Part 10 (1/2)

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).