala/Art Resource, NY

Fig. 76: The Metropolitan Museum of Art, Bequest of Stephen C. Clark, 1960.(61.101.17)

Fig. 77: Philadelphia Museum of Art: The A. E. Gallatin Collection, 1952. © 2002 Artists Rights Society (ARS), New York/ADAGP, Paris

Fig. 78: Private Collection, Rome. © 2002 Artists Rights Society (ARS), New York/ADAGP, Paris

Fig. 79: © 2002 Artists Rights Society (ARS), New York/ADAGP, Paris/FLC

Figs. 80, 81: © 2002 Artists Rights Society (ARS), New York/ADAGP, Paris/FLC

Fig. 82: Private Collection. From “Module Proportion, Symmetry, Rhythm” by Gyorgy Kepes, George Braziller. © 2002 Artists Rights Society (ARS), New York/DACS, London

Fig. 83: The Museum of Modern Art/Licensed by Scala/Art Resource, NY. © 2002 Mondrian/Holtzman Trust, c/o Beeldrecht/Artists Rights Society (ARS), New York

Fig. 84: Reprinted with permission from G. Markowsky, The College Mathematics Journal, 23, 2 (1992).

Fig. 85: Reprinted with permission from Denis Arnold, ed., The New Oxford Companion to Music, Vol. 2 (Oxford: Oxford University Press, 1984).

Figs. 87, 88: Reprinted with permission from Ernö Lendvai, Béla Bartók, An Analysis of His Music (London: Kahn & Averill, 1971).

Fig. 89: Reprinted with permission from Roy Howat, Debussy in Proportion (Cambridge: Cambridge University Press, 1983).

Fig. 90: Reprinted with permission from Joseph Schillinger, The Schillinger System of Musical Composition (New York: Carl Fischer, LLC, 1946).

Fig. 92: Rijksmuseum, Amsterdam

Fig. 93: Kunsthistorisches Museum, Vienna

Fig. 94: Reprinted with permission from Ivars Peterson, The Mathematical Tourist (New York: W. H. Freeman, 1988).

Fig. 95: Courtesy Ricardo Villa-Real. From “The Alhambra and the Generalife” by Ricardo Villa-Real

Fig. 96: © 2002 Cordon Art-Baard, Holland. All rights reserved

Fig. 99: Reprinted with permission from Richard A. Dunlap, The Golden Ratio and Fibonacci Numbers (Singapore: World Scientific, 1997).

Fig. 100: Reprinted with permission from Richard A. Dunlap, The Golden Ratio and Fibonacci Numbers (Singapore: World Scientific, 1997).

Fig. 104: Reprinted with permission from Martin Gardner, Penrose Tiles to Trapdoor Ciphers (New York: W H. Freeman, 1988).

Fig. 108–109: Reprinted with permission from Paul Steinhardt.

Fig. 110: Reprinted with permission from Pat Thiel.

Fig. 111: Erich Lessing/Art Resource, NY

Figs. 115–122: Reprinted with permission from Hans Walser, The Golden Section (Washington: The Mathematical Association of America, 2001).

Figs. 123–124: Reprinted with permission from Alan H. Guth, The Inflationary Universe (Reading: Addison-Wesley, 1997).

Figs. 125–126: Reprinted with permission from R. R. Prechter and A. J. Frost, The Elliott Wave Principle (Gainesville: New Classic Library, 1998).

Fig. 127: Reprinted with permission from Robert Fischer, Fibonacci Applications and Strategies for Traders (New York: John Wiley & Sons, 1993).

Fig. 128: The Pierpont Morgan Library/Art Resource, NY

TEXT:

Page 39: Poem on Pythagoras: Reprinted with permission from Steven Cushing.

Page 223: Poem on William Blake: Reprinted with permission from Jasper Memory.

Page 81, Poem “Constantly Mean”: Reprinted with permission from The Fibonacci Quarterly, 15.3 (1977) p. 236.

Page 194: First stanza of “The Statues”: Reprinted with permission of Scribner, an imprint of Simon & Schuster Adult Publishing Group, from The Collected Works of W. B. Yeats: Volume I, The Poems, Revised, edited by Richard J. Finneran. © 1940 by Georgie Yeats; copyright renewed 1968 by Bertha Georgie Yeats, Michael Butler Yeats, Anne Yeats.

Page 198: Limerick with number of beats indicated: Reprinted with permission from Trudi Hammel Garland, Fascinating Fibonaccis Mystery and Magic in Numbers © 1987 by Dale Seymour Publications, an imprint of Pearson Learning, a division of Pearson Education, Inc.

ADDITIONAL CREDITS:

Fig. 8: Reprinted from Robert Lawlor, Sacred Geometry (London: Thames and Hudson, 1982).

Figs. 15–16: Reprinted from Robert Lawlor, Sacred Geometry (London: Thames and Hudson, 1982).

Page 195: Poem by Katherine O'Brien: Reprinted from Robert L. Weber, Science with a Smile (Bristol: Institute of Physics Publishing, 1992). All best effort was made to track down the source holder.

Good faith efforts have been made to contact the copyright holders of the art in this book, but in a few instances the author has been unable to locate them. Such copyright holders should contact Broadway Books, a division of Random House, Inc. 1540Broadway, New York, NY 10036.

Page 195: Poem by J. A. Lindon: Reprinted from Martin Gardner, Mathematical Circus, (New York: Alfred A. Knopf, 1979). All best effort was made to track down the source holder.