The Golden Ratio Part 9

where a, b, c a, b, c are arbitrary numbers. For example, in the equation are arbitrary numbers. For example, in the equation 2x 2x2+ 3x+1 = 0, a = 2, b = a = 2, b = 3, 3, c c= 1.

The general formula for the two solutions of the equation is

In the above example

In the equation we obtained for the Golden Ratio,

we have a = 1, b = -1, c = - a = 1, b = -1, c = - 1. The two solutions therefore are: 1. The two solutions therefore are:

APPENDIX 6.

The problem of the inheritance can be solved as follows. Let us denote the entire estate by E E and the share (in bezants) of each son by and the share (in bezants) of each son by x. x. (They all shared the inheritance equally.) (They all shared the inheritance equally.) The first son received:

The second son received:

Equating the two shares:

and arranging:

Therefore, each son received 6 bezants.

Subst.i.tuting in the first equation we have:

The total estate was 36 bezants. The number of sons was therefore 36/6 = 6. Fibonacci's solution reads as follows: The total inheritance has to be a number such that when 1 times 6 is added to it, it will be divisible by 1 plus 6, or 7; when 2 times 6 is added to it, it is divisible by 2 plus 6, or 8; when 3 times 6 is added, it is divisible by 3 plus 6, or 9, and so forth. The number is of 36 minus of 36 minus is is plus 1 is plus 1 is or 6; and this is the amount each son received; the total inheritance divided by the share of each son equals the number of sons, or or 6; and this is the amount each son received; the total inheritance divided by the share of each son equals the number of sons, or equals 6. equals 6.

APPENDIX 7.

The relation between the number of subobjects, n n, the length reduction factor, f f, and the dimension, D D, is

If a positive number A A is written as is written as A = A = 10 10L, then we call L the logarithm logarithm (base 10) of (base 10) of A A, and we write it as log A. A. In other words, the two equations In other words, the two equations A = A = 10 10L and L = log and L = log A A are entirely equivalent to each other. The rules of logarithms are: are entirely equivalent to each other. The rules of logarithms are: (i)The logarithm of a product of a product is the is the sum sum of the logarithms of the logarithms

(ii)The logarithm of a ratio ratio is the is the difference difference of the logarithms of the logarithms

(iii)The logarithm of a power of a number a power of a number is the power is the power times times the logarithm of the number the logarithm of the number

Since 100 = 1, we have from the definition of the logarithm that log 1 = 0. Since 10 = 1, we have from the definition of the logarithm that log 1 = 0. Since 101 = 10, 10 = 10, 102 = 100, and so on, we have that log 10 = 1, log 100 = 2, and so on. Consequently, the logarithm of any number between 1 and 10 is a number between 0 and 1; the logarithm of any number between 10 and 100 is a number between 1 and 2; and so on. = 100, and so on, we have that log 10 = 1, log 100 = 2, and so on. Consequently, the logarithm of any number between 1 and 10 is a number between 0 and 1; the logarithm of any number between 10 and 100 is a number between 1 and 2; and so on.

If we take the logarithm (base 10) of both sides in the above equation (describing the relation between n, f n, f and and D) D), we obtain

Therefore, dividing both sides by log f f

In the case of the Koch snowflake, for example, each curve contains four "subcurves" that are one-third in size; therefore n n = 4, = 4,f = and we obtain = and we obtain

APPENDIX 8.

If we examine Figure 116 Figure 116(a), we see that the condition for the two branches to touch amounts to the simple requirement that the sum of all the horizontal horizontal lengths of the ever-decreasing branches with lengths starting with lengths of the ever-decreasing branches with lengths starting with f f3 would be equal to the horizontal component of the large branch of length would be equal to the horizontal component of the large branch of length f. f. All the horizontal components are given by the total length multiplied by the cosine of 30 degrees. We therefore obtain: All the horizontal components are given by the total length multiplied by the cosine of 30 degrees. We therefore obtain:

Dividing by cos 30 we obtain

The sum on the right-hand side is the sum of an infinite geometric geometric series (each term is equal to the previous term multiplied by a constant factor) in which the first term is series (each term is equal to the previous term multiplied by a constant factor) in which the first term is f f, and the ratio of two consecutive terms is f. f. In general, the sum In general, the sum S S of an infinite geometric sequence in which the first term is of an infinite geometric sequence in which the first term is a a, and the ratio of consecutive terms q q, is equal to

For example, the sum of the sequence

in which a = a = 1 and 1 and q q = is equal to = is equal to

In our case we find from the equation above:

Dividing both sides by f f, we get

Multiplying by (1-f) and arranging, we obtain the quadratic equation:

with the positive solution

which is 1/.

APPENDIX 9.

Benford's law states that the probability P P that digit that digit D D appears in appears in the first place the first place is given by (logarithm base 10): is given by (logarithm base 10):

Therefore, for D = D = 1 1

For D = 2 D = 2

And so on. For D = D = 9, 9,

The more general law says, for example, that the probability that the first three digits are 1, 5, and 8 is:

APPENDIX 10.

Euclid's proof that infinitely many primes exist is based on the method of reductio ad absurdum. He began by a.s.suming the contradictory-that only a finite number of primes exist. If that is true, however, then one of them must be the largest prime. Let us denote that prime by P. P. Euclid then constructed a new number by the following process: He multiplied together all the primes from 2 up to (and including) Euclid then constructed a new number by the following process: He multiplied together all the primes from 2 up to (and including) P P, and then he added 1 to the product. The new number is therefore

By the original a.s.sumption, this number must be composite (not a prime), because it is obviously larger than P P, which was a.s.sumed to be the largest prime. Consequently, this number must be divisible by at least one of the existing primes. However, from its construction, we see that if we divide this number by any of the primes up to P P, this will leave a remainder 1. The implication is, that if the number is indeed composite, some prime larger than P P must divide it. However, this conclusion contradicts the a.s.sumption that must divide it. However, this conclusion contradicts the a.s.sumption that P P is the largest prime, thus completing the proof that there are infinitely many primes. is the largest prime, thus completing the proof that there are infinitely many primes.

FURTHER READINGIt is only shallow people who do not judge by appearances. The mystery of the world is the visible, not the invisible.-OSCAR W WILDE (18541900) (18541900)Most of the books and articles selected here are from the nontechnical literature. The few that are more technical were chosen on the basis of some special features. I have also listed a few websites that contain interesting material.

1: PRELUDE TO A NUMBERAckermann, F. "The Golden Section," Mathematical Monthly Mathematical Monthly 2 (1895): 260264 2 (1895): 260264Dunlap, R.A. The Golden Ratio and Fibonacci Numbers. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific, 1997. Singapore: World Scientific, 1997.Fowler, D.H. "A Generalization of the Golden Section," The Fibonacci Quarterly, The Fibonacci Quarterly, 20 (1982): 146158 20 (1982): 146158Gardner, M. The Second Scientific American Book of Mathematical Puzzles & Diversions. The Second Scientific American Book of Mathematical Puzzles & Diversions. Chicago: University of Chicago Press, 1987. Chicago: University of Chicago Press, 1987.Ghyka, M. The Geometry of Art and Life. The Geometry of Art and Life. New York: Dover Publications, 1977. New York: Dover Publications, 1977.Grattan-Guinness, I. The Norton History of the Mathematical Sciences. The Norton History of the Mathematical Sciences. New York: W W Norton & Company, 1997. New York: W W Norton & Company, 1997.Herz-Fischler, R. A Mathematical History of the Golden Number. A Mathematical History of the Golden Number. Mineola, NY: Dover Publications, 1998. Mineola, NY: Dover Publications, 1998.Hoffer W "A Magic Ratio Occurs Throughout Art and Nature," Smithsonian (December 1975): 110120.Hoggatt, V.E., Jr. "Number Theory: The Fibonacci Sequence," in Yearbook of Science and the Future. Yearbook of Science and the Future. Chicago: Encyclopaedia Britannica, 1977, 178191. Chicago: Encyclopaedia Britannica, 1977, 178191.Huntley, H.E. The Divine Proportion. The Divine Proportion. New York: Dover Publications, 1970. New York: Dover Publications, 1970.Knott, R. www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html.Knott, R. www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnet2.html.Markowski, G. "Misconceptions about the Golden Ratio," College Mathematics Journal, College Mathematics Journal, 23 (1992): 219 23 (1992): 219Ohm, M. Die reine Elementar-Mathematik. Die reine Elementar-Mathematik. Berlin: Jonas Veilags-Buchhandlung, 1835. Berlin: Jonas Veilags-Buchhandlung, 1835.Runion, G.E. The Golden Section. The Golden Section. Palo Alto: Dale Seymour Publications, 1990. Palo Alto: Dale Seymour Publications, 1990.2: THE PITCH AND THE PENTAGRAMlibrary.thinkquest.org/27890/mainIndex.html. search.britannica.com/search?query=fibonacci. search.britannica.com/search?query=fibonacci.Barrow, J.D. Pi in the Sky. Pi in the Sky. Boston: Little, Brown and Company, 1992. Boston: Little, Brown and Company, 1992.Beckmann, P. A History of . A History of . Boulder, CO: Golem Press, 1977. Boulder, CO: Golem Press, 1977.Boulger, W. "Pythagoras Meets Fibonacci," Mathematics Teacher, Mathematics Teacher, 82 (1989): 277282 82 (1989): 277282Boyer, C.B. A History of Mathematics. A History of Mathematics. New York: John Wiley & Sons, 1991. New York: John Wiley & Sons, 1991.Burkert, W. Lore and Science in Ancient Pythagoreanism. Lore and Science in Ancient Pythagoreanism. Cambridge, MA: Harvard University Press, 1972. Cambridge, MA: Harvard University Press, 1972.Conway, J.H., and Guy, R.K. The Book of Numbers. The Book of Numbers. New York: Copernicus, 1996. New York: Copernicus, 1996.Dantzig, T. Number: The Language of Science. Number: The Language of Science. New York: The Free Press, 1954. New York: The Free Press, 1954.de la Fuye, A. Le Pentagramme Pythagoricien, Sa Diffusion, Son Emploi dans le Syllaboire Cuneiform. Le Pentagramme Pythagoricien, Sa Diffusion, Son Emploi dans le Syllaboire Cuneiform. Paris: Geuthner, 1934. Paris: Geuthner, 1934.Guthrie, K.S. The Pythagorean Sourcebook and Library. The Pythagorean Sourcebook and Library. Grand Rapids, MI: Phanes Press, 1988. Grand Rapids, MI: Phanes Press, 1988.Lfrah, G. The Universal History of Numbers. The Universal History of Numbers. New York: John Wiley & Sons, 2000. New York: John Wiley & Sons, 2000.Maor, E. e: The Story of a Number. e: The Story of a Number. Princeton, NJ: Princeton University Press, 1994. Princeton, NJ: Princeton University Press, 1994.Paulos, J.A., Innumeracy. Innumeracy. New York: Vintage Books, 1988. New York: Vintage Books, 1988.Pickover, C.A. Wonders of Numbers. Wonders of Numbers. Oxford: Oxford University Press, 2001. Oxford: Oxford University Press, 2001.Schimmel, A. The Mystery of Numbers. The Mystery of Numbers. Oxford: Oxford University Press, 1994. Oxford: Oxford University Press, 1994.Schmandt-Besserat, D. "The Earliest Precursor of Writing," Scientific American (June 1978): 3847.Schmandt-Besserat, D. "Reckoning Before Writing," Archaeology, Archaeology, 3233 (1979): 2231 3233 (1979): 2231Singh, S. Fermat's Enigma. Fermat's Enigma. New York: Anchor Books, 1997. New York: Anchor Books, 1997.Stanley, T. Pythagoras. Pythagoras. Los Angeles: The Philosophical Research Society, 1970. Los Angeles: The Philosophical Research Society, 1970.Strohmeier, J., and Westbrook, P. Divine Harmony. Divine Harmony. Berkeley, CA: Berkeley Hills Books, 1999. Berkeley, CA: Berkeley Hills Books, 1999.Turnbull, H.W. The Great Mathematicians. The Great Mathematicians. New York: Barnes & n.o.ble, 1993. New York: Barnes & n.o.ble, 1993.von Fritz, K. "The Discovery of Incommensurability of Hipposus of Metapontum," Annals of Mathematics, Annals of Mathematics, 46 (1945): 242264 46 (1945): 242264Wells, D. Curious and Interesting Numbers. Curious and Interesting Numbers. London: Penguin Books, 1986. London: Penguin Books, 1986.Wells, D. Curious and Interesting Mathematics. Curious and Interesting Mathematics. London: Penguin Books, 1997. London: Penguin Books, 1997.3: UNDER A STAR-Y-POINTING PYRAMID?Beard, R.S. "The Fibonacci Drawing Board Design of the Great Pyramid of Gizeh," The Fibonacci Quarterly, The Fibonacci Quarterly, 6 (1968): 8587 6 (1968): 8587Burton, D.M. The History of Mathematics: An Introduction. The History of Mathematics: An Introduction. Boston: Allyn and Bacon, 1985. Boston: Allyn and Bacon, 1985.Doczi, O. The Power of Limits. The Power of Limits. Boston: Shambhala, 1981. Boston: Shambhala, 1981.Fischler, R. "Theories Mathematiques de la Grande Pyramide," Crux Mathematicorum, Crux Mathematicorum, 4 (1978): 122129 4 (1978): 122129Fischler, R. "What Did Herodotus Really Say? or How to Build (a Theory of) the Great Pyramid," Environment and Planning B, Environment and Planning B, 6 (1979): 8993 6 (1979): 8993Gardner, M. Fads and Fallacies in the Name of Science. Fads and Fallacies in the Name of Science. New York: Dover Publications, 1957. New York: Dover Publications, 1957.Gazale, M.J., Gnomon. Gnomon. Princeton, NJ: Princeton University Press, 1999. Princeton, NJ: Princeton University Press, 1999.Gillings, R.J. Mathematics in the Time of the Pharaohs. Mathematics in the Time of the Pharaohs. New York: Dover Publications, 1972. New York: Dover Publications, 1972.Goff, B. Symbols of Prehistoric Mesopotamia. Symbols of Prehistoric Mesopotamia. New Haven, CT Yale University Press, 1963. New Haven, CT Yale University Press, 1963.Hedian, H. "The Golden Section and the Artist," The Fibonacci Quarterly, The Fibonacci Quarterly, 14 (1976): 406418 14 (1976): 406418Lawlor, R. Sacred Geometry. Sacred Geometry. London: Thames and Hudson, 1982. London: Thames and Hudson, 1982.Mendelssohn, K. The Riddle of the Pyramids. The Riddle of the Pyramids. New York: Praeger Publishers, 1974. New York: Praeger Publishers, 1974.Petrie, W. The Pyramids and Temples of Gizeh. The Pyramids and Temples of Gizeh. London: Field and Tuer, 1883. London: Field and Tuer, 1883.Piazzi Smyth, C. The Great Pyramid. The Great Pyramid. New York: Gramercy Books, 1978. New York: Gramercy Books, 1978.Schneider, M.S. A Beginner's Guide to Constructing the Universe. A Beginner's Guide to Constructing the Universe. New York: Harper Perennial, 1995. New York: Harper Perennial, 1995.Spence, K. "Ancient Egyptian Chronology and the Astronomical Orientation of the Pyramids," Nature, Nature, 408 (2000): 320324 408 (2000): 320324Stewart, I. "Counting the Pyramid Builders," Scientific American Scientific American (September 1998): 98100. (September 1998): 98100.Verheyen, H.F. "The Icosahedral Design of the Great Pyramid," in Fivefold Symmetry. Fivefold Symmetry. Singapore: World Scientific, 1992, 333360. Singapore: World Scientific, 1992, 333360.Wier, S.K. "Insights from Geometry and Physics into the Construction of Egyptian Old Kingdom Pyramids," Cambridge Archaeological Journal, Cambridge Archaeological Journal, 6 (1996): 150163 6 (1996): 1501634: THE SECOND TREASUREBorissavlievitch, M. The Golden Number and the Scientific Aesthetics of Architecture. The Golden Number and the Scientific Aesthetics of Architecture. London: Alec Tiranti, 1958. London: Alec Tiranti, 1958.Bruckman, P.S. "Constantly Mean," The Fibonacci Quarterly, The Fibonacci Quarterly, 15 (1977): 236 15 (1977): 236c.o.xeter, H. S. M. Introduction to Geometry. Introduction to Geometry. New York: John Wiley & Sons, 1963. New York: John Wiley & Sons, 1963.Cromwell, P.R. Polyhedra. Polyhedra. Cambridge: Cambridge University Press, 1997. Cambridge: Cambridge University Press, 1997.Dixon, K. Mathographics. Mathographics. New York: Dover Publications, 1987. New York: Dover Publications, 1987.Ghyka, M. L'Esthetique des proportions dans la nature et dans les arts. L'Esthetique des proportions dans la nature et dans les arts. Paris: Gallimard, 1927. Paris: Gallimard, 1927.Heath, T. A History of Greek Mathematics. A History of Greek Mathematics. New York: Dover Publications, 1981. New York: Dover Publications, 1981.Heath, T. The Thirteen Books of Euclid's Elements. The Thirteen Books of Euclid's Elements. New York: Dover Publications, 1956. New York: Dover Publications, 1956.Jowett, B. The Dialogues of Plato. The Dialogues of Plato. Oxford: Oxford University Press, 1953. Oxford: Oxford University Press, 1953.Kraut, R. The Cambridge Companion to Plato. The Cambridge Companion to Plato. Cambridge: Cambridge University Press, 1992. Cambridge: Cambridge University Press, 1992.La.s.serre, F. The Birth of Mathematics in the Age of Plato. The Birth of Mathematics in the Age of Plato. London: Hutchinson, 1964. London: Hutchinson, 1964.Pappas, T. The Joy of Mathematics. The Joy of Mathematics. San Carlos, CA: Wide World Publis.h.i.+ng, 1989. San Carlos, CA: Wide World Publis.h.i.+ng, 1989.Trachtenberg, M., andHyman, I. Architecture: From Prehistory to Post Modernism/The Western Tradition. Architecture: From Prehistory to Post Modernism/The Western Tradition. New York: Harry N. Abrams, 1986. New York: Harry N. Abrams, 1986.Zeising, A. Der goldne Schnitt. Der goldne Schnitt. Halle: Druck von E. Blochmann & Son in Dresden, 1884. Halle: Druck von E. Blochmann & Son in Dresden, 1884.5: SON OF GOOD NATUREcedar.evansville.eduh ck6/index.html.Adler, I., Barabe, D., andJean, R.V. "A History of the Study of Phyllotaxis," Annals of Botany, Annals of Botany, 80 (1997): 231244 80 (1997): 231244Basin, S.L. "The Fibonacci Sequence as It Appears in Nature," The Fibonacci Quarterly, The Fibonacci Quarterly, 1 (1963): 5364 1 (1963): 5364Brousseau, Brother A., An Introduction to Fibonacci Discovery. An Introduction to Fibonacci Discovery. Aurora, SD: The Fibonacci a.s.sociation, 1965. Aurora, SD: The Fibonacci a.s.sociation, 1965.Bruckman, P.S. "Constantly Mean," The Fibonacci Quarterly, The Fibonacci Quarterly, 15 (1977): 236 15 (1977): 236c.o.xeter, H. S. M. "The Golden Section, Phyllotaxis, and Wythoffs Game," Scripta Mathematica, Scripta Mathematica, 19 (1953): 135143 19 (1953): 135143c.o.xeter, H. S. M. Introduction to Geometry. Introduction to Geometry. New York: John Wiley & Sons, 1963. New York: John Wiley & Sons, 1963.Cook, T A., The Curves of Life. The Curves of Life. New York: Dover Publications, 1979. New York: Dover Publications, 1979.Devlin, K. Mathematics. Mathematics. New York: Columbia University Press, 1999. New York: Columbia University Press, 1999.Douady, S., andCouder, Y., "Phyllotaxis as a Physical Self-Organized Process," Physical Review Letters, Physical Review Letters, 68 (1992): 20982101 68 (1992): 20982101Dunlap, R.A. The Golden Ratio and Fibonacci Numbers. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific, 1997. Singapore: World Scientific, 1997.Fibonacci, L.P. The Book of Squares. The Book of Squares. Orlando, FL: Academic Press, 1987. Orlando, FL: Academic Press, 1987."The Fibonacci Numbers," Time, April 4, 1969, 4950.Gardner, M. Mathematical Circus. Mathematical Circus. New York: Alfred A. Knopf, 1979. New York: Alfred A. Knopf, 1979.Gardner, M. "The Multiple Fascination of the Fibonacci Sequence," Scientific American (March 1969): 116120.Garland, T H., Fascinating Fibonaccis. Fascinating Fibonaccis. White Plains, NY: Dale Seymour Publications, 1987. White Plains, NY: Dale Seymour Publications, 1987.Gies, J., andGies, F., Leonard of Pisa and the New Mathematics of the Middle Ages. Leonard of Pisa and the New Mathematics of the Middle Ages. New York: Thomas Y Crowell Company, 1969. New York: Thomas Y Crowell Company, 1969.Hoggatt,V. E. Jr. "Number Theory: The Fibonacci Sequence," Chicago: Encyclopaedia Britannica, Yearbook of Science and the Future, 1977, 178191Hoggatt, V.E. Jr., and Bicknell-Johnson, M. "Reflections Across Two and Three Gla.s.s Plates," The Fibonacci Quarterly, The Fibonacci Quarterly, 17 (1979): 118142 17 (1979): 118142Horadam, A.F. "Eight Hundred Years Young," The Australian Mathematics Teacher, The Australian Mathematics Teacher, 31 (1975): 123134 31 (1975): 123134Jean, R.V. Mathematical Approach to Pattern and Form in Plant Growth. Mathematical Approach to Pattern and Form in Plant Growth. New York: John Wiley & Sons, 1984. New York: John Wiley & Sons, 1984.O'Connor, J.J., and Robertson, E.F. www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fibonacci.html. www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fibonacci.html.Pickover, C.A. Keys to Infinity. Keys to Infinity. New York: John Wiley & Sons, 1995. New York: John Wiley & Sons, 1995.Rivier, N., Occelli, R., Pantaloni, J., and Lissowdki, A. "Structure of Binard Convection Cells, Phyllotaxis and Crystallography in Cylindrical Symmetry," Journal Physique, Journal Physique, 45 (1984): 4963 45 (1984): 4963Singh, P. "The So-Called Fibonacci Numbers in Ancient and Medieval India," Historia Mathematica, Historia Mathematica, 12 (1985): 229244 12 (1985): 229244Smith, D.E. History of Mathematics. History of Mathematics. New York: Dover Publications, 1958. New York: Dover Publications, 1958.Stewart, I. "Fibonacci Forgeries," Scientific American (May 1995): 102105.Stewart, I. Life's Other Secret. Life's Other Secret. New York: John Wiley & Sons, 1998. New York: John Wiley & Sons, 1998.Thompson, D.W. On Growth and Form. On Growth and Form. New York: Dover Publications, 1992. New York: Dover Publications, 1992.Vajda, S. Fibonacci & Lucas Numbers, and the Golden Section. Fibonacci & Lucas Numbers, and the Golden Section. Chichester: Ellis Horwood Limited, 1989. Chichester: Ellis Horwood Limited, 1989.Vorob'ev, N.N., Fibonacci Numbers. Fibonacci Numbers. New York: Blaisdell, 1961. New York: Blaisdell, 1961.6: THE DIVINE PROPORTIONAra.s.se, D. Leonardo Da Vinci. Leonardo Da Vinci. New York: Konecky & Konecky, 1998. New York: Konecky & Konecky, 1998.Beer, A. and Beer, eds., P., Kepler: Four Hundred Years. Vistas in Astronomy, Kepler: Four Hundred Years. Vistas in Astronomy, vol. 18, New York: Pergamon Press, 1975. vol. 18, New York: Pergamon Press, 1975.Calvesi, M. Piero Della Francesca. Piero Della Francesca. New York: Rizzoli, 1998. New York: Rizzoli, 1998.Caspar, M. Kepler. Kepler. New York: Dover Publications, 1993. New York: Dover Publications, 1993.Cromwell, P.R. Polyhedra. Polyhedra. Cambridge: Cambridge University Press, 1997. Cambridge: Cambridge University Press, 1997.Gingerich, O. "Kepler, Galilei, and the Harmony of the World," in Music and Science in the Age of Galileo, ed. V. Coeltho, 4563. Dordrecht: Kluwer, 1992.Gingerich, O. "Kepler, Johannes," in Dictionary of Scientific Biography, in Dictionary of Scientific Biography, ed. Charles Coulston Gillespie, vol. 7, 289312. New York: Scribners, 1973. ed. Charles Coulston Gillespie, vol. 7, 289312. New York: Scribners, 1973.Ginzburg, C. The Enigma of Piero. The Enigma of Piero. London: Verso, 2000. London: Verso, 2000.James J. The Music of the Spheres. The Music of the Spheres. New York: Copernicus, 1993. New York: Copernicus, 1993.Jardine, N. The Birth of History and Philosophy of Science: Kepler's "A Defense of Tycho against Ursus" with Essays on Its Provenance and Significance. The Birth of History and Philosophy of Science: Kepler's "A Defense of Tycho against Ursus" with Essays on Its Provenance and Significance. Cambridge: Cambridge University Press, 1984. Cambridge: Cambridge University Press, 1984.Kepler, J. The Harmony of the World. The Harmony of the World. Philadelphia: American Philosophical Society, 1997. Philadelphia: American Philosophical Society, 1997.Kepler, J. Mysterium Cosmographic.u.m. Mysterium Cosmographic.u.m. New York: Abaris Books, 1981. NY: Artabras/Reynal and Company, Leonardo Da Vinci. 1938. New York: Abaris Books, 1981. NY: Artabras/Reynal and Company, Leonardo Da Vinci. 1938.MacKinnon, N. "The Portrait of Fra Luca Pacioli," Mathematical Gazette, Mathematical Gazette, 77 (1993): 130219 77 (1993): 130219Martens, R. Kepler's Philosophy and the New Astronomy. Kepler's Philosophy and the New Astronomy. Princeton, NJ: Princeton University Press, 2000. Princeton, NJ: Princeton University Press, 2000.O'Connor, J.J., andRobertson, E.F. www-history.mcs.st-andrews.ac.uk/history/Mathe-maticians/Pacioli.html/Durer.html.Pacioli L. Divine Proportion. Divine Proportion. Paris: Librairie du Compagnonnage, 1988. Paris: Librairie du Compagnonnage, 1988.Pauli, W. "The Influence of Archetypal Ideas on the Scientific Theories of Kepler," in The Interpretation of Nature and the Psyche, The Interpretation of Nature and the Psyche, 147240. New York: Parthenon, 1955. 147240. New York: Parthenon, 1955.Stephenson, B. The Music of the Spheres: Kepler's Harmonic Astronomy. The Music of the Spheres: Kepler's Harmonic Astronomy. Princeton, NJ: Princeton University Press, 1994. Princeton, NJ: Princeton University Press, 1994.Strieder, P. Albrecht Durer. Albrecht Durer. New York: Abaris Books, 1982. New York: Abaris Books, 1982.Taylor, R.E. No Royal Road. No Royal Road. Chapel Hill: University of North Carolina Press, 1942. Chapel Hill: University of North Carolina Press, 1942.Voelkel, J.R. Johannes Kepler. Johannes Kepler. Oxford: Oxford University Press, 1999. Oxford: Oxford University Press, 1999.Westman, R.A. "The Astronomer's Role in the Sixteenth Century: A Preliminary Survey," History of Science, History of Science, 18 (1980): 105147 18 (1980): 1051477: PAINTERS AND POETS HAVE EQUAL LICENSEAltschuler, E.L. Bacha.n.a.lia. Bacha.n.a.lia. Boston: Little, Brown and Company, 1994. Boston: Little, Brown and Company, 1994.d'Arcais, F F Giotto. Giotto. New York: Abbeville Press Publishers, 1995. New York: Abbeville Press Publishers, 1995.Bellosi, L. Cimabue. Cimabue. New York: Abbeville Press Publishers, 1998. New York: Abbeville Press Publishers, 1998.Bergamini, D. Mathematics. Mathematics. New York: Time Incorporated, 1963. New York: Time Incorporated, 1963.Bois, Y.-A., Joosten, J., Rudenstine, A.Z., andJanssen, H. Piet Mondrian. Piet Mondrian. Boston: Little, Brown and Company, 1995. Boston: Little, Brown and Company, 1995.Boring, E.G. A History of Experimental Psychology. A History of Experimental Psychology. New York: Appleton-Century-Crofts, 1957. New York: Appleton-Century-Crofts, 1957.Bouleau, C. The Painter's Secret Geometry. The Painter's Secret Geometry. New York: Harcourt, Brace & World, 1963. New York: Harcourt, Brace & World, 1963.Curchin, L., and Fischler, R. "Hero of Alexandria's Numerical Treatment of Division in Extreme and Mean Ratio and Its Implications," Phoenix, Phoenix, 35 (1981): 129133 35 (1981): 129133Curtis, W. J. R. Le Corbusier: Ideas and Forms. Le Corbusier: Ideas and Forms. Oxford: Phaidon, 1986. Oxford: Phaidon, 1986.Duckworth, G.E. Structural Patterns and Proportions in Vergil's Aeneid. Structural Patterns and Proportions in Vergil's Aeneid. Ann Arbor: University of Michigan Press, 1962. Ann Arbor: University of Michigan Press, 1962.Emmer, M. The Visual Mind. The Visual Mind. Cambridge, MA: MIT Press, 1993. Cambridge, MA: MIT Press, 1993.Fancher, R.E. Pioneers of Psychology. Pioneers of Psychology. New York: W. W. Norton & Company, 1990. New York: W. W. Norton & Company, 1990.Fechner, G. T Vorschule der Aesthetik. Vorschule der Aesthetik. Leipzig: Breitkopf & Hartel, 1876. Leipzig: Breitkopf & Hartel, 1876.Fischler, R. "How to Find the 'Golden Number' Without Really Trying," The Fibonacci Quarterly, The Fibonacci Quarterly, 19 (1981): 406410 19 (1981): 406410Fischler, R. "On the Application of the Golden Ratio in the Visual Arts," Leonardo, Leonardo, 14 (1981): 3132 14 (1981): 3132Fischler, R. "The Early Relations.h.i.+p of Le Corbusier to the Golden Number," Environment and Planning B, Environment and Planning B, 6 (1979): 95103 6 (1979): 95103G.o.dkewitsch, M. "The Golden Section: An Artifact of Stimulus Range and Measure of Preference," American Journal of Psychology, American Journal of Psychology, 87 (1974): 269277 87 (1974): 269277Hambidge, J. The Elements of Dynamic Symmetry. The Elements of Dynamic Symmetry. New York: Dover Publications, 1967. New York: Dover Publications, 1967.Herz-Fischler, R. "An Examination of Claims Concerning Seurat and the Golden Number," Gazette des Beaux-Arts, Gazette des Beaux-Arts, 125 (1983): 109112 125 (1983): 109112Herz-Fischler, R. "Le Corbusier's 'regulating lines' for the villa at Garches (1927) and other early works," Journal of the Society of Architectural Historians, Journal of the Society of Architectural Historians, 43 (1984): 5359 43 (1984): 5359Herz-Fischler, R. "Le Nombre d'or en France de 1896 a 1927," Revue de l'Art, Revue de l'Art, 118 (1997): 916 118 (1997): 916Hockney D. Secret Knowledge. Secret Knowledge. New York: Viking Studio, 2001. New York: Viking Studio, 2001.Howat, R. Debussy in Proportion. Debussy in Proportion. Cambridge: Cambridge University Press, 1983. Cambridge: Cambridge University Press, 1983.Kepes, G. Module, Proportion, Symmetry, Rhythm. Module, Proportion, Symmetry, Rhythm. New York: George Braziller, 1966. New York: George Braziller, 1966.Larson, P. "The Golden Section in the Earliest Notated Western Music," The Fibonacci Quarterly, The Fibonacci Quarterly, 16 (1978): 513515 16 (1978): 513515Le Courbusier. Modulor I and II. Modulor I and II. Cambridge, MA: Harvard University Press, 1980. Cambridge, MA: Harvard University Press, 1980.Lendvai, E. Bela Bartok: An a.n.a.lysis of His Music. Bela Bartok: An a.n.a.lysis of His Music. London: Kahn & Averill, 1971. London: Kahn & Averill, 1971.Lowman, E.A. "Some Striking Proportions in the Music of Bela Bartok," The Fibonacci Quarterly, The Fibonacci Quarterly, 9 (1971): 527537 9 (1971): 527537Marevna. Life with the Painters of La Ruche. Life with the Painters of La Ruche. New York: Macmillan Publis.h.i.+ng Co., 1974. New York: Macmillan Publis.h.i.+ng Co., 1974.McMa.n.u.s, I.C. "The Aesthetics of Simple Figures," British Journal of Psychology, British Journal of Psychology, 71 (1980): 505524 71 (1980): 505524Nims, J. F Western Wind. Western Wind. New York: McGraw-Hill, 1992. New York: McGraw-Hill, 1992.Nuland, S.B. Leonardo da Vinci. Leonardo da Vinci. New York: Viking, 2000. New York: Viking, 2000.Osborne, H. ed. The Oxford Companion to Art. The Oxford Companion to Art. Oxford: Oxford University Press, 1970. Oxford: Oxford University Press, 1970.Putz, J. F "The Golden Section and the Piano Sonatas of Mozart," Mathematics Magazine, Mathematics Magazine, 68 (1995): 275282 68 (1995): 275282Sadie, S. The New Grove Dictionary of Music and Musicians. The New Grove Dictionary of Music and Musicians. New York: Grove, 2001. New York: Grove, 2001.Schiffman, H.R., and Bobka, D.J. "Preference in Linear Part.i.tioning: The Golden Section Reexamined," Perception & Psychophysics, Perception & Psychophysics, 24 (1978): 102103 24 (1978): 102103Schillinger, F. Joseph Schillinger. Schillinger. New York: Da Capo Press, 1976. New York: Da Capo Press, 1976.Schillinger, J. The Mathematical Basis of the Arts. The Mathematical Basis of the Arts. New York: Philosophical Library, 1948. New York: Philosophical Library, 1948.Schwarz, L. "The Art Historian's Computer," Scientific American (April 1995): 106111.Somfai, L. Bela Bartok: Compositions, Concepts, and Autograph Sources. Bela Bartok: Compositions, Concepts, and Autograph Sources. Berkeley: University of California Press, 1996. Berkeley: University of California Press, 1996.Svensson, L. T "Note on the Golden Section," Scandinavian Journal of Psychology, Scandinavian Journal of Psychology, 18 (1977): 7980 18 (1977): 7980Tatlow, R., and Griffiths, P. "Numbers and Music," in The New Grove Dictionary of Music and Musicians, The New Grove Dictionary of Music and Musicians, 18 (2001): 231236 18 (2001): 231236Watson, R.I. The Great Psychologists. The Great Psychologists. Philadelphia: J. B. Lippincott Company, 1978. Philadelphia: J. B. Lippincott Company, 1978.White, M. Leonardo the First Scientist. Leonardo the First Scientist. London: Little, Brown and Company, 2000. London: Little, Brown and Company, 2000.Woodworth, R.S., andSchlosberg, H. Experimental Psychology. Experimental Psychology. New York: Holt, Rinehart and Winston, 1965. New York: Holt, Rinehart and Winston, 1965.Zusne, L. Visual Perception of Form. Visual Perception of Form. New York: Academic Press, 1970. New York: Academic Press, 1970.8: FROM THE TILES TO THE HEAVENSCohen, J., andStewart, I. The Collapse of Chaos. Discovering Simplicity in a Complex World. The Collapse of Chaos. Discovering Simplicity in a Complex World. New York: Penguin Books, 1995. New York: Penguin Books, 1995.Fischer, R. Financial Applications and Strategies for Traders. Financial Applications and Strategies for Traders. New York: John Wiley & Sons, 1993. New York: John Wiley & Sons, 1993.Gardner, M. Penrose Tiles to Trapdoor Ciphers. Penrose Tiles to Trapdoor Ciphers. New York: W H. Freeman and Company, 1989. New York: W H. Freeman and Company, 1989.Gleick, J. Chaos. Chaos. New York: Penguin Books, 1987. New York: Penguin Books, 1987.Lesmoir-Gordon, N., Rood, W, andEdney, R. Introducing Fractal Geometry. Introducing Fractal Geometry. Cambridge: Icon Books, 2000. Cambridge: Icon Books, 2000.Mandelbrot, B.B. Fractal Geometry of Nature. Fractal Geometry of Nature. New York: W H. Freeman and Company, 1988. New York: W H. Freeman and Company, 1988.Mandelbrot, B.B. "A Multifractal Walk Down Wall Street," Scientific American Scientific American (February 1999): 7073. (February 1999): 7073.Matthews, R. "The Power of One," New Scientist, July 10, 1999, 2730.Peitgen, H.-O., Jurgens, H., andSaupe, D. Chaos and Fractals. Chaos and Fractals. New York: Springer-Verlag, 1992. New York: Springer-Verlag, 1992.Peterson, I. "Fibonacci at Random," Science News, Science News, 155 (1999): 376377 155 (1999): 376377Peterson, I. The Mathematical Tourist. The Mathematical Tourist. New York: W H. Freeman and Company, 1988. New York: W H. Freeman and Company, 1988.Peterson, I. "A Quasicrystal Construction Kit," Science News, Science News, 155 (1999): 6061 155 (1999): 6061Prechter, R.R. Jr., and Frost, A.J. Elliot Wave Principle. Elliot Wave Principle. Gainesville, GA: New Cla.s.sics Library, 1978. Gainesville, GA: New Cla.s.sics Library, 1978.Schroeder, M. Fractals, Chaos, Power Laws. Fractals, Chaos, Power Laws. New York: W H. Freeman and Company, 1991. New York: W H. Freeman and Company, 1991.Steinhardt, P.J., Jeong, H.-C, Saitoh, K., Tanaka, M., Abe, E., andTsai, A.P. "Experimental Verification of the Quasi-Unit-Cell Model of Quasicrystal Structure," Nature, Nature, 396 (1998): 5557 396 (1998): 5557Stewart, I. Does G.o.d Play Dice? Does G.o.d Play Dice? London: Penguin Books, 1997. London: Penguin Books, 1997.Walser, H. The Golden Section. The Golden Section. Was.h.i.+ngton, DC: The Mathematical a.s.sociation of America, 2001. Was.h.i.+ngton, DC: The Mathematical a.s.sociation of America, 2001.9: IS G.o.d A MATHEMATICIAN?Baierlein, R. Newton to Einstein: The Trail of Light. Newton to Einstein: The Trail of Light. Cambridge: Cambridge University Press, 1992. Cambridge: Cambridge University Press, 1992.Barrow, J.D. Impossibility. Impossibility. Oxford: Oxford University Press, 1998. Oxford: Oxford University Press, 1998.Chandrasekhar, S. Truth and Beauty. Truth and Beauty. Chicago: University of Chicago Press, 1987. Chicago: University of Chicago Press, 1987.Chown, M. "Principia Mathematica III," New Scientist, New Scientist, August 25, 2001, 4447. August 25, 2001, 4447.Davis, P.J., and Hersh, R. The Mathematical Experience. The Mathematical Experience. Boston: Houghton Mifflin Company, 1998. Boston: Houghton Mifflin Company, 1998.Dehaene, S. The Number Sense. The Number Sense. Oxford: Oxford University Press, 1997. Oxford: Oxford University Press, 1997.Deutsch, D. The Fabric of Reality. The Fabric of Reality. New York: Penguin Books, 1997. New York: Penguin Books, 1997.Hersh, R. What Is Mathematics, Really? What Is Mathematics, Really? New York: Oxford University Press, 1997. New York: Oxford University Press, 1997.Hill, T.P. "The First Digit Phenomenon," American Scientist, American Scientist, 86 (1998): 358363 86 (1998): 358363Kleene, S.C. "Foundations of Mathematics," Chicago: Encyclopaedia Britannica Encyclopaedia Britannica (1971), 10971103. (1971), 10971103.Lakatos, I. Mathematics, Science and Epistemology. Mathematics, Science and Epistemology. Cambridge: Cambridge University Press, 1978. Cambridge: Cambridge University Press, 1978.Lakoff, G., andNunez, R., Where Mathematics Comes From. Where Mathematics Comes From. New York: Basic Books, 2000. New York: Basic Books, 2000.Livio, M. The Accelerating Universe. The Accelerating Universe. New York: John Wiley & Sons, 2000. New York: John Wiley & Sons, 2000.Maor, E. To Infinity and Beyond: A Cultural History of the Infinite. To Infinity and Beyond: A Cultural History of the Infinite. Princeton, NJ: Princeton University Press, 1987. Princeton, NJ: Princeton University Press, 1987.Matthews, R. "The Power of One," New Scientist, New Scientist, July 10, 1999, 2630. July 10, 1999, 2630.Penrose, R. The Emperor's New Mind. The Emperor's New Mind. Oxford: Oxford University Press, 1989. Oxford: Oxford University Press, 1989.Penrose, R. Shadows of the Mind. Shadows of the Mind. Oxford: Oxford University Press, 1994. Oxford: Oxford University Press, 1994.Pickover, C.A. The Loom of G.o.d. The Loom of G.o.d. Cambridge, MA: Perseus Books, 1997. Cambridge, MA: Perseus Books, 1997.Popper, K.R., and Eccles, J.C. The Self and Its Brain. The Self and Its Brain. New York: Springer International, 1977. New York: Springer International, 1977.Raimi, R. "The Peculiar Distribution of the First Digit," Scientific American Scientific American (December 1969): 109119. (December 1969): 109119.Raskin, J. www.jefraskin.com/forjef2/jefweb-compiled/unpublished/effectiveness_ mathematics/Robinson, A. "From a Formalist's Point of View," Dialectica, Dialectica, 23, (1969): 4549. 23, (1969): 4549.Russell, B. A History of Western Philosophy. A History of Western Philosophy. New York: Simon and Schuster, 1945. New York: Simon and Schuster, 1945.Russell, B. Human Knowledge, Its Scope and Its Limits. Human Knowledge, Its Scope and Its Limits. New York: Simon and Schuster, 1948. New York: Simon and Schuster, 1948.Weisstein, E. matworld.wolfram.com/BenfordsLaw.html.Wolfram, S., A New Kind of Science. A New Kind of Science. Champaign, IL: Wolfram Media, 2002. Champaign, IL: Wolfram Media, 2002.

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 Sh.e.l.l Museum Fig. 5: Chester Dale Collection, Photograph 2002 Board of Trustees, National Gallery of Art, Was.h.i.+ngton, 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 Pi In the Sky (Oxford: Oxford University Press, 1992). (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 Mathographics (Mineola: Dover Publications, 1987). (Mineola: Dover Publications, 1987).

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

Fig. 23, top: Alison Frantz Photographic Collection, American School of Cla.s.sical Studies at Athens Fig. 28: 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.

Figs. 3132: 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.

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

Fig. 36: Reprinted with permission from N. Rivier et al., J. Physique 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 Pattern and Design with Dynamic Symmetry (Mineola: Dover Publications, 1967). (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, Mich.e.l.le Emmer Editor, The Visual Mind: Art and Mathematics The Visual Mind: Art and Mathematics (Cambridge: the MIT Press,1993). (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, d.i.c.k Fund, 1943 Fig. 57: Reprinted with permission from David Wells, The Penguin Book of Curious and Interesting Mathematics The Penguin Book of Curious and Interesting Mathematics (London: The Penguin Group, 1997), copyright David Wells, 1997. (London: The Penguin Group, 1997), copyright David Wells, 1997.

Figs. 6869: Kindly provided by the Inst.i.tute 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 The College Mathematics Journal, 23, 2 (1992).