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The Golden Relationships: an Exploration of Fibonacci Numbers and Phi

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The Golden Relationships: an Exploration of Fibonacci Numbers and Phi The Golden Relationships: An Exploration of Fibonacci Numbers and Phi Anthony Rayvon Watson Faculty Advisor: Paul Manos Duke University Biology Department April, 2017 This project was submitted in partial fulfillment of the requirements for the degree of Master of Arts in the Graduate Liberal Studies Program in the Graduate School of Duke University. Copyright by Anthony Rayvon Watson 2017 Abstract The Greek letter Ø (Phi), represents one of the most mysterious numbers (1.618…) known to humankind. Historical reverence for Ø led to the monikers “The Golden Number” or “The Devine Proportion”. This simple, yet enigmatic number, is inseparably linked to the recursive mathematical sequence that produces Fibonacci numbers. Fibonacci numbers have fascinated and perplexed scholars, scientists, and the general public since they were first identified by Leonardo Fibonacci in his seminal work Liber Abacci in 1202. These transcendent numbers which are inextricably bound to the Golden Number, seemingly touch every aspect of plant, animal, and human existence. The most puzzling aspect of these numbers resides in their universal nature and our inability to explain their pervasiveness. An understanding of these numbers is often clouded by those who seemingly find Fibonacci or Golden Number associations in everything that exists. Indeed, undeniable relationships do exist; however, some represent aspirant thinking from the observer’s perspective. My work explores a number of cases where these relationships appear to exist and offers scholarly sources that either support or refute the claims. By analyzing research relating to biology, art, architecture, and other contrasting subject areas, I paint a broad picture illustrating the extensive nature of these numbers. In particular, the role of Fibonacci Numbers in the plant and animal kingdoms is analyzed within the context of quantification. Likewise, art, architecture, and aesthetics is examined for Fibonacci and Golden relationships to determine if an innate human preference for these associations exists and to what degree it can be measured. Finally, I draw conclusions supporting the existence of these unique relationships, and offer theories supporting adaptive advantages in living entities. Watson iii Table of Contents Abstract ................................................................................................................................................. iii List of Illustrations .................................................................................................................................. vi Acknowledgements ................................................................................................................................ ix Introduction ............................................................................................................................................ 1 Chapter 1 ................................................................................................................................................ 5 The History of Ø and Leonardo Fibonacci ......................................................................................... 5 Fibonacci Numbers Defined ............................................................................................................. 7 Defining the Golden Section and Golden Rectangle.......................................................................... 8 An Approximation of Ø (Phi) .......................................................................................................... 11 The Golden Section and Golden Rectangle ..................................................................................... 12 The Golden Spiral .......................................................................................................................... 13 The Rabbit Question ...................................................................................................................... 14 Interdependent Concepts .............................................................................................................. 15 Chapter 2 .............................................................................................................................................. 16 Plants and Fibonacci Numbers ....................................................................................................... 16 Phyllotaxis and Light Capture ......................................................................................................... 23 Structural Integrity......................................................................................................................... 24 Chapter 3 .............................................................................................................................................. 26 Phi and the Animal Kingdom .......................................................................................................... 26 Microbial Organisms ...................................................................................................................... 26 Earthworms and Arthropoda ......................................................................................................... 27 The Human Skeletal System ........................................................................................................... 28 The Human Cochlea ....................................................................................................................... 28 The Hand, Golden Spiral, and Fibonacci Numbers .......................................................................... 29 Thackeray’s Species Constant ........................................................................................................ 35 Watson iv Chapter 4 .............................................................................................................................................. 37 Aesthetics and Commercial Applications of Ø ................................................................................ 37 Aesthetic Theory and Logo Design ................................................................................................. 38 Facial Aesthetics ............................................................................................................................ 39 Preference for Golden Features? ................................................................................................... 41 Financial Markets and Fibonacci Numbers ..................................................................................... 42 Commercial Impact ........................................................................................................................ 43 Chapter 5 .............................................................................................................................................. 45 Art and Architecture ...................................................................................................................... 45 Art and the Golden Relationships ................................................................................................... 46 Did Michelangelo Utilize the Golden Ratio? ................................................................................... 48 Golden Relationships and Leonardo Da Vinci ................................................................................. 49 Great Pyramid of Egypt .................................................................................................................. 51 The Egyptian Calendar and Platonic Solids ..................................................................................... 53 The Greek Parthenon ..................................................................................................................... 54 Conclusion ............................................................................................................................................. 56 Bibliography .......................................................................................................................................... 61 Watson v List of Illustrations Chapter 1 Figure 1: First 10 Fibonacci Numbers and Derivations. Watson (2017). Figure 2a: The Golden Rectangle. Debnath (2011) pg. 342. Figure 2b: The Golden Triangle. Debnath (2011) pg. 342. Figure 3: Calculation of Phi (Ø). Watson (2017). Figure 6: Golden Section. Watson (2017). Figure 7: Golden Rectangle. Watson (2017) Figure 8: Golden Spiral. Reich [Date Unknown] pg. 9. Figure 9: The Rabbit Problem. Wille (2012) pg. 216. Figure 10: Phi and Related Concepts. Watson (2017). Chapter 2 Figure 11: Sunflower with clockwise spirals. Photo by Scott Hotton. Rehmeyer, J. (2007) The Mathematical Lives of Plants. Available at: http://www.sciencenews.org/article/mathematical- lives-plants (Accessed: 31 January 2017). Figure 12: Sunflower with counterclockwise spirals. Photo by Scott Hotton. Rehmeyer, J. (2007) The Mathematical Lives of Plants. Available at: http://www.sciencenews.org/article/mathematical-lives-plants (Accessed: 31 January 2017). Figure 13: Pinecone with spiral pattern. Knott, R. (2005) Fibonacci Pine Cone [Online]. Available at: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html (Accessed: 31 January 2017). Figure 14: Resin Canals in Lodgepole Pine. Fredeen et al. (2011) pg. 1541. Figure 15: Spiral patterns in pinecone and pineapple. Atela (2002) pg.642. Figure 16: Pi used as divergence
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