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.

Knott, R.

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.


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.

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.

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. 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.

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