FURTHER READING - The Golden Ratio: The Story of Phi, the World's Most Astonishing Number - Mario Livio

The Golden Ratio: The Story of Phi, the World's Most Astonishing Number - Mario Livio (2003)


It is only shallow people who do not judge by
appearances. The mystery of the world is the visible,
not the invisible.

—OSCAR WILDE (1854-1900)

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.


Ackermann, F. “The Golden Section,” Mathematical Monthly 2 (1895): 260-264

Dunlap, R.A. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific, 1997.

Fowler, D.H. “A Generalization of the Golden Section,” The Fibonacci Quarterly, 20 (1982): 146-158

Gardner, M. The Second Scientific American Book of Mathematical Puzzles & Diversions. Chicago: University of Chicago Press, 1987.

Ghyka, M. The Geometry of Art and Life. New York: Dover Publications, 1977.

Grattan-Guinness, I. The Norton History of the Mathematical Sciences. New York: W W Norton & Company, 1997.

Herz-Fischler, R. A Mathematical History of the Golden Number. Mineola, NY: Dover Publications, 1998.

Hoffer W “A Magic Ratio Occurs Throughout Art and Nature,” Smithsonian (December 1975): 110-120.

Hoggatt, V.E., Jr. “Number Theory: The Fibonacci Sequence,” in Yearbook of Science and the Future. Chicago: Encyclopaedia Britannica, 1977, 178-191.

Huntley, H.E. The Divine Proportion. 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, 23 (1992): 2-19

Ohm, M. Die reine Elementar-Mathematik. Berlin: Jonas Veilags-Buchhandlung, 1835.

Runion, G.E. The Golden Section. Palo Alto: Dale Seymour Publications, 1990.


library.thinkquest.org/27890/mainIndex.html. search.britannica.com/search?query=fibonacci.

Barrow, J.D. Pi in the Sky. Boston: Little, Brown and Company, 1992.

Beckmann, P. A History of π. Boulder, CO: Golem Press, 1977.

Boulger, W. “Pythagoras Meets Fibonacci,” Mathematics Teacher, 82 (1989): 277-282

Boyer, C.B. A History of Mathematics. New York: John Wiley & Sons, 1991.

Burkert, W. Lore and Science in Ancient Pythagoreanism. Cambridge, MA: Harvard University Press, 1972.

Conway, J.H., and Guy, R.K. The Book of Numbers. New York: Copernicus, 1996.

Dantzig, T. Number: The Language of Science. New York: The Free Press, 1954.

de la Füye, A. Le Pentagramme Pythagoricien, Sa Diffusion, Son Emploi dans le Syllaboire Cuneiform. Paris: Geuthner, 1934.

Guthrie, K.S. The Pythagorean Sourcebook and Library. Grand Rapids, MI: Phanes Press, 1988.

Lfrah, G. The Universal History of Numbers. New York: John Wiley & Sons, 2000.

Maor, E. e: The Story of a Number. Princeton, NJ: Princeton University Press, 1994.

Paulos, J.A., Innumeracy. New York: Vintage Books, 1988.

Pickover, C.A. Wonders of Numbers. Oxford: Oxford University Press, 2001.

Schimmel, A. The Mystery of Numbers. Oxford: Oxford University Press, 1994.

Schmandt-Besserat, D. “The Earliest Precursor of Writing,” Scientific American (June 1978): 38-47.

Schmandt-Besserat, D. “Reckoning Before Writing,” Archaeology, 32-33 (1979): 22-31

Singh, S. Fermat's Enigma. New York: Anchor Books, 1997.

Stanley, T. Pythagoras. Los Angeles: The Philosophical Research Society, 1970.

Strohmeier, J., and Westbrook, P. Divine Harmony. Berkeley, CA: Berkeley Hills Books, 1999.

Turnbull, H.W. The Great Mathematicians. New York: Barnes & Noble, 1993.

von Fritz, K. “The Discovery of Incommensurability of Hipposus of Metapontum,” Annals of Mathematics, 46 (1945): 242-264

Wells, D. Curious and Interesting Numbers. London: Penguin Books, 1986.

Wells, D. Curious and Interesting Mathematics. London: Penguin Books, 1997.


Beard, R.S. “The Fibonacci Drawing Board Design of the Great Pyramid of Gizeh,” The Fibonacci Quarterly, 6 (1968): 85-87

Burton, D.M. The History of Mathematics: An Introduction. Boston: Allyn and Bacon, 1985.

Doczi, O. The Power of Limits. Boston: Shambhala, 1981.

Fischler, R. “Théories Mathématiques de la Grande Pyramide,” Crux Mathematicorum, 4 (1978): 122-129

Fischler, R. “What Did Herodotus Really Say? or How to Build (a Theory of) the Great Pyramid,” Environment and Planning B, 6 (1979): 89-93

Gardner, M. Fads and Fallacies in the Name of Science. New York: Dover Publications, 1957.

Gazalé, M.J., Gnomon. Princeton, NJ: Princeton University Press, 1999.

Gillings, R.J. Mathematics in the Time of the Pharaohs. New York: Dover Publications, 1972.

Goff, B. Symbols of Prehistoric Mesopotamia. New Haven, CT Yale University Press, 1963.

Hedian, H. “The Golden Section and the Artist,” The Fibonacci Quarterly, 14 (1976): 406-418

Lawlor, R. Sacred Geometry. London: Thames and Hudson, 1982.

Mendelssohn, K. The Riddle of the Pyramids. New York: Praeger Publishers, 1974.

Petrie, W. The Pyramids and Temples of Gizeh. London: Field and Tuer, 1883.

Piazzi Smyth, C. The Great Pyramid. New York: Gramercy Books, 1978.

Schneider, M.S. A Beginner's Guide to Constructing the Universe. New York: Harper Perennial, 1995.

Spence, K. “Ancient Egyptian Chronology and the Astronomical Orientation of the Pyramids,” Nature, 408 (2000): 320-324

Stewart, I. “Counting the Pyramid Builders,” Scientific American (September 1998): 98-100.

Verheyen, H.F. “The Icosahedral Design of the Great Pyramid,” in Fivefold Symmetry. Singapore: World Scientific, 1992, 333-360.

Wier, S.K. “Insights from Geometry and Physics into the Construction of Egyptian Old Kingdom Pyramids,” Cambridge Archaeological Journal, 6 (1996): 150-163


Borissavlievitch, M. The Golden Number and the Scientific Aesthetics of Architecture. London: Alec Tiranti, 1958.

Bruckman, P.S. “Constantly Mean,” The Fibonacci Quarterly, 15 (1977): 236

Coxeter, H. S. M. Introduction to Geometry. New York: John Wiley & Sons, 1963.

Cromwell, P.R. Polyhedra. Cambridge: Cambridge University Press, 1997.

Dixon, K. Mathographics. New York: Dover Publications, 1987.

Ghyka, M. L'Esthetique des proportions dans la nature et dans les arts. Paris: Gallimard, 1927.

Heath, T. A History of Greek Mathematics. New York: Dover Publications, 1981.

Heath, T. The Thirteen Books of Euclid's Elements. New York: Dover Publications, 1956.

Jowett, B. The Dialogues of Plato. Oxford: Oxford University Press, 1953.

Kraut, R. The Cambridge Companion to Plato. Cambridge: Cambridge University Press, 1992.

Lasserre, F. The Birth of Mathematics in the Age of Plato. London: Hutchinson, 1964.

Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publishing, 1989.

Trachtenberg, M., andHyman, I. Architecture: From Prehistory to Post Modernism/The Western Tradition. New York: Harry N. Abrams, 1986.

Zeising, A. Der goldne Schnitt. Halle: Druck von E. Blochmann & Son in Dresden, 1884.


cedar.evansville.eduh ck6/index.html.

Adler, I., Barabe, D., andJean, R.V. “A History of the Study of Phyllotaxis,” Annals of Botany, 80 (1997): 231-244

Basin, S.L. “The Fibonacci Sequence as It Appears in Nature,” The Fibonacci Quarterly, 1 (1963): 53-64

Brousseau, Brother A., An Introduction to Fibonacci Discovery. Aurora, SD: The Fibonacci Association, 1965.

Bruckman, P.S. “Constantly Mean,” The Fibonacci Quarterly, 15 (1977): 236

Coxeter, H. S. M. “The Golden Section, Phyllotaxis, and Wythoffs Game,” Scripta Mathematica, 19 (1953): 135-143

Coxeter, H. S. M. Introduction to Geometry. New York: John Wiley & Sons, 1963.

Cook, T A., The Curves of Life. New York: Dover Publications, 1979.

Devlin, K. Mathematics. New York: Columbia University Press, 1999.

Douady, S., andCouder, Y., “Phyllotaxis as a Physical Self-Organized Process,” Physical Review Letters, 68 (1992): 2098-2101

Dunlap, R.A. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific, 1997.

Fibonacci, L.P. The Book of Squares. Orlando, FL: Academic Press, 1987.

“The Fibonacci Numbers,” Time, April 4, 1969, 49-50.

Gardner, M. Mathematical Circus. New York: Alfred A. Knopf, 1979.

Gardner, M. “The Multiple Fascination of the Fibonacci Sequence,” Scientific American (March 1969): 116-120.

Garland, T H., Fascinating Fibonaccis. White Plains, NY: Dale Seymour Publications, 1987.

Gies, J., andGies, F., Leonard of Pisa and the New Mathematics of the Middle Ages. 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, 178-191

Hoggatt, V.E. Jr., and Bicknell-Johnson, M. “Reflections Across Two and Three Glass Plates,” The Fibonacci Quarterly, 17 (1979): 118-142

Horadam, A.F. “Eight Hundred Years Young,” The Australian Mathematics Teacher, 31 (1975): 123-134

Jean, R.V. Mathematical Approach to Pattern and Form in Plant Growth. 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.

Pickover, C.A. Keys to Infinity. 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, 45 (1984): 49-63

Singh, P. “The So-Called Fibonacci Numbers in Ancient and Medieval India,” Historia Mathematica, 12 (1985): 229-244

Smith, D.E. History of Mathematics. New York: Dover Publications, 1958.

Stewart, I. “Fibonacci Forgeries,” Scientific American (May 1995): 102-105.

Stewart, I. Life's Other Secret. New York: John Wiley & Sons, 1998.

Thompson, D.W. On Growth and Form. New York: Dover Publications, 1992.

Vajda, S. Fibonacci & Lucas Numbers, and the Golden Section. Chichester: Ellis Horwood Limited, 1989.

Vorob'ev, N.N., Fibonacci Numbers. New York: Blaisdell, 1961.


Arasse, D. Leonardo Da Vinci. New York: Konecky & Konecky, 1998.

Beer, A. and Beer, eds., P., Kepler: Four Hundred Years. Vistas in Astronomy, vol. 18, New York: Pergamon Press, 1975.

Calvesi, M. Piero Della Francesca. New York: Rizzoli, 1998.

Caspar, M. Kepler. New York: Dover Publications, 1993.

Cromwell, P.R. Polyhedra. 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, 45-63. Dordrecht: Kluwer, 1992.

Gingerich, O. “Kepler, Johannes,” in Dictionary of Scientific Biography, ed. Charles Coulston Gillespie, vol. 7, 289-312. New York: Scribners, 1973.

Ginzburg, C. The Enigma of Piero. London: Verso, 2000.

James J. The Music of the Spheres. 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. Cambridge: Cambridge University Press, 1984.

Kepler, J. The Harmony of the World. Philadelphia: American Philosophical Society, 1997.

Kepler, J. Mysterium Cosmographicum. New York: Abaris Books, 1981. NY: Artabras/Reynal and Company, Leonardo Da Vinci. 1938.

MacKinnon, N. “The Portrait of Fra Luca Pacioli,” Mathematical Gazette, 77 (1993): 130-219

Martens, R. Kepler's Philosophy and the New Astronomy. 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. 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, 147-240. New York: Parthenon, 1955.

Stephenson, B. The Music of the Spheres: Kepler's Harmonic Astronomy. Princeton, NJ: Princeton University Press, 1994.

Strieder, P. Albrecht Dürer. New York: Abaris Books, 1982.

Taylor, R.E. No Royal Road. Chapel Hill: University of North Carolina Press, 1942.

Voelkel, J.R. Johannes Kepler. Oxford: Oxford University Press, 1999.

Westman, R.A. “The Astronomer's Role in the Sixteenth Century: A Preliminary Survey,” History of Science, 18 (1980): 105-147


Altschuler, E.L. Bachanalia. Boston: Little, Brown and Company, 1994.

d’Arcais, F F Giotto. New York: Abbeville Press Publishers, 1995.

Bellosi, L. Cimabue. New York: Abbeville Press Publishers, 1998.

Bergamini, D. Mathematics. New York: Time Incorporated, 1963.

Bois, Y.-A., Joosten, J., Rudenstine, A.Z., andJanssen, H. Piet Mondrian. Boston: Little, Brown and Company, 1995.

Boring, E.G. A History of Experimental Psychology. New York: Appleton-Century-Crofts, 1957.

Bouleau, C. The Painter's Secret Geometry. 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, 35 (1981): 129-133

Curtis, W. J. R. Le Corbusier: Ideas and Forms. Oxford: Phaidon, 1986.

Duckworth, G.E. Structural Patterns and Proportions in Vergil's Aeneid. Ann Arbor: University of Michigan Press, 1962.

Emmer, M. The Visual Mind. Cambridge, MA: MIT Press, 1993.

Fancher, R.E. Pioneers of Psychology. New York: W. W. Norton & Company, 1990.

Fechner, G. T Vorschule der Aesthetik. Leipzig: Breitkopf & Härtel, 1876.

Fischler, R. “How to Find the ‘Golden Number’ Without Really Trying,” The Fibonacci Quarterly, 19 (1981): 406-410

Fischler, R. “On the Application of the Golden Ratio in the Visual Arts,” Leonardo, 14 (1981): 31-32

Fischler, R. “The Early Relationship of Le Corbusier to the Golden Number,” Environment and Planning B, 6 (1979): 95-103

Godkewitsch, M. “The Golden Section: An Artifact of Stimulus Range and Measure of Preference,” American Journal of Psychology, 87 (1974): 269-277

Hambidge, J. The Elements of Dynamic Symmetry. New York: Dover Publications, 1967.

Herz-Fischler, R. “An Examination of Claims Concerning Seurat and the Golden Number,” Gazette des Beaux-Arts, 125 (1983): 109-112

Herz-Fischler, R. “Le Corbusier's ‘regulating lines’ for the villa at Garches (1927) and other early works,” Journal of the Society of Architectural Historians, 43 (1984): 53-59

Herz-Fischler, R. “Le Nombre d'or en France de 1896 à 1927,” Revue de l’Art, 118 (1997): 9-16

Hockney D. Secret Knowledge. New York: Viking Studio, 2001.

Howat, R. Debussy in Proportion. Cambridge: Cambridge University Press, 1983.

Kepes, G. Module, Proportion, Symmetry, Rhythm. New York: George Braziller, 1966.

Larson, P. “The Golden Section in the Earliest Notated Western Music,” The Fibonacci Quarterly, 16 (1978): 513-515

Le Courbusier. Modulor I and II. Cambridge, MA: Harvard University Press, 1980.

Lendvai, E. Béla Bartók: An Analysis of His Music. London: Kahn & Averill, 1971.

Lowman, E.A. “Some Striking Proportions in the Music of Bela Bartók,” The Fibonacci Quarterly, 9 (1971): 527-537

Marevna. Life with the Painters of La Ruche. New York: Macmillan Publishing Co., 1974.

McManus, I.C. “The Aesthetics of Simple Figures,” British Journal of Psychology, 71 (1980): 505-524

Nims, J. F Western Wind. New York: McGraw-Hill, 1992.

Nuland, S.B. Leonardo da Vinci. New York: Viking, 2000.

Osborne, H. ed. The Oxford Companion to Art. Oxford: Oxford University Press, 1970.

Putz, J. F “The Golden Section and the Piano Sonatas of Mozart,” Mathematics Magazine, 68 (1995): 275-282

Sadie, S. The New Grove Dictionary of Music and Musicians. New York: Grove, 2001.

Schiffman, H.R., and Bobka, D.J. “Preference in Linear Partitioning: The Golden Section Reexamined,” Perception & Psychophysics, 24 (1978): 102-103

Schillinger, F. Joseph Schillinger. New York: Da Capo Press, 1976.

Schillinger, J. The Mathematical Basis of the Arts. New York: Philosophical Library, 1948.

Schwarz, L. “The Art Historian's Computer,” Scientific American (April 1995): 106-111.

Somfai, L. Béla Bartók: Compositions, Concepts, and Autograph Sources. Berkeley: University of California Press, 1996.

Svensson, L. T “Note on the Golden Section,” Scandinavian Journal of Psychology, 18 (1977): 79-80

Tatlow, R., and Griffiths, P. “Numbers and Music,” in The New Grove Dictionary of Music and Musicians, 18 (2001): 231-236

Watson, R.I. The Great Psychologists. Philadelphia: J. B. Lippincott Company, 1978.

White, M. Leonardo the First Scientist. London: Little, Brown and Company, 2000.

Woodworth, R.S., andSchlosberg, H. Experimental Psychology. New York: Holt, Rinehart and Winston, 1965.

Zusne, L. Visual Perception of Form. New York: Academic Press, 1970.


Cohen, J., andStewart, I. The Collapse of Chaos. Discovering Simplicity in a Complex World. New York: Penguin Books, 1995.

Fischer, R. Financial Applications and Strategies for Traders. New York: John Wiley & Sons, 1993.

Gardner, M. Penrose Tiles to Trapdoor Ciphers. New York: W H. Freeman and Company, 1989.

Gleick, J. Chaos. New York: Penguin Books, 1987.

Lesmoir-Gordon, N., Rood, W, andEdney, R. Introducing Fractal Geometry. Cambridge: Icon Books, 2000.

Mandelbrot, B.B. Fractal Geometry of Nature. New York: W H. Freeman and Company, 1988.

Mandelbrot, B.B. “A Multifractal Walk Down Wall Street,” Scientific American (February 1999): 70-73.

Matthews, R. “The Power of One,” New Scientist, July 10, 1999, 27-30.

Peitgen, H.-O., Jürgens, H., andSaupe, D. Chaos and Fractals. New York: Springer-Verlag, 1992.

Peterson, I. “Fibonacci at Random,” Science News, 155 (1999): 376-377

Peterson, I. The Mathematical Tourist. New York: W H. Freeman and Company, 1988.

Peterson, I. “A Quasicrystal Construction Kit,” Science News, 155 (1999): 60-61

Prechter, R.R. Jr., and Frost, A.J. Elliot Wave Principle. Gainesville, GA: New Classics Library, 1978.

Schroeder, M. Fractals, Chaos, Power Laws. 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, 396 (1998): 55-57

Stewart, I. Does God Play Dice? London: Penguin Books, 1997.

Walser, H. The Golden Section. Washington, DC: The Mathematical Association of America, 2001.


Baierlein, R. Newton to Einstein: The Trail of Light. Cambridge: Cambridge University Press, 1992.

Barrow, J.D. Impossibility. Oxford: Oxford University Press, 1998.

Chandrasekhar, S. Truth and Beauty. Chicago: University of Chicago Press, 1987.

Chown, M. “Principia Mathematica III,” New Scientist, August 25, 2001, 44-47.

Davis, P.J., and Hersh, R. The Mathematical Experience. Boston: Houghton Mifflin Company, 1998.

Dehaene, S. The Number Sense. Oxford: Oxford University Press, 1997.

Deutsch, D. The Fabric of Reality. New York: Penguin Books, 1997.

Hersh, R. What Is Mathematics, Really? New York: Oxford University Press, 1997.

Hill, T.P. “The First Digit Phenomenon,” American Scientist, 86 (1998): 358-363

Kleene, S.C. “Foundations of Mathematics,” Chicago: Encyclopaedia Britannica (1971), 1097-1103.

Lakatos, I. Mathematics, Science and Epistemology. Cambridge: Cambridge University Press, 1978.

Lakoff, G., andNúñez, R., Where Mathematics Comes From. New York: Basic Books, 2000.

Livio, M. The Accelerating Universe. New York: John Wiley & Sons, 2000.

Maor, E. To Infinity and Beyond: A Cultural History of the Infinite. Princeton, NJ: Princeton University Press, 1987.

Matthews, R. “The Power of One,” New Scientist, July 10, 1999, 26-30.

Penrose, R. The Emperor's New Mind. Oxford: Oxford University Press, 1989.

Penrose, R. Shadows of the Mind. Oxford: Oxford University Press, 1994.

Pickover, C.A. The Loom of God. Cambridge, MA: Perseus Books, 1997.

Popper, K.R., and Eccles, J.C. The Self and Its Brain. New York: Springer International, 1977.

Raimi, R. “The Peculiar Distribution of the First Digit,” Scientific American (December 1969): 109-119.

Raskin, J. www.jefraskin.com/forjef2/jefweb-compiled/unpublished/effectiveness_ mathematics/

Robinson, A. “From a Formalist's Point of View,” Dialectica, 23, (1969): 45-49.

Russell, B. A History of Western Philosophy. New York: Simon and Schuster, 1945.

Russell, B. Human Knowledge, Its Scope and Its Limits. New York: Simon and Schuster, 1948.

Weisstein, E. matworld.wolfram.com/BenfordsLaw.html.

Wolfram, S., A New Kind of Science. Champaign, IL: Wolfram Media, 2002.