Golden ratio (English Wikipedia)

Livio 2002, pp. 203–209
Gratias, Denis; Quiquandon, Marianne (2019). "Discovery of quasicrystals: The early days". Comptes Rendus Physique. 20 (7–8): 803–816. Bibcode:2019CRPhy..20..803G. doi:10.1016/j.crhy.2019.05.009. S2CID 182005594.
Jaric, Marko V. (1989). Introduction to the Mathematics of Quasicrystals. Academic Press. p. x. ISBN 9780120406029. Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major influence on the new field.
Goldman, Alan I.; Anderegg, James W.; Besser, Matthew F.; Chang, Sheng-Liang; Delaney, Drew W.; Jenks, Cynthia J.; Kramer, Matthew J.; Lograsso, Thomas A.; Lynch, David W.; McCallum, R. William; Shield, Jeffrey E.; Sordelet, Daniel J.; Thiel, Patricia A. (1996). "Quasicrystalline materials". American Scientist. 84 (3): 230–241. Bibcode:1996AmSci..84..230G. JSTOR 29775669. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
King, S.; Beck, F.; Lüttge, U. (2004). "On the mystery of the golden angle in phyllotaxis". Plant, Cell and Environment. 27 (6): 685–695. Bibcode:2004PCEnv..27..685K. doi:10.1111/j.1365-3040.2004.01185.x.
Coldea, R.; Tennant, D.A.; Wheeler, E.M.; Wawrzynksa, E.; Prabhakaran, D.; Telling, M.; Habicht, K.; Smeibidl, P.; Keifer, K. (2010). "Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry". Science. 327 (5962): 177–180. arXiv:1103.3694. Bibcode:2010Sci...327..177C. doi:10.1126/science.1180085. PMID 20056884. S2CID 206522808.

ams.org (Global: 434^th place; English: 249^th place)

mathscinet.ams.org

Sizer, Walter S. (1986). "Continued roots". Mathematics Magazine. 59 (1): 23–27. doi:10.1080/0025570X.1986.11977215. JSTOR 2690013. MR 0828417.

archim.org.uk (Global: low place; English: low place)

Penrose, Roger (1978). "Pentaplexity". Eureka. Vol. 39. p. 32. (original PDF Archived 2022-08-09 at the Wayback Machine)

archive.org (Global: 6^th place; English: 6^th place)

Livio 2002, pp. 3, 81. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Summerson, John (1963). Heavenly Mansions and Other Essays on Architecture. New York: W.W. Norton. p. 37. And the same applies in architecture, to the rectangles representing these and other ratios (e.g., the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design.
Herz-Fischler 1998. Herz-Fischler, Roger (1998) [1987]. A Mathematical History of the Golden Number. Dover. ISBN 9780486400075. (Originally titled A Mathematical History of Division in Extreme and Mean Ratio.)
Herz-Fischler 1998, pp. 20–25. Herz-Fischler, Roger (1998) [1987]. A Mathematical History of the Golden Number. Dover. ISBN 9780486400075. (Originally titled A Mathematical History of Division in Extreme and Mean Ratio.)
Livio 2002, p. 6. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Livio 2002, p. 4: "... line division, which Euclid defined for ... purely geometrical purposes ..." Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Livio 2002, pp. 7–8. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Livio 2002, pp. 4–5. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Livio 2002, p. 78. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. pp. 20–21. ISBN 9781402735226.
Livio 2002, p. 3. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Livio 2002, pp. 88–96. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Livio 2002, pp. 131–132. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Livio 2002, pp. 134–135. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Livio 2002, p. 141. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Livio 2002, pp. 151–152. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Fink, Karl (1903). A Brief History of Mathematics. Translated by Beman, Wooster Woodruff; Smith, David Eugene (2nd ed.). Chicago: Open Court. p. 223. (Originally published as Geschichte der Elementar-Mathematik.)
Cain, Alan J. (2024). Form & Number: A History of Mathematical Beauty. Lisbon: Ebook. p. 551.
Cook, Theodore Andrea (1914). The Curves of Life. London: Constable. p. 420.
Livio 2002, p. 5. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Gardner, Martin (2001). "7. Penrose Tiles". The Colossal Book of Mathematics. Norton. pp. 73–93.
Livio 2002, pp. 203–209
Gratias, Denis; Quiquandon, Marianne (2019). "Discovery of quasicrystals: The early days". Comptes Rendus Physique. 20 (7–8): 803–816. Bibcode:2019CRPhy..20..803G. doi:10.1016/j.crhy.2019.05.009. S2CID 182005594.
Jaric, Marko V. (1989). Introduction to the Mathematics of Quasicrystals. Academic Press. p. x. ISBN 9780120406029. Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major influence on the new field.
Goldman, Alan I.; Anderegg, James W.; Besser, Matthew F.; Chang, Sheng-Liang; Delaney, Drew W.; Jenks, Cynthia J.; Kramer, Matthew J.; Lograsso, Thomas A.; Lynch, David W.; McCallum, R. William; Shield, Jeffrey E.; Sordelet, Daniel J.; Thiel, Patricia A. (1996). "Quasicrystalline materials". American Scientist. 84 (3): 230–241. Bibcode:1996AmSci..84..230G. JSTOR 29775669. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Hailperin, Max; Kaiser, Barbara K.; Knight, Karl W. (1999). Concrete Abstractions: An Introduction to Computer Science Using Scheme. Brooks/Cole. p. 63.
Hardy, G. H.; Wright, E. M. (1960) [1938]. "§11.8. The measure of the closest approximations to an arbitrary irrational". An Introduction to the Theory of Numbers (4th ed.). Oxford University Press. pp. 163–164. ISBN 978-0-19-853310-8. {{cite book}}: ISBN / Date incompatibility (help)
Tattersall, James Joseph (1999). Elementary number theory in nine chapters. Cambridge University Press. p. 28.
Parker, Matt (2014). Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux. p. 284. ISBN 9780374275655.
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Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. pp. 537–547. ISBN 9780716711933.
Penrose, Roger (1978). "Pentaplexity". Eureka. Vol. 39. p. 32. (original PDF Archived 2022-08-09 at the Wayback Machine)
Livio (2002, pp. 70–72) Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Urwin, Simon (2003). Analysing Architecture (2nd ed.). Routledge. pp. 154–155.
Tschichold, Jan (1991). The Form of the Book. Hartley & Marks. p. 43 Fig 4. ISBN 0-88179-116-4. Framework of ideal proportions in a medieval manuscript without multiple columns. Determined by Jan Tschichold 1953. Page proportion 2:3. margin proportions 1:1:2:3, Text area proportioned in the Golden Section. The lower outer corner of the text area is fixed by a diagonal as well.
Tschichold, Jan (1991). The Form of the Book. Hartley & Marks. pp. 27–28. ISBN 0-88179-116-4.
Jones, Ronald (1971). "The golden section: A most remarkable measure". The Structurist. 11: 44–52. Who would suspect, for example, that the switch plate for single light switches are standardized in terms of a Golden Rectangle?
Johnson, Art (1999). Famous problems and their mathematicians. Teacher Ideas Press. p. 45. ISBN 9781563084461. The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.
Stakhov, Alexey P.; Olsen, Scott (2009). "§1.4.1 A Golden Rectangle with a Side Ratio of τ". The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science. World Scientific. pp. 20–21. A credit card has a form of the golden rectangle
Cox, Simon (2004). Cracking the Da Vinci Code. Barnes & Noble. p. 62. ISBN 978-1-84317-103-4. The Golden Ratio also crops up in some very unlikely places: widescreen televisions, postcards, credit cards and photographs all commonly conform to its proportions.
Livio 2002, p. 190. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Howat, Roy (1983). "1. Proportional structure and the Golden Section". Debussy in Proportion: A Musical Analysis. Cambridge University Press. pp. 1–10.
Livio 2002, p. 154. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Zeising, Adolf (1854). "Einleitung [preface]". Neue Lehre von den Proportionen des menschlichen Körpers [New doctrine of the proportions of the human body] (in German). Weigel. pp. 1–10.
Pheasant, Stephen (1986). Bodyspace. Taylor & Francis. ISBN 9780850663402.
Man, John (2002). Gutenberg: How One Man Remade the World with Word. Wiley. pp. 166–167. ISBN 9780471218234. The half-folio page (30.7 × 44.5 cm) was made up of two rectangles—the whole page and its text area—based on the so called 'golden section', which specifies a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8.
Fechner, Gustav (1876). Vorschule der Ästhetik [Preschool of Aesthetics] (in German). Leipzig: Breitkopf & Härtel. pp. 190–202.
Livio 2002, p. 7. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. ISBN 0-88920-324-5. The entire book surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle.
Rossi, Corinna (2004). Architecture and Mathematics in Ancient Egypt. Cambridge University Press. pp. 67–68. there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to ⁠ $\varphi$ ⁠, and ⁠ $\varphi$ ⁠ itself as a number, do not fit with the extant Middle Kingdom mathematical sources; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56
Rossi, Corinna; Tout, Christopher A. (2002). "Were the Fibonacci series and the Golden Section known in ancient Egypt?". Historia Mathematica. 29 (2): 101–113. doi:10.1006/hmat.2001.2334. hdl:11311/997099.
Markowsky, George (1992). "Misconceptions about the Golden Ratio" (PDF). The College Mathematics Journal. 23 (1). Mathematical Association of America: 2–19. doi:10.2307/2686193. JSTOR 2686193. Retrieved 2012-06-29. It does not appear that the Egyptians even knew of the existence of ⁠ $\varphi$ ⁠ much less incorporated it in their buildings
Livio 2002, pp. 74–75. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Devlin, Keith J. (2005). The Math Instinct. New York: Thunder's Mouth Press. p. 108.
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Livio 2002, p. 169. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
Green, Christopher (1992). Juan Gris. Yale. pp. 37–38. ISBN 9780300053746.
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Bouleau, Charles (1963). The Painter's Secret Geometry: A Study of Composition in Art. Harcourt, Brace & World. pp. 247–248.
Livio 2002, pp. 177–178. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.

archives-ouvertes.fr (Global: 1,143^rd place; English: 973^rd place)

hal.archives-ouvertes.fr

Livio 2002, pp. 203–209
Gratias, Denis; Quiquandon, Marianne (2019). "Discovery of quasicrystals: The early days". Comptes Rendus Physique. 20 (7–8): 803–816. Bibcode:2019CRPhy..20..803G. doi:10.1016/j.crhy.2019.05.009. S2CID 182005594.
Jaric, Marko V. (1989). Introduction to the Mathematics of Quasicrystals. Academic Press. p. x. ISBN 9780120406029. Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major influence on the new field.
Goldman, Alan I.; Anderegg, James W.; Besser, Matthew F.; Chang, Sheng-Liang; Delaney, Drew W.; Jenks, Cynthia J.; Kramer, Matthew J.; Lograsso, Thomas A.; Lynch, David W.; McCallum, R. William; Shield, Jeffrey E.; Sordelet, Daniel J.; Thiel, Patricia A. (1996). "Quasicrystalline materials". American Scientist. 84 (3): 230–241. Bibcode:1996AmSci..84..230G. JSTOR 29775669. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.

arxiv.org (Global: 49^th place; English: 31^st place)

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Posamentier & Lehmann 2011, p. 8. Posamentier, Alfred S.; Lehmann, Ingmar (2011). The Glorious Golden Ratio. Prometheus Books. ISBN 9-781-61614-424-1.
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Posamentier & Lehmann 2011, chapter 4, footnote 12: "The Togo flag was designed by the artist Paul Ahyi (1930–2010), who claims to have attempted to have the flag constructed in the shape of a golden rectangle". Posamentier, Alfred S.; Lehmann, Ingmar (2011). The Glorious Golden Ratio. Prometheus Books. ISBN 9-781-61614-424-1.
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bridgesmathart.org (Global: low place; English: low place)

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bibliothequekandinsky.centrepompidou.fr

Jeunes Peintres ne vous frappez pas !, La Section d'Or: Numéro spécial consacré à l'Exposition de la "Section d'Or", première année, no. 1, 9 octobre 1912, pp. 1–7 Archived 2020-10-30 at the Wayback Machine, Bibliothèque Kandinsky

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Batchelor, Roy; Ramyar, Richard (2005). Magic numbers in the Dow (Report). Cass Business School. pp. 13, 31. Popular press summaries can be found in: Stevenson, Tom (2006-04-10). "Not since the 'big is beautiful' days have giants looked better". The Daily Telegraph. "Technical failure". The Economist. 2006-09-23.

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doi.org (Global: 2^nd place; English: 2^nd place)

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Livio 2002, pp. 203–209
Gratias, Denis; Quiquandon, Marianne (2019). "Discovery of quasicrystals: The early days". Comptes Rendus Physique. 20 (7–8): 803–816. Bibcode:2019CRPhy..20..803G. doi:10.1016/j.crhy.2019.05.009. S2CID 182005594.
Jaric, Marko V. (1989). Introduction to the Mathematics of Quasicrystals. Academic Press. p. x. ISBN 9780120406029. Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major influence on the new field.
Goldman, Alan I.; Anderegg, James W.; Besser, Matthew F.; Chang, Sheng-Liang; Delaney, Drew W.; Jenks, Cynthia J.; Kramer, Matthew J.; Lograsso, Thomas A.; Lynch, David W.; McCallum, R. William; Shield, Jeffrey E.; Sordelet, Daniel J.; Thiel, Patricia A. (1996). "Quasicrystalline materials". American Scientist. 84 (3): 230–241. Bibcode:1996AmSci..84..230G. JSTOR 29775669. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
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Sizer, Walter S. (1986). "Continued roots". Mathematics Magazine. 59 (1): 23–27. doi:10.1080/0025570X.1986.11977215. JSTOR 2690013. MR 0828417.
King, S.; Beck, F.; Lüttge, U. (2004). "On the mystery of the golden angle in phyllotaxis". Plant, Cell and Environment. 27 (6): 685–695. Bibcode:2004PCEnv..27..685K. doi:10.1111/j.1365-3040.2004.01185.x.
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Busard, Hubert L. L. (1968). "L'algèbre au Moyen Âge : le "Liber mensurationum" d'Abû Bekr". Journal des Savants (in French and Latin). 1968 (2): 65–124. doi:10.3406/jds.1968.1175. See problem 51, reproduced on p. 98
Bruce, Ian (1994). "Another instance of the golden right triangle" (PDF). Fibonacci Quarterly. 32 (3): 232–233. doi:10.1080/00150517.1994.12429219.
Loeb, Arthur L.; Varney, William (March 1992). "Does the golden spiral exist, and if not, where is its center?". In Hargittai, István; Pickover, Clifford A. (eds.). Spiral Symmetry. World Scientific. pp. 47–61. doi:10.1142/9789814343084_0002. ISBN 978-981-02-0615-4.
Diedrichs, Danilo R. (February 2019). "Archimedean, Logarithmic and Euler spirals – intriguing and ubiquitous patterns in nature". The Mathematical Gazette. 103 (556): 52–64. doi:10.1017/mag.2019.7. S2CID 127189159.
Hume, Alfred (1900). "Some propositions on the regular dodecahedron". The American Mathematical Monthly. 7 (12): 293–295. doi:10.2307/2969130. JSTOR 2969130.
Berndt, Bruce C.; Chan, Heng Huat; Huang, Sen-Shan; Kang, Soon-Yi; Sohn, Jaebum; Son, Seung Hwan (1999). "The Rogers–Ramanujan Continued Fraction" (PDF). Journal of Computational and Applied Mathematics. 105 (1–2): 9–24. doi:10.1016/S0377-0427(99)00033-3. Archived from the original (PDF) on 2022-10-06. Retrieved 2022-11-29.
Duffin, Richard J. (1978). "Algorithms for localizing roots of a polynomial and the Pisot Vijayaraghavan numbers". Pacific Journal of Mathematics. 74 (1): 47–56. doi:10.2140/pjm.1978.74.47.
Frings, Marcus (2002). "The Golden Section in Architectural Theory". Nexus Network Journal. 4 (1): 9–32. doi:10.1007/s00004-001-0002-0. S2CID 123500957.
Mongoven, Casey (2010). "A style of music characterized by Fibonacci and the golden ratio" (PDF). Congressus Numerantium. 201: 127–138.
Hasegawa, Robert (2011). "Gegenstrebige Harmonik in the Music of Hans Zender". Perspectives of New Music. 49 (1). Project Muse: 207–234. doi:10.1353/pnm.2011.0000. JSTOR 10.7757/persnewmusi.49.1.0207.
Smethurst, Reilly (2016). "Two Non-Octave Tunings by Heinz Bohlen: A Practical Proposal". In Torrence, Eve; et al. (eds.). Proceedings of Bridges 2016. Jyväskylä, Finland. Tessellations Publishing. pp. 519–522.
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Coldea, R.; Tennant, D.A.; Wheeler, E.M.; Wawrzynksa, E.; Prabhakaran, D.; Telling, M.; Habicht, K.; Smeibidl, P.; Keifer, K. (2010). "Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry". Science. 327 (5962): 177–180. arXiv:1103.3694. Bibcode:2010Sci...327..177C. doi:10.1126/science.1180085. PMID 20056884. S2CID 206522808.
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Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. ISBN 0-88920-324-5. The entire book surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle.
Rossi, Corinna (2004). Architecture and Mathematics in Ancient Egypt. Cambridge University Press. pp. 67–68. there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to ⁠ $\varphi$ ⁠, and ⁠ $\varphi$ ⁠ itself as a number, do not fit with the extant Middle Kingdom mathematical sources; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56
Rossi, Corinna; Tout, Christopher A. (2002). "Were the Fibonacci series and the Golden Section known in ancient Egypt?". Historia Mathematica. 29 (2): 101–113. doi:10.1006/hmat.2001.2334. hdl:11311/997099.
Markowsky, George (1992). "Misconceptions about the Golden Ratio" (PDF). The College Mathematics Journal. 23 (1). Mathematical Association of America: 2–19. doi:10.2307/2686193. JSTOR 2686193. Retrieved 2012-06-29. It does not appear that the Egyptians even knew of the existence of ⁠ $\varphi$ ⁠ much less incorporated it in their buildings
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Osler, Carol (2000). "Support for Resistance: Technical Analysis and Intraday Exchange Rates" (PDF). Federal Reserve Bank of New York Economic Policy Review. 6 (2): 53–68. Archived (PDF) from the original on 2007-05-12. 38.2 percent and 61.8 percent retracements of recent rises or declines are common,

hdl.handle.net (Global: 66^th place; English: 34^th place)

Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. ISBN 0-88920-324-5. The entire book surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle.
Rossi, Corinna (2004). Architecture and Mathematics in Ancient Egypt. Cambridge University Press. pp. 67–68. there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to ⁠ $\varphi$ ⁠, and ⁠ $\varphi$ ⁠ itself as a number, do not fit with the extant Middle Kingdom mathematical sources; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56
Rossi, Corinna; Tout, Christopher A. (2002). "Were the Fibonacci series and the Golden Section known in ancient Egypt?". Historia Mathematica. 29 (2): 101–113. doi:10.1006/hmat.2001.2334. hdl:11311/997099.
Markowsky, George (1992). "Misconceptions about the Golden Ratio" (PDF). The College Mathematics Journal. 23 (1). Mathematical Association of America: 2–19. doi:10.2307/2686193. JSTOR 2686193. Retrieved 2012-06-29. It does not appear that the Egyptians even knew of the existence of ⁠ $\varphi$ ⁠ much less incorporated it in their buildings

hkust.edu.hk (Global: low place; English: low place)

sciencefocus.hkust.edu.hk

"MythBusters: The Golden Ratio Around Us | Science Focus". sciencefocus.hkust.edu.hk. Retrieved 2026-04-17.

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Berndt, Bruce C.; Chan, Heng Huat; Huang, Sen-Shan; Kang, Soon-Yi; Sohn, Jaebum; Son, Seung Hwan (1999). "The Rogers–Ramanujan Continued Fraction" (PDF). Journal of Computational and Applied Mathematics. 105 (1–2): 9–24. doi:10.1016/S0377-0427(99)00033-3. Archived from the original (PDF) on 2022-10-06. Retrieved 2022-11-29.

jstor.org (Global: 23^rd place; English: 15^th place)

Schielack, Vincent P. (1987). "The Fibonacci Sequence and the Golden Ratio". The Mathematics Teacher. 80 (5): 357–358. doi:10.5951/MT.80.5.0357. JSTOR 27965402. This source contains an elementary derivation of the golden ratio's value.
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Mackinnon, Nick (1993). "The Portrait of Fra Luca Pacioli". The Mathematical Gazette. 77 (479): 130–219. doi:10.2307/3619717. JSTOR 3619717. S2CID 195006163.
Livio 2002, pp. 203–209
Gratias, Denis; Quiquandon, Marianne (2019). "Discovery of quasicrystals: The early days". Comptes Rendus Physique. 20 (7–8): 803–816. Bibcode:2019CRPhy..20..803G. doi:10.1016/j.crhy.2019.05.009. S2CID 182005594.
Jaric, Marko V. (1989). Introduction to the Mathematics of Quasicrystals. Academic Press. p. x. ISBN 9780120406029. Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major influence on the new field.
Goldman, Alan I.; Anderegg, James W.; Besser, Matthew F.; Chang, Sheng-Liang; Delaney, Drew W.; Jenks, Cynthia J.; Kramer, Matthew J.; Lograsso, Thomas A.; Lynch, David W.; McCallum, R. William; Shield, Jeffrey E.; Sordelet, Daniel J.; Thiel, Patricia A. (1996). "Quasicrystalline materials". American Scientist. 84 (3): 230–241. Bibcode:1996AmSci..84..230G. JSTOR 29775669. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
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Frettlöh, D.; Harriss, E.; Gähler, F. "Robinson Triangle". Tilings Encyclopedia.
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Odom, George; van de Craats, Jan (1986). "E3007: The golden ratio from an equilateral triangle and its circumcircle". Problems and solutions. The American Mathematical Monthly. 93 (7): 572. doi:10.2307/2323047. JSTOR 2323047.
Hume, Alfred (1900). "Some propositions on the regular dodecahedron". The American Mathematical Monthly. 7 (12): 293–295. doi:10.2307/2969130. JSTOR 2969130.
Mongoven, Casey (2010). "A style of music characterized by Fibonacci and the golden ratio" (PDF). Congressus Numerantium. 201: 127–138.
Hasegawa, Robert (2011). "Gegenstrebige Harmonik in the Music of Hans Zender". Perspectives of New Music. 49 (1). Project Muse: 207–234. doi:10.1353/pnm.2011.0000. JSTOR 10.7757/persnewmusi.49.1.0207.
Smethurst, Reilly (2016). "Two Non-Octave Tunings by Heinz Bohlen: A Practical Proposal". In Torrence, Eve; et al. (eds.). Proceedings of Bridges 2016. Jyväskylä, Finland. Tessellations Publishing. pp. 519–522.
Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. ISBN 0-88920-324-5. The entire book surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle.
Rossi, Corinna (2004). Architecture and Mathematics in Ancient Egypt. Cambridge University Press. pp. 67–68. there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to ⁠ $\varphi$ ⁠, and ⁠ $\varphi$ ⁠ itself as a number, do not fit with the extant Middle Kingdom mathematical sources; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56
Rossi, Corinna; Tout, Christopher A. (2002). "Were the Fibonacci series and the Golden Section known in ancient Egypt?". Historia Mathematica. 29 (2): 101–113. doi:10.1006/hmat.2001.2334. hdl:11311/997099.
Markowsky, George (1992). "Misconceptions about the Golden Ratio" (PDF). The College Mathematics Journal. 23 (1). Mathematical Association of America: 2–19. doi:10.2307/2686193. JSTOR 2686193. Retrieved 2012-06-29. It does not appear that the Egyptians even knew of the existence of ⁠ $\varphi$ ⁠ much less incorporated it in their buildings

lhup.edu (Global: low place; English: low place)

Simanek, Donald E. "Fibonacci Flim-Flam". Archived from the original on January 9, 2010. Retrieved April 9, 2013.

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Devlin, Keith (2007). "The Myth That Will Not Go Away". Archived from the original on November 12, 2020. Retrieved September 26, 2013. Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like 'which the ancient Greeks and others believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the lines of 'Leonardo Da Vinci believed that the human form displays the golden ratio.' There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on.

maine.edu (Global: low place; English: 9,584^th place)

umcs.maine.edu

Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. ISBN 0-88920-324-5. The entire book surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle.
Rossi, Corinna (2004). Architecture and Mathematics in Ancient Egypt. Cambridge University Press. pp. 67–68. there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to ⁠ $\varphi$ ⁠, and ⁠ $\varphi$ ⁠ itself as a number, do not fit with the extant Middle Kingdom mathematical sources; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56
Rossi, Corinna; Tout, Christopher A. (2002). "Were the Fibonacci series and the Golden Section known in ancient Egypt?". Historia Mathematica. 29 (2): 101–113. doi:10.1006/hmat.2001.2334. hdl:11311/997099.
Markowsky, George (1992). "Misconceptions about the Golden Ratio" (PDF). The College Mathematics Journal. 23 (1). Mathematical Association of America: 2–19. doi:10.2307/2686193. JSTOR 2686193. Retrieved 2012-06-29. It does not appear that the Egyptians even knew of the existence of ⁠ $\varphi$ ⁠ much less incorporated it in their buildings

maths.org (Global: low place; English: low place)

plus.maths.org

Livio, Mario (2002). "The golden ratio and aesthetics". Plus Magazine. Retrieved November 26, 2018.

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"A Disco Ball in Space". NASA. 2001-10-09. Archived from the original on 2020-12-22. Retrieved 2007-04-16.

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Coldea, R.; Tennant, D.A.; Wheeler, E.M.; Wawrzynksa, E.; Prabhakaran, D.; Telling, M.; Habicht, K.; Smeibidl, P.; Keifer, K. (2010). "Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry". Science. 327 (5962): 177–180. arXiv:1103.3694. Bibcode:2010Sci...327..177C. doi:10.1126/science.1180085. PMID 20056884. S2CID 206522808.

numberworld.org (Global: low place; English: low place)

Yee, Alexander J. (2021-03-13). "Records Set by y-cruncher". numberword.org. Two independent computations done by Clifford Spielman.

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Strogatz, Steven (2012-09-24). "Me, Myself, and Math: Proportion Control". The New York Times.

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Sloane, N. J. A. (ed.). "Sequence A001622 (Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
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persee.fr (Global: 310^th place; English: 770^th place)

Busard, Hubert L. L. (1968). "L'algèbre au Moyen Âge : le "Liber mensurationum" d'Abû Bekr". Journal des Savants (in French and Latin). 1968 (2): 65–124. doi:10.3406/jds.1968.1175. See problem 51, reproduced on p. 98

projecteuclid.org (Global: 4,781^st place; English: 3,117^th place)

Duffin, Richard J. (1978). "Algorithms for localizing roots of a polynomial and the Pisot Vijayaraghavan numbers". Pacific Journal of Mathematics. 74 (1): 47–56. doi:10.2140/pjm.1978.74.47.

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Peterson, Ivars (1 April 2005). "Sea shell spirals". Science News. Archived from the original on 3 October 2012. Retrieved 10 November 2008.

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Kurt Hofstetter: A Simple Construction from: the Golden Section Forum Geometricorum, Volume 2 (2002), pages 65–66 (PDF)

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Peters, J. M. H. (1978). "An Approximate Relation between π and the Golden Ratio". The Mathematical Gazette. 62 (421): 197–198. doi:10.2307/3616690. JSTOR 3616690. S2CID 125919525.
Mackinnon, Nick (1993). "The Portrait of Fra Luca Pacioli". The Mathematical Gazette. 77 (479): 130–219. doi:10.2307/3619717. JSTOR 3619717. S2CID 195006163.
Livio 2002, pp. 203–209
Gratias, Denis; Quiquandon, Marianne (2019). "Discovery of quasicrystals: The early days". Comptes Rendus Physique. 20 (7–8): 803–816. Bibcode:2019CRPhy..20..803G. doi:10.1016/j.crhy.2019.05.009. S2CID 182005594.
Jaric, Marko V. (1989). Introduction to the Mathematics of Quasicrystals. Academic Press. p. x. ISBN 9780120406029. Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major influence on the new field.
Goldman, Alan I.; Anderegg, James W.; Besser, Matthew F.; Chang, Sheng-Liang; Delaney, Drew W.; Jenks, Cynthia J.; Kramer, Matthew J.; Lograsso, Thomas A.; Lynch, David W.; McCallum, R. William; Shield, Jeffrey E.; Sordelet, Daniel J.; Thiel, Patricia A. (1996). "Quasicrystalline materials". American Scientist. 84 (3): 230–241. Bibcode:1996AmSci..84..230G. JSTOR 29775669. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
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Frettlöh, D.; Harriss, E.; Gähler, F. "Robinson Triangle". Tilings Encyclopedia.
Clason, Robert G (1994). "A family of golden triangle tile patterns". The Mathematical Gazette. 78 (482): 130–148. doi:10.2307/3618569. JSTOR 3618569. S2CID 126206189.
Diedrichs, Danilo R. (February 2019). "Archimedean, Logarithmic and Euler spirals – intriguing and ubiquitous patterns in nature". The Mathematical Gazette. 103 (556): 52–64. doi:10.1017/mag.2019.7. S2CID 127189159.
Frings, Marcus (2002). "The Golden Section in Architectural Theory". Nexus Network Journal. 4 (1): 9–32. doi:10.1007/s00004-001-0002-0. S2CID 123500957.
Coldea, R.; Tennant, D.A.; Wheeler, E.M.; Wawrzynksa, E.; Prabhakaran, D.; Telling, M.; Habicht, K.; Smeibidl, P.; Keifer, K. (2010). "Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry". Science. 327 (5962): 177–180. arXiv:1103.3694. Bibcode:2010Sci...327..177C. doi:10.1126/science.1180085. PMID 20056884. S2CID 206522808.
Falbo, Clement (March 2005). "The golden ratio—a contrary viewpoint". The College Mathematics Journal. 36 (2): 123–134. doi:10.1080/07468342.2005.11922119. S2CID 14816926.
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O'Connor, John J.; Robertson, Edmund F. (2001). "The Golden Ratio". MacTutor History of Mathematics archive. Retrieved 2007-09-18.

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Batchelor, Roy; Ramyar, Richard (2005). Magic numbers in the Dow (Report). Cass Business School. pp. 13, 31. Popular press summaries can be found in: Stevenson, Tom (2006-04-10). "Not since the 'big is beautiful' days have giants looked better". The Daily Telegraph. "Technical failure". The Economist. 2006-09-23.

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Mongoven, Casey (2010). "A style of music characterized by Fibonacci and the golden ratio" (PDF). Congressus Numerantium. 201: 127–138.
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Smethurst, Reilly (2016). "Two Non-Octave Tunings by Heinz Bohlen: A Practical Proposal". In Torrence, Eve; et al. (eds.). Proceedings of Bridges 2016. Jyväskylä, Finland. Tessellations Publishing. pp. 519–522.

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Gunn, Charles; Sullivan, John M. (2008). "The Borromean Rings: A video about the New IMU logo". In Sarhangi, Reza; Séquin, Carlo H. (eds.). Proceedings of Bridges 2008. Leeuwarden, the Netherlands. Tarquin Publications. pp. 63–70.; Video at "The Borromean Rings: A new logo for the IMU". International Mathematical Union. Archived from the original on 2021-03-08.

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Frettlöh, D.; Harriss, E.; Gähler, F. "Robinson Triangle". Tilings Encyclopedia.
Clason, Robert G (1994). "A family of golden triangle tile patterns". The Mathematical Gazette. 78 (482): 130–148. doi:10.2307/3618569. JSTOR 3618569. S2CID 126206189.

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Penrose, Roger (1978). "Pentaplexity". Eureka. Vol. 39. p. 32. (original PDF Archived 2022-08-09 at the Wayback Machine)
Gunn, Charles; Sullivan, John M. (2008). "The Borromean Rings: A video about the New IMU logo". In Sarhangi, Reza; Séquin, Carlo H. (eds.). Proceedings of Bridges 2008. Leeuwarden, the Netherlands. Tarquin Publications. pp. 63–70.; Video at "The Borromean Rings: A new logo for the IMU". International Mathematical Union. Archived from the original on 2021-03-08.
Berndt, Bruce C.; Chan, Heng Huat; Huang, Sen-Shan; Kang, Soon-Yi; Sohn, Jaebum; Son, Seung Hwan (1999). "The Rogers–Ramanujan Continued Fraction" (PDF). Journal of Computational and Applied Mathematics. 105 (1–2): 9–24. doi:10.1016/S0377-0427(99)00033-3. Archived from the original (PDF) on 2022-10-06. Retrieved 2022-11-29.
Devlin, Keith (2007). "The Myth That Will Not Go Away". Archived from the original on November 12, 2020. Retrieved September 26, 2013. Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like 'which the ancient Greeks and others believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the lines of 'Leonardo Da Vinci believed that the human form displays the golden ratio.' There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on.
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Salvador Dalí (2008). The Dali Dimension: Decoding the Mind of a Genius. Media 3.14-TVC-FGSD-IRL-AVRO. Archived from the original (DVD) on 2005-02-09. Retrieved 2010-12-09.
"A Disco Ball in Space". NASA. 2001-10-09. Archived from the original on 2020-12-22. Retrieved 2007-04-16.
Peterson, Ivars (1 April 2005). "Sea shell spirals". Science News. Archived from the original on 3 October 2012. Retrieved 10 November 2008.
Osler, Carol (2000). "Support for Resistance: Technical Analysis and Intraday Exchange Rates" (PDF). Federal Reserve Bank of New York Economic Policy Review. 6 (2): 53–68. Archived (PDF) from the original on 2007-05-12. 38.2 percent and 61.8 percent retracements of recent rises or declines are common,
Jeunes Peintres ne vous frappez pas !, La Section d'Or: Numéro spécial consacré à l'Exposition de la "Section d'Or", première année, no. 1, 9 octobre 1912, pp. 1–7 Archived 2020-10-30 at the Wayback Machine, Bibliothèque Kandinsky

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