Part 9 (2/2)

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.

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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. /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. 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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. 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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. /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).

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