Master Fibonacci: Citations

Citations from Master Fibonacci: The Man Who Changed Math

PART 1: ORIGINS OF THE FIBONACCI SEQUENCE

I. Calculating in Medieval Europe
II. Leonardo Pisano, a.k.a Fibonacci
III. Liber Abaci
IV. Fibonacci’s Intellectual Legacy

“Arabic numerals in Europe.” Italia Medievale. Medieval Italian Cultural Association, 3 April 2015, http:// www.italiamedievale.org/portale/numerazione-araba-in-europa/?lang=en. Accessed 23 July 2018.

“Béla Bartók – The Golden Ratio in Music.” ETH Zurich. ETH Library, 2018, http://www.library.ethz.ch/en/ ms/Virtual-exhibitions/Fibonacci.-Un-ponte-sul-Mediterraneo/Reception-of-Fibonacci-numbers-and-the- golden-ratio/Bela-Bartok-the-golden-ratio-in-music. Accessed 20 July 2018.

“Biography of Leonardo of Pisa.” ETH Zurich. ETH Library, 2018, http://www.library.ethz.ch/ms/Virtuelle- Ausstellungen/Fibonacci.-Un-ponte-sul-Mediterraneo/Biographie-von-Leonardo-Pisano. Accessed 20 July 2018.

“Calculating with Fingers.” ETH Zurich. ETH Library, 2018, http://www.library.ethz.ch/en/ms/Virtual- exhibitions/Fibonacci.-Un-ponte-sul- Mediterraneo/Fibonacci-s-significance-for-the-present-day/ Calculating-with-fingers. Accessed 20 July 2018.

Devlin, K. Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World. Princeton University Press, 2017.

Devlin, K. The Man of Numbers: Fibonacci’s Arithmetic Revolution. Walker Publishing Company, New York, 2011.

The Editors of Encyclopaedia Britannica. “Pisa.” Britannica.com. Encyclopædia Britannica, Inc., 2018, https:// www.britannica.com/place/Pisa-Italy, Accessed 27 July 2018.

“Education in Europe – Medieval Education.” Science Encyclopedia – Jrank. Net Industries, 2018, http:// science.jrank.org/pages/9077/Education-in-Europe-Medieval-Education.html. Accessed 20 July 2018.

“Fibonacci.” Famous Scientists. famousscientists.org, 17 July 2018, www.famousscientists.org/fibonacci- leonardo-of-pisa/. Accessed 20 July 2018.

Fitzpatrick, Richard, “Euclid’s Elements of Geometry.” utexas.edu. The University of Texas at Austin, 2008, http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf, Accessed 29 July 2018.

Gies, F. C. “Fibonacci Biography & Facts.” Encyclopaedia Britannica. Encyclopædia Britannica, Inc.,11 June 2018, https://www.britannica.com/biography/Fibonacci. Accessed 20 July 2018.

Horadam, A. F. “Eight Hundred Years Young.” Vol 31, pp 123-134. The Australian Mathematics Teacher, 1975, Evansville.edu. 2018, http://faculty.evansville.edu/ck6/bstud/fibo.html. Accessed 19 July 2018.

Knott, Ron. “A Brief Biographical Sketch of Fibonacci, His Life, Times and Mathematical Achievements.” Surrey.ac.uk. University of Surrey, 25 July 2005, http://personal.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibBio.html. Accessed 25 July 2018.

Livio, M. The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number. Broadway Books, New York, 2002.

McClenon, R. “Leonardo of Pisa and His Liber Quadratorum.” The American Mathematical Monthly, Official Journal of the Mathematical Association of America. Volume XXVI. Jan. 1919, https://archive.org/details/jstor-2974039. Accessed July 2018.

O’Connor, J. J. and E. F. Robertson. “Leonardo Pisano Fibonacci.” MacTutor History of Mathematics Archive, School of Mathematics and Statistics University of St Andrews, Scotland, 1998, http://www.history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html. Accessed 20 July 2018.

O’Shea, Thomas. “Typus Arithmeticae.” Dr. Thomas O’Shea. 2018. https://www.drthomasoshea.com/typus-arithmeticae.html. Accessed 21 July 2018.

“The Rabbit Problem.” ETH Zurich. ETH Library, 2018, http://www.library.ethz.ch/en/ms/Virtuelle-Ausstellungen/Fibonacci.-Un-ponte-sul-Mediterraneo/Bedeutung-Fibonaccis-fuer-die-Gegenwart/ Kaninchenaufgabe. Accessed 20 July 2018.

Radford, L. “Reviews for Mathematics Education across Time and Place.” Dr. Thomas O’Shea. 2018, https://www.drthomasoshea.com/reviews.html, Accessed 21 July 2018.

Roselaar, S. “Economic Networks and Cultural Change in the Antique Mediterranean.” The Blue Web, 2018, http://bluenetworks2014.weebly.com/antiquity.html. Accessed 20 July 2018.

Scotta, T. C. and P. Marketos. “On the Origin of the Fibonacci Sequence.” History. MCS. 23 Mar. 2014, http://www-history.mcs.st-and.ac.uk/~history/Publications/fibonacci.pdf. Accessed 19 July 2018.

Sesiano, Jacques (1997-07-31). “Abū Kāmil”. Encyclopaedia of the history of science, technology, and medicine in non-western cultures. Springer. pp. 4–5.

Smith, David Eugene and Louis Charles Karpinski. “The Project Gutenberg EBook of the Hindu-Arabic Numerals.” Gutenberg. Project Gutenberg, 14 Sept. 2007, https://www.gutenberg.org/files/22599/22599-h/22599-h.htm. Accessed 20 July 2018.

“The Transition from Roman to Hindu-Arabic Numerals.” The Blue Web. 6 June 2014, http:// bluenetworks2014.weebly.com/the-transition-from-roman-to-hindu-arabic-numerals.html. Accessed 20 July 2018.

Citations from Master Fibonacci: The Man Who Changed Math

PART 2: THE FIBONACCI SEQUENCE

I. Fibonacci Sequence
II. The Golden Ratio
III. Phi
IV. Geometric Constructions Involving Phi
V. Chart of Terms

“Calculating.” ETH Zurich. ETH Library, 2018, http://www.library.ethz.ch/en/ms/Virtual-exhibitions/Fibonacci.-Un-ponte-sul-Mediterraneo/Fibonacci-s-significance-for-the-present-day/Calculating-with-fingers. Accessed 08 Aug 2018.

“Development of Numeral Systems.” ETH Zurich. ETH Library, 2018, http://www.library.ethz.ch/en/ms/ Virtual-exhibitions/Fibonacci.-Un-ponte-sul-Mediterraneo/Fibonacci-s-significance-for-the-present-day/ Development-of-numeral-systems. Accessed 20 July 2018.

Devlin, K. Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World. Princeton University Press, 2017.

Devlin, K. The Man of Numbers: Fibonacci’s Arithmetic Revolution. Walker Publishing Company, New York, 2011.

“Dispute Between Abacists and Algorists.” ETH Zurich. ETH Library, 2018, http://www.library.ethz.ch/ms/ Virtuelle-Ausstellungen/Fibonacci.-Un-ponte-sul-Mediterraneo/Bedeutung-Fibonaccis-fuer-die-Gegenwart/ Streit-zwischen-Abakisten-und-Algoristen. Accessed 10 Aug 2018.

Donnegan, James, M.D. “A New Greek and English Lexicon: Principally on the Plan of the Greek and German Lexicon of Schneider, the Words Alphabetically Arranged.” Hilliard, Gray, Boston, 1834, https://archive.org/ details/newgreekenglishl00donnuoft. Accessed 23 July 2018.

“Education in Europe – Medieval Education.” Science Encyclopedia – Jrank. Net Industries, 2018, http:// science.jrank.org/pages/9077/Education-in-Europe-Medieval-Education.html. Accessed 20 July 2018.

Eppstein. “ICS 161: Design and Analysis of Algorithms Lecture notes for January 9, 1996.” geeksforgeeks. UCI Donald Bren School of Information & Computer Sciences. https://www.ics.uci.edu/~eppstein/ 161/960109.html. Accessed 09 Aug 2018.

“Euclid.” ETH Zurich. ETH Library, 2018, http://www.library.ethz.ch/en/ms/Virtual-exhibitions/Fibonacci.- Un-ponte-sul-Mediterraneo/Reception-of-Fibonacci-numbers-and-the-golden-ratio/Euclid-construction-of- the-golden-ratio. Accessed 9 Aug 2018.

“Fibonacci.” Famous Scientists. famousscientists.org, 10 Sept 2018, www.famousscientists.org/fibonacci-leonardo-of-pisa/. Accessed 20 July 2018.

Gascueña, Dory. “Fibonacci and the Golden Ratio: Divine Geometry?” Open Mind. BBVA Group, 15 March 2017, https://www.bbvaopenmind.com/en/fibonacci-and-the-golden-ratio-divine-geometry/. Accessed 06 August 2018.

Ghusayni, Badih. “Favorite mathematics topics from the 12th Century to the 21st Century.” Academic Journal Academic Journal. International Journal of Mathematics & Computer Science. Vol. 13 Issue 1, p83-104. 22p. 2018, http://content.ebscohost.com/ContentServer.asp? EbscoContent=dGJyMMTo50SeprE4v%2BvlOLCmr1Cep7JSsqe4S7KWxWXS&ContentCust omer=dGJyMPGqskm1r7ZJuePfgeyx9Yvf5ucA&T=P&P=AN&S=R&D=a9h&K=127054609. Accessed 3 August 2018.

Horadam, A.F. “Book Review: Leonardo Pisano (Fibonacci)—The Book of Squares (an annotated translation into modern English)—L.E. Sigler, Academic Press 1987.” University of New England, Armidale, Australia 2351, p 382. https://www.fq.math.ca/Scanned/26-4/review.pdf. Accessed 13 Aug 2018.

Horadam, A. F. “Eight Hundred Years Young.” Vol 31, pp 123-134. The Australian Mathematics Teacher, 1975, Evansville.edu. 2018, http://faculty.evansville.edu/ck6/bstud/fibo.html. Accessed 19 July 2018.

Knott, Ron. “A Brief Biographical Sketch of Fibonacci, His Life, Times and Mathematical Achievements.” Surrey.ac.uk. University of Surrey, 25 July 2005, http://personal.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/ fibBio.html. Accessed 25 July 2018.

Knott, Ron. “Dudeney’s Cows.” Surrey.ac.uk. University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom, 25 September 2016, http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat. Accessed 08 August 2018.

Levy, Joel. A Curious History of Mathematics: The (BIG) Ideas from Early Number Concepts to Chaos Theory. Andre Deutsch, London, 2013.

Lines, Malcolm E. Think of a Number. Adam Hilger, New York, 1990.

Livio, M. The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number. Broadway Books, New York, 2002.

McClenon, R. “Leonardo of Pisa and His Liber Quadratorum.” The American Mathematical Monthly. Official Journal of the Mathematical Association of America, Volume XXVI. Jan. 1919, https://archive.org/details/jstor-2974039. Accessed 2 July 2018.

Meisner, Gary. “Spirals and the Golden Ratio.” GoldenNumber.net. PhiPoint Solutions, LLC, 25 Aug 2012, https://www.goldennumber.net/spirals/. Accessed 13 Aug 2018.

O’Connor, J. J. and E. F. Robertson. “Leonardo Pisano Fibonacci.” MacTutor History of Mathematics Archive, School of Mathematics and Statistics University of St Andrews, Scotland, 1998, http://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html. Accessed 20 July 2018.

O’Shea, Thomas. “Typus Arithmeticae.” Dr. Thomas O’Shea. 2018. https://www.drthomasoshea.com/typus-arithmeticae.html. Accessed 21 July 2018.

“Phi in Atomic Structure.” Sacred Geometry. Drupal, 2018, http://www.sacredgeometry.es/?q=en/content/phi-atomic-structure, Accessed 10 Aug 2018.

“The Rabbit Problem.” ETH Zurich. ETH Library, 2018, http://www.library.ethz.ch/en/ms/Virtuelle- Ausstellungen/Fibonacci.-Un-ponte-sul-Mediterraneo/Bedeutung-Fibonaccis-fuer-die-Gegenwart/ Kaninchenaufgabe. Accessed 20 July 2018.

Radford, L. “Reviews for Mathematics Education across Time and Place.” Dr. Thomas O’Shea. 2018, https://www.drthomasoshea.com/reviews.html, Accessed 21 July 2018.

Scotta, T. C. and P. Marketos. “On the Origin of the Fibonacci Sequence.” History. MCS. 23 Mar. 2014, http://www-history.mcs.st-and.ac.uk/~history/Publications/fibonacci.pdf. Accessed 19 July 2018.

Seewald, Nick. “The Myth of the Golden Ratio.” Golden Ratio Myth. Weebly, 2010, https:// goldenratiomyth.weebly.com/a-brief-history-of-phi.html. Accessed 6 Aug 2018.

Seife, Charles. Zero: The Biography of a Dangerous Idea. Viking, 2000.

Sesiano, Jacques (1997-07-31). “Abū Kāmil.” Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer. pp. 4–5.

Smith and Karpinski. “The Hindu-Arabic Numerals, Boston and London, 1911.” Archibald, Euclid’s Book on Divisions of Figures; with a Restoration based on Woepcke’s Text and on the Practica Geometrice, of Leonardo Pisano. Cambridge, England, 1915.

TallBloke. “A Remarkable Discovery: All Solar System Periods Fit the Fibonacci Series and the Golden Ratio. Why Phi?” TallBloke’s Talkshop, 20 Feb 2013, https://tallbloke.wordpress.com/2013/02/20/a-remarkable- discovery-all-solar-system- periods-fit-the-fibonacci-series-and-the-golden-ratio-why-phi. Accessed 07 Aug 2018.

“The Transition from Roman to Hindu-Arabic Numerals” The Blue Web. Bluenetworks2014, 6 June 2014, http://bluenetworks2014.weebly.com/the-transition-from-roman-to-hindu-arabic-numerals.html. Accessed 10 Aug 2018.

Velasquez, Robert. “5 Reasons Using The Fibonacci Sequence Makes You Better At Agile Development.” eLearningIndustry. 7 Nov 2017, https://elearningindustry.com/using-the-fibonacci-sequence-makes-better- agile-development-5-reasons. Accessed 13 Aug 2018.

Watkins, John J. Number Theory: A Historical Approach. Princeton University Press, 2013.

Wolchover, Natalie. “Strange Stars Pulse to the Golden Mean.” Quanta Magazine. Quanta, 10 Mar 2015, https://www.quantamagazine.org/variable-stars-have-strange-nonchaotic-attractors-20150310/. Accessed10 Aug 2018.

PART 3: FIBONACCI IN ART & MUSIC

I. Fibonacci in Art & Architecture
II. Fibonacci in Music
III. Fibonacci in Film
IV. Fibonacci in Applied Arts
V. Disputation of Fibonacci in Art

Akhtaruzzaman, Md. and Amir A. Shafie. “Geometrical Substantiation of Phi, the Golden Ratio and the Baroque of Nature, Architecture, Design and Engineering,” International Journal of Arts, Vol. 1 No. 1, 2011, pp. 1-22. doi: 10.5923/j.arts.20110101.01.

Armienti, Pietro. “The Medieval Roots of Modern Scientific Thought. A Fibonacci Abacus in the Facade of the Church of San Nicola in Pisa Ietro.” 2015, Journal of Cultural Heritage (2015), http://dx.doi.org/10.1016/j.culher.2015.07.015. Accessed 20 Aug 2018.

“As Easy as 1, 1, 2, 3 ….” The Guardian. Guardian News and Media Limited, 12 May 2005, https:// www.theguardian.com/music/2005/may/12/classicalmusicandopera1. Accessed 22 Aug 2018.

Beer, M., 2008. “Mathematics and Music: Relating Science to Arts?” Mathematical Spectrum, 41(1):36-42.  https://www.michael.beer.name/File+download/1/mathandmusic.pdf Accessed 20 Aug 2018.

Bidwell, Eugene Boyd. “Violin Design Method of Stradivari.” Violincad.com. 2018, http:// www.violincad.com/violin/betts/article.htm. Accessed 05 Sep 2018.

Birkhead, Mike. “Tips from a Combat Designer: Fibonacci Game Design.” Gamasutra.com. UBM, 15 Feb 2012, https://www.gamasutra.com/view/news/129642/ Tips_from_a_combat_designer_Fibonacci_game_design.php. Accessed 21 Aug 2018.

Brandon, James. “Divine Composition with Fibonacci’s Ratio (The Rule of Thirds on Steroids).” Digital Photography School. Digital Photography School, 2018, https://digital-photography-school.com/divine- composition-with-fibonaccis-ratio-the-rule-of-thirds-on-steroids/. Accessed 19 Aug 2018.

Cartwright, Mark. “Vitruvius.” Ancient History Encyclopedia. Ancient History Encyclopedia Limited, 22 April 2015, https://www.ancient.eu/Vitruvius/. Accessed 20 Aug 2018.

Cohen, J. -L. (2014). Le Corbusier’s Modulor and the Debate on Proportion in France. Architectural Histories, 2(1), Art. 23. DOI: http://doi.org/10.5334/ah.by, https://journal.eahn.org/articles/10.5334/ah.by/. Accessed 22 Aug 2018.

Du Sautoy, Marcus. “Listen by Numbers: Music and Maths.” The Guardian. Guardian News, 27 Jun 2011, https://www.theguardian.com/music/2011/jun/27/music-mathematics-fibonacci. Accessed 18 Aug 2018.

Edmark, John. “John Edmark/Work/Golden Angle.” John Edmark. John Edmark, 2018, http:// www.johnedmark.com/work/golden-angle/. Accessed 22 Aug 2018.

“The Fibonacci Numbers Have Been Discovered on a Church in Pisa.” 18 Sep 2015, https://sandrozicari.com/ 2015/09/18/the-fibonacci-numbers-have-been-discovered-on-a-church-in-pisa/. Accessed 20 Aug 2018.

“Fibonacci – The Flight of the Numbers.” Arte.it. ART.it, 23 Aug 2018. http://www.arte.it/work_of_art/ fibonacci-the-flight-of-the-numbers-5739. Accessed 23 Aug 2018.

Fischler, Roger. “On the Application of the Golden Ratio in the Visual Arts.” Leonardo, vol. 14, no. 1, 1981, pp. 31–32. JSTOR, JSTOR, www.jstor.org/stable/1574475. Accessed 19 Aug 2018.

Frampton, Hollis, and Scott MacDonald. “Interview with Hollis Frampton: The Early Years.” October, vol. 12, 1980, pp. 103–126. JSTOR, JSTOR, www.jstor.org/stable/778577. Accessed 20 Aug. 2018.

Grandinetti, Mary-Jane. “The Fib Review.” Muse-Pie Press. 30 Jun 2018, http://www.musepiepress.com/ fibreview/index.html. Accessed 21 Aug 2018.

Haylock, Derek. “Golden Ratio and Beethoven’s 5th.” Derek-Haylock.blogspot.com http://derek-haylock.blogspot.com/2014/08/golden-ratio-and-beethovens-5th.html. Accessed 20 Aug 2018.

Hoijer, Natalie. “Unleashing Music’s Hidden Blueprint: An Analysis of Mathematical Symmetries Used in Music.” IWU.edu, Digital Commons @ Illinois Wesleyan University. 2015, https://digitalcommons.iwu.edu/ cgi/viewcontent.cgi?article=1007&context=music_papers. Accessed 20 Aug 2018.

Hom, Elaine J. “What Is the Golden Ratio?” LiveScience.com. Purch, 24 Jun 2013, https:// www.livescience.com/37704-phi-golden-ratio.html. Accessed 21 Aug 2018.

Horadam, A. F. “Eight Hundred Years Young.” Vol 31, pp 123-134. The Australian Mathematics Teacher, 1975, Evansville.edu. 2018, http://faculty.evansville.edu/ck6/bstud/fibo.html. Accessed 19 July 2018.

“How We Built the Core.” Eden Project.com. Eden Project, https://www.edenproject.com/eden-story/behind- the-scenes/how-we-built-the-core. Accessed 20 Aug 2018.

Hunt, Patrick. “Mozart and Mathematics.” Electrum Magazine. Electrum Magazine, 30 June 2013, http:// www.electrummagazine.com/2013/06/mozart-and-mathematics/. Accessed 20 Aug 2018.

Iyer, Vijay. “Strength in Numbers: How Fibonacci Taught Us How to Swing.” The Guardian. Guardian News and Media Limited, 15 Oct 2009, https://www.theguardian.com/music/2009/oct/15/fibonacci-golden-ratio. Accessed 22 Aug 2018.

“Journey to the Core.” Eden Project.com. Eden Project, 2018, https://www.edenproject.com/sites/default/ files/documents/journey-to-the-core.pdf. Accessed 20 Aug 2018.

Klykavka, Oleksandr. “Solar Observatory in Ukraine.” Ancient History Et Cetera. Ancient History et cetera, 15 Dec 2016, http://etc.ancient.eu/education/solar-observatory-ukraine/. Accessed 21 Aug 2018.

“Le Corbusier – the Modulor.” ETH Library. ETH Library, 2018, http://www.library.ethz.ch/en/ms/Virtual- exhibitions/Fibonacci.-Un-ponte-sul-Mediterraneo/Reception-of-Fibonacci-numbers-and-the-golden-ratio/Le- Corbusier-the-Modulor. Accessed 22 Aug 2018.

Livio, Mario. The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number. Broadway Books; Reprint edition, 2003.

Lobo, Carmen. “Fibonacci Met Reinfried Marass, but Who is Fibonacci?” ART & Thoughts. ART & Thoughts, 10 Feb 2013, https://articulosparapensar.wordpress.com/2013/02/10/fibonacci-met-reinfried- marass-but-who-is-fibonacci/. Accessed 22 Aug 2018.

McNally, Jess. “Earth’s Most Stunning Natural Fractal Patterns.” Wired.com. Conde Nast, 9 Oct 2010, https://www.wired.com/2010/09/fractal-patterns-in-nature/. Accessed 22 Aug 2018.

Meisner, Gary. “Acoustics and the Golden Ratio.” Golden Number.net. PhiPoint Solutions, LLC, 2018, https:// www.goldennumber.net/acoustics/. Accessed 21 Aug 2018.

Miller, Peter H. “The Neuroscience of Traditional Architecture: Looking at Architectural Ornament Soothes Our Senses and Lifts Our Spirits.” Traditional Building.com. Active Interest Media, 10 May 2018, https:// www.traditionalbuilding.com/opinions/neuroscience-architecture. Accessed 18 Aug 2018.

Paphides, Pete. “The Music of Mathematics.” Times, the (United Kingdom), n.d. https://www.thetimes.co.uk/ article/delia-derbyshire-the-music-of-mathematics-mjbt5ts6g5z. Accessed 18 Aug 2018.

Parkin, Simon. “The Video Game that Maps the Galaxy.” The New Yorker. Conde Nast, 9 Jul 2014, https:// www.newyorker.com/tech/elements/the-video-game-that-maps-the-galaxy. Accessed 21 Aug 2018.

Pascoe, Clive B. “Motivational Pacing in Musical Design an Active Force in Music Education.” Contributions to Music Education, no. 3, 1974, pp. 106–112. JSTOR, JSTOR, www.jstor.org/stable/24126929. Accessed 20 Aug 2018.

Pincus, Greg. “Greg Pincus – Writer Guy.” Greg Pincus. Greg Pincus, 2013, http://www.gregpincus.com/ the-14-fibs.html. Accessed 24 Aug 2018.

Posamentier, Alfred S. and Ingmar Lehmann. The (Fabulous) Fibonacci Numbers. Prometheus Books, NY, 2007. Rogers, Michael R. “Chopin, Prelude in A Minor, Op. 28, No. 2.” 19th-Century Music, vol. 4, no. 3, 1981, pp. 245–250. JSTOR, JSTOR, www.jstor.org/stable/746697. Accessed 20 Aug 2018.

Rory PQ, “Music and the Fibonacci Sequence w/ Rory PQ.” Dubspot. DS14, Inc. – Dubspot. 1 Jun 2016. http://blog.dubspot.com/fibonacci-sequence-in-music/ Accessed 29 Oct 2018.

Salingaros, Nikos A. Applications of the Golden Mean to Architecture in Meandering through Mathematics, 2012.

Salingaros, Nikos A. Why Monotonous Repetition is Unsatisfying in Meandering through Mathematics, 2011.

Sartwell, Crispin, “Beauty”, The Stanford Encyclopedia of Philosophy (Winter 2017 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/win2017/entries/beauty/. Accessed 19 Aug 2018.

Shamsian, Jacob and Carl Mueller. “An Artist Uses Mathematical Formulas to Make Otherworldly Geometric Sculptures.” Business Insider. Insider Inc. https://www.businessinsider.com/artist-john-edmark-uses-math- to-make-otherworldly-geometric-sculptures-2016-4?r=US&IR=T&IR=T. Accessed 19 Aug 2018.

Sinha, Sudipta. “The Fibonacci Numbers and Its Amazing Applications.” International Journal of Engineering Science Invention ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726 www.ijesi.org ||Volume 6 Issue 9|| September 2017 || PP. 07-14, http://www.ijesi.org/papers/Vol(6)9/Version-3/B0609030714.pdf. Accessed 22 Aug 2018.

Spinak, Mike. The Golden Section Hypothesis: A Critical Look in Naturography. 2011.

Stakhov, Alexey. “The Mathematics of Harmony: Euclid to Contemporary Mathematics and Computer Science.” World Scientific.com. World Scientific Publishing Co. Pte. Ltd., 2018, http:// www.worldscibooks.com/mathematics/6635.html. Accessed 23 Aug 2018.

Stefanovic, Konstantin. “Dynamic Symmetry and Golden Section.” Eng trans: Ana Smiljanic. Sanu.ac. n.d. http://www.mi.sanu.ac.rs/vismath/jadrbookhtml/part42.html. Accessed 21 Aug 2018.

Sylvestre, Loïc, and Marco Costa. “The Mathematical Architecture of Bach’s ‘The Art of Fugue.’” Il Saggiatore Musicale, vol. 17, no. 2, 2010, pp. 175–195. JSTOR, JSTOR, www.jstor.org/stable/43030058. Accessed 19 Aug 2018.

Van Gend, Robert. “The Fibonacci Sequence and the Golden Ratio in Music.” Notes on Number Theory and Discrete Mathematics ISSN 1310–5132 Vol. 20, 2014, No. 1, 72–77, http://www.nntdm.net/papers/ nntdm-20/NNTDM-20-1-72-77.pdf. Accessed 05 Sep 2018.

Weisstein, Eric W. “Golden Spiral.” MathWorld. A Wolfram Web Resource. 31 Jul 2018, http:// mathworld.wolfram.com/GoldenSpiral.html. Accessed 21 Aug 2018.

Wu, Feng. “Stradivarius: Music of the Golden Ratio.” WufengEngineering. Wordpress, 18 Jan 2015, https:// wufengengineering.wordpress.com/2015/01/18/stradivarius-music-of-the-golden-ratio/. Accessed 21 Aug 2018.

PART 4: FIBONACCI IN NATURE

I. Fibonacci in Nature
II. Fibonacci and Insects
III. Fibonacci and Animals
IV. Fibonacci in Space & Geography
V. Fibonacci in Humans

Akhtaruzzaman, Md. and Amir A. Shafie. “Geometrical Substantiation of Phi, the Golden Ratio and the Baroque of Nature, Architecture, Design and Engineering.” International Journal of Arts p-ISSN: 2168-4995 e-ISSN: 2168-5002. 2011. 1(1): 1-22. doi: 10.5923/j.arts.20110101.01. Scientific & Academic Publishing. 2012, http://article.sapub.org/10.5923.j.arts.20110101.01.html. Accessed 19 Aug 2018.

Ashrafian H, Athanasiou T. “Fibonacci Series and Coronary Anatomy.” Heart Lung Circ 2011 Jul;20(7):483–4.

Bellos, Alex. “Golden ratio discovered in uterus.” The Guardian. Guardian News and Media Limited, 14 Aug 2012. https://www.theguardian.com/science/alexs-adventures-in-numberland/2012/aug/14/ golden-ratio- uterus. Accessed 06 Sep 2018.

Britton, Jill. “Fibonacci Numbers in Nature.” 20 Jun 2011. https://archive.li/t2RY http:// britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm. Accessed 29 Aug 2018.

Boeyens JCA, Thackeray JF. “Number Theory and the Unity of Science.” S Afr J Sci. 2014;110(11/12), Art. #a0084, 2 pages. http://dx.doi. org/10.1590/sajs.2014/a0084. Accessed 05 Sep 2018.

Cartwright, Jon. “Fibonacci Spirals in Nature Could Be Stress-Related.” SOTT.net: Signs of the Times. PhysicsWeb. 26 Apr 2007 07:13 UTC, https://sott.net/en130945, Accessed 1 Sep 2018.

Chin, Gilbert J. “Organismal Biology: Flying Along a Logarithmic Spiral.” Science, 290 (5498): 1857, 8 Dec 2000, doi:10.1126/science.290.5498.1857c Coxeter, H. S. M. Introduction to Geometry. Wiley: New York, 1961, p. 172.

D’Agnese, Joseph Blockhead: The Life of Fibonacci. Henry Holt and Co. (BYR); First edition. 30 Mar 2010, p. 40.

Davis, T. Antony. “Why Fibonacci Sequence for Palm Leaf Spirals?.” (1971), http://library.isical.ac.in:8080/ jspui/bitstream/10263/897/1/TFQ-9-3-1971- P237-244.pdf. Accessed 04 Sep 2018.

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