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.

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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. 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. 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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. 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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, 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 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).