The Golden Ratio Part 10

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

Figs. 87, 88: Reprinted with permission from Erno Lendvai, Bela Bartok, An a.n.a.lysis Bela Bartok, An a.n.a.lysis of His Music of His Music (London: Kahn & Averill, 1971). (London: Kahn & Averill, 1971).

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

Fig. 90: Reprinted with permission from Joseph Schillinger, The Schillinger System of Musical Composition The Schillinger System of Musical Composition (New York: Carl Fischer, LLC, 1946). (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 The Mathematical Tourist (New York: W. H. Freeman, 1988). (New York: W. H. Freeman, 1988).

Fig. 95: Courtesy Ricardo Villa-Real. From "The Alhambra and the Generalife" by Ricardo Villa-Real Fig. 96: 96: 2002 Cordon Art-Baard, Holland. All rights reserved 2002 Cordon Art-Baard, Holland. All rights reserved Fig. 99: 99: Reprinted with permission from Richard A. Dunlap, Reprinted with permission from Richard A. Dunlap, The Golden Ratio and Fibonacci Numbers The Golden Ratio and Fibonacci Numbers (Singapore: World Scientific, 1997). (Singapore: World Scientific, 1997).

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

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

Fig. 108109: Reprinted with permission from Paul Steinhardt.

Fig. 110: Reprinted with permission from Pat Thiel.

Fig. 111: Erich Lessing/Art Resource, NY Figs. 115122: Reprinted with permission from Hans Walser, The Golden Section The Golden Section (Was.h.i.+ngton: The Mathematical a.s.sociation of America, 2001). (Was.h.i.+ngton: The Mathematical a.s.sociation of America, 2001).

Figs. 123124: Reprinted with permission from Alan H. Guth, The Inflationary Universe (Reading: The Inflationary Universe (Reading: Addison-Wesley, 1997). Addison-Wesley, 1997).

Figs. 125126: Reprinted with permission from R. R. Prechter and A. J. Frost, The Elliott Wave Principle The Elliott Wave Principle (Gainesville: New Cla.s.sic Library, 1998). (Gainesville: New Cla.s.sic Library, 1998).

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

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

Page 81, Poem "Constantly Mean": Reprinted with permission from The Fibonacci Quarterly 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 Publis.h.i.+ng Group, from The Collected Works of W. B. Yeats: Volume I, The Poems, Revised 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 Fascinating Fibonaccis Mystery and Magic in Numbers 1987 by Dale Seymour Publications, an imprint of Pearson Learning, a division of Pearson Education, Inc. 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 Sacred Geometry (London: Thames and Hudson, 1982). (London: Thames and Hudson, 1982).

Figs. 1516: Reprinted from Robert Lawlor, Sacred Geometry Sacred Geometry (London: Thames and Hudson, 1982). (London: Thames and Hudson, 1982).

Page 195: Poem by Katherine O'Brien: Reprinted from Robert L. Weber, Science with a Smile Science with a Smile (Bristol: Inst.i.tute of Physics Publis.h.i.+ng, 1992). All best effort was made to track down the source holder. (Bristol: Inst.i.tute of Physics Publis.h.i.+ng, 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 Mathematical Circus, (New York: Alfred A. Knopf, 1979). 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. 1540 Broadway, New York, NY 10036.