The information in this book comes from a mixture of original research, contributions by visitors to my websites at www.goldennumber.net and www.phimatrix.com, original interviews, online sources, and books. Wikipedia.com is a great starting point for further research on a variety of mathematics-related subjects, but there are also a number of good sources focused more solidly on mathematics and its history, including the MacTutor History of Mathematics archive from the University of St. Andrews, Scotland (http://www-groups.dcs.st-and.ac.uk/~history/index.html), and Wolfram MathWorld (http://mathworld.wolfram.com/).
Herz-Fischler, Roger. A Mathematical History of the Golden Number. New York: Dover Publications, 1998.
Huntley, H. E., The Divine Proportion: A Study in Mathematical Beauty. New York: Dover Publications, 1970.
Lawlor, Robert. Sacred Geometry: Philosophy and Practice. London: Thames and Hudson, 1982.
Livio, Mario. The Golden Ratio: The Story of Phi. The World’s Most Astonishing Number. New York: Broadway Books, 2002.
Olsen, Scott A. The Golden Section: Nature’s Greatest Secret. Glastonbury: Wooden Books, 2009.
Skinner, Stephen. Sacred Geometry: Deciphering the Code. New York: Sterling, 2006.
1. “Internet users per 100 inhabitants 1997 to 2007,” ICT Indicators Database, International Telecommunication Union (ITU), http://www.itu.int/ITU-D/ict/statistics/ict/.
2. “ICT Facts and Figures 2017,” Telecommunication Development Bureau, International Telecommunication Union (ITU), https://www.itu.int/en/ITU-D/Statistics/Pages/facts/default.aspx.
3. “History of Wikipedia,” Wikipedia, https://en.wikipedia.org/wiki/History_of_Wikipedia.
4. Roger Nerz-Fischler, A Mathematical History of the Golden Number (New York: Dover, 1987), 167.
5. Mario Livio, The Golden Ratio: The Story of Phi. The World’s Most Astonishing Number (New York: Broadway Books, 2002), 7.
6. David E. Joyce, “Euclid’s Elements: Book VI: Definition 3,” Department of Mathematics and Computer Science, Clark University, https://mathcs.clarku.edu/~djoyce/elements/bookVI/defVI3.html.
1. As quoted by Karl Fink, Geschichte der Elementar-Mathematik (1890), translated as “A Brief History of Mathematics” (Chicago: Open Court Publishing Company, 1900) by Wooster Woodruff Beman and David Eugene Smith. Also see Carl Benjamin Boyer, A History of Mathematics (New York: Wiley, 1968).
2. “Timaeus by Plato,” translated by Benjamin Jowett, The Internet Classics Archive, http://classics.mit.edu/Plato/timaeus.html.
3. These passages and illustrations were recreated and edited based on the translations and content at David E. Joyce, “Euclid’s Elements,” Department of Mathematics and Computer Science, Clark University, https://mathcs.clarku.edu/~djoyce/elements/elements.html.
4. Roger Nerz-Fischler, A Mathematical History of the Golden Number (New York: Dover, 1987), 159.
5. Eric W. Weisstein, “Icosahedral Group,” MathWorld—A Wolfram Web Resource, http://mathworld.wolfram.com/IcosahedralGroup.html.
6. Ibid.
1. As quoted at “Quotations: Galilei, Galileo (1564-1642),” Convergence, Mathematical Association of America, https://www.maa.org/press/periodicals/convergence/quotations/galilei-galileo-1564-1642-1.
2. Jacques Sesiano, “Islamic mathematics,” in Selin, Helaine; D’Ambrosio, Ubiratan, eds., Mathematics Across Cultures: The History of Non-Western Mathematics (Dordrecht: Springer Netherlands, 2001), 148.
3. J.J. O’Connor and E.F. Robertson, “The Golden Ratio,” School of Mathematics and Statistics, University of St Andrews, Scotland, http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Golden_ratio.html.
4. French-born mathematician Albert Girard (1595-1632) was the first to formulate the algebraic expression that describes the Fibonacci sequence (fn+2 = fn+1 + fn) and link it to the golden ratio, according to Scottish mathematician Robert Simson, “An Explication of an Obscure Passage in Albert Girard’s Commentary upon Simon Stevin’s Works (Vide Les Oeuvres Mathem. de Simon Stevin, a Leyde, 1634, p. 169, 170),” Philosophical Transactions of the Royal Society of London 48 (1753-1754), 368-377.
5. James Joseph Tattersall, Elementary Number Theory in Nine Chapters (2nd ed.), (Cambridge: Cambridge University Press, 2005), 28.
6. Mario Livio, The Golden Ratio: The Story of Phi. The World’s Most Astonishing Number (New York: Broadway Books, 2002), 7.
7. Many interesting patterns associated with the Fibonacci sequence can be found at Dr. Ron Knott, “The Mathematical Magic of the Fibonacci Numbers,” Department of Mathematics, University of Surrey, http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#section13.1.
8. Jain 108, “Divine Phi Proportion,” Jain 108 Mathemagics, https://jain108.com/2017/06/25/divine-phi-proportion/.
9. This pattern was first described and illustrated by Lucien Khan, and the graphic below was recreated based on his original design.
10. J.J. O’Connor and E.F.
Robertson, “The Golden Ratio.”
1. This is possibly a paraphrase of his philosophical reflections on the prime importance of mathematics.
2. As quoted in Mario Livio, The Golden Ratio: The Story of Phi. The World’s Most Astonishing Number (New York: Broadway Books, 2002), 131.
3. Richard Owen, “Piero della Francesca masterpiece ‘holds clue to 15th-century murder’,” The Times, January 23, 2008.
4. “The Ten Books on Architecture, 3.1,” translated by Joseph Gwilt, Lexundria, https://lexundria.com/vitr/3.1/gw.
5. Jackie Northam, “Mystery Solved: Saudi Prince is Buyer of $450M DaVinci Painting,” The Two-Way, December 7, 2017, https://www.npr.org/sections/thetwo-way/2017/12/07/569142929/mystery-solved-saudi-prince-is-buyer-of-450m-davinci-painting.
6. J.J. O’Connor and E.F. Robertson, “Quotations by Leonardo da Vinci,” School of Mathematics and Statistics, University of St Andrews, Scotland, http://www-history.mcs.st-andrews.ac.uk/Quotations/Leonardo.html. Quoted in Des MacHale, Wisdom (London: Prion, 2002).
7. “Nascita di Venere,” Le Gallerie degli Uffizi, https://www.uffizi.it/opere/nascita-di-venere.
1. “Georges-Pierre Seurat: Grandcamp, Evening,” MoMA.org, https://www.moma.org/collection/works/79409.
2. deIde, “allRGB,” https://allrgb.com/
3. Mark Lehner, The Complete Pyramids (London: Thames & Hudson, 2001), 108.
4. H. C. Agnew, A Letter from Alexandria on the Evidence of the Practical Application of the Quadrature of the Circle in the Configuration of the Great Pyramids of Gizeh (London: R. and J.E. Taylor, 1838).
5. John Taylor, The Great Pyramid: Why Was It Built? And Who Built It? (Cambridge: Cambridge University Press, 1859).
6. The Palermo Stone, which is dated to the Fifth Dynasty of Egypt (c. 2392–2283 BCE), contains the first known use of the Egyptian royal cubit to describe Nile flood levels during the First Dynasty of Egypt (c. 3150–c. 2890 BCE).
7. D. I. Lightbody, “Biography of a Great Pyramid Casing Stone,” Journal of Ancient Egyptian Architecture 1, 2016, 39–56.
8. Glen R. Dash, “Location, Location, Location: Where, Precisely, are the Three Pyramids of Giza?” Dash Foundation Blog, February 13, 2014, http://glendash.com/blog/2014/02/13/location-location-location-where-precisely-are-the-three-pyramids-of-giza/.
9. Leland M. Roth, Understanding Architecture: Its Elements, History, and Meaning (3rd ed.) (New York: Routledge, 2018).
10. Chris Tedder, “Giza Site Layout,” last modified 2002, https://web.archive.org/web/20090120115741/http:/www.kolumbus.fi/lea.tedder/OKAD/Gizaplan.htm.
11. Henutsen was described as a “king’s daughter” by the Inventory Stela discovered in 1858, but most Egyptologists consider it a fake.
12. Theodore Andrea Cook, The Curves of Life (New York: Dover Publications, 1979).
13. “Statue of Zeus at Olympia, Greece,” 7 Wonders, http://www.7wonders.org/europe/greece/olympia/zeus-at-olympia/
14. Guido Zucconi, Florence: An Architectural Guide (San Giovanni Lupatoto, Italy: Arsenale Editrice, 2001).
15. PBS, “Birth of a Dynasty,” The Medici: Godfathers of the Renaissance, March 30, 2009, https://www.youtube.com/watch?v=9FFDJK8jmms.
16. Matila Ghyka, The Geometry of Art and Life (2nd ed.) (New York: Dover Publications, 1977), 156.
17. Michael J. Ostwald, “Review of Modulor and Modulor 2 by Le Corbusier (Charles Edouard Jeanneret),” Nexus Network Journal, vol. 3, no. 1 (Winter 2001), http://www.nexusjournal.com/reviews_v3n1-Ostwald.html.
18. “United Nations Secretariat Building,” Emporis, https://www.emporis.com/buildings/114294/united-nations-secretariat-building-new-york-city-ny-usa.
19. Richard Padovan, Proportion: Science, Philosophy, Architecture (New York: Routledge, 1999).
20. “Fact Sheet: History of the United Nations Headquarters,” Public Inquiries, UN Visitors Centre, February 20, 2013, https://visit.un.org/sites/visit.un.org/files/FS_UN_Headquarters_History_English_Feb_2013.pdf.
21. “DB9,” Aston Martin. Last modified 2014. https://web.archive.org/web/20140817055237/http:/www.astonmartin.com/en/cars/the-new-db9/db9-design.
22. “Star Trek: Designing the Enterprise,” Walter “Matt” Jeffries, http://www.mattjefferies.com/start.html.
23. Darrin Crescenzi, “Why the Golden Ratio Matters,” Medium, April 21, 2015, https://medium.com/@quick_brown_fox/why-the-golden-ratio-matters-583f6737c10c.
24. Ibid.
1. Stephen Marquardt, Lecture to the American Academy of Cosmetic Dentistry, April 29, 2004
2. Richard Padovan, Proportion: Science, Philosophy, Architecture (New York: Routledge, 1999).
3. Scott Olsen, The Golden Section: Nature’s Greatest Secret (Glastonbury: Wooden Books, 2009).
4. Alex Bellos, “The golden ratio has spawned a beautiful new curve: the Harriss spiral,” The Guardian, January 13, 2015, https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jan/13/golden-ratio-beautiful-new-curve-harriss-spiral.
5. “Insects, Spiders, Centipedes, Millipedes,” National Park Service, last updated October 17, 2017, https://www.nps.gov/ever/learn/nature/insects.htm.
6. Eva Bianconi, Allison Piovesan, Federica Facchin, Alina Beraudi, et al, “An estimation of the number of cells in the human body,” Annals of Human Biology 40, no. 6 (2013): 463-471, https://www.tandfonline.com/doi/full/10.3109/03014460.2013.807878.
7. Richard R. Sinden, DNA Structure and Function (San Diego: Academic Press, 1994), 398.
8. “Chromatin,” modENCODE Project, last updated 2018, http://modencode.sciencemag.org/chromatin/introduction.
9. Edwin I. Levin, “The updated application of the golden proportion to dental aesthetics,” Aesthetic Dentistry Today 5, no. 3 (May 2011).
1. Ari Sihvola, “Ubi materia, ibi geometria,” Helsinki University of Technology, Electromagnetics Laboratory Report Series, No. 339, September 2000, https://users.aalto.fi/~asihvola/umig.pdf.
2. J. P. Luminet, “Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background,” Nature 425 (October 9, 2003) 593-595.
3. Dr. David R. Williams, “Moon Fact Sheet,” NASA, last updated July 3, 2017, https://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html.
4. Dr. David R. Williams, “Venus Fact Sheet,” NASA, last updated December 23, 2016, https://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html.
5. Mercury, the innermost planet, has an orbital period of 87.97 days, about .2408 of one Earth year. This number varies only 2.0% from 1/Ф3. Saturn, the outermost visible planet, has an orbital period of 10759.22 days, which is 29.4567 times one Earth year. This number varies only 1.5% from Ф7. These are, perhaps, just coincidences, but while we’re at it here’s one more: Take the ratio of the mean distance from the sun of each planet from Mercury to Pluto (yes, we know) to the one before it. Start with Mercury as 1 and throw in Ceres to represent the asteroid belt. The average of these relative distances is 1.6196, a variance of less than 0.1% from Ф.
6. John F. Lindner, “Strange Nonchaotic Stars,” Physical Review Letters 114, no. 5 (February 6, 2015).
7. P. C. W. Davies, “Thermodynamic phase transitions of Kerr-Newman black holes in de Sitter space,” Classical and Quantum Gravity 6, no. 12 (1989): 1909-1914. DOI: 10.1088/0264-9381/6/12/018.
8. N. Cruz, M. Olivares, & J. R. Villanueva, European Physical Journal C, no 77 (2017): 123. https://doi.org/10.1140/epjc/s10052-017-4670-7
9. J.A. Nieto, “A link between black holes and the golden ratio” (2011), https://arxiv.org/abs/1106.1600v1.
10. L. Bindi, J. M. Eiler, Y. Guan et al., “Evidence for the extraterrestrial origin of a natural quasicrystal,” Proceedings of the National Academy of Sciences 109, no. 5 (January 1, 2012): 1396-1401, https://doi.org/10.1073/pnas.1111115109.
11. Eric W. Weisstein, “Icosahedral Group,” MathWorld—A Wolfram Web Resource, http://mathworld.wolfram.com/IcosahedralGroup.html.
12. R. Coldea, D. A. Tennant, E. M. Wheeler et al., “Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry,” Science 327 (2010): 177-180.
13. See “2004 Dow Jones Industrial Average Historical Prices / Charts” at http://futures.tradingcharts.com/historical/DJ/2004/0/continuous.html.
14. See “2008 Dow Jones Industrial Average Historical Prices / Charts” at http://futures.tradingcharts.com/historical/DJ/2008/0/continuous.html.
15. Vladimir A Lefebvre, A Psychological Theory of Bipolarity and Reflexivity (Lewiston, NY: Edwin Mellen Press, 1992).
1. “Apophenia,” Merriam-Webster Online, https://www.merriam-webster.com/dictionary/apophenia.