By Dalia Zidan
Introduction
Zero, One, One, Two, Three, Five, Eight, Thirteen, Twenty-One, Thirty-Four, Fifty-Five, Eighty-Nine, Infinity. A recurring natural pattern that represents nature at its core, and widely governs many concepts within society today: The Fibonacci Sequence. While the Fibonacci Numbers may look like a simple set of integers, the sequence extends to representing the structures and physical models we see in reality, from the satisfying composition of music, human anatomy, petals on a flower, and branches on a tree, to most importantly, the visual concept of mathematics: an arc-based spiral that can describe many naturally-ocurring today.
The Fibonacci Sequence
First described by early Indian mathematician and poet, Acharya Pingala, in pieces of Sanskrit literature, the Fibonacci Sequence is generally agreed to have been actively discovered by Leonardo Fibonacci, an Italian mathematician, in the early 13th century (Lakshmi). At its core, each number in the Fibonacci Sequence is created by adding the two numbers preceding it (see Diagram 1). Fibonacci largely traveled throughout the Middle East, Northern Africa, and India for the majority of his mathematical career, finding within the core of these communities traders and merchants that desired to track their transactions through the use of arithmetic (O’Conner & Robertson). The mathematical systems set in place around them, specifically the Roman numeral system, limited them from using addition, subtraction, multiplication, and division to complete this goal. It was Fibonacci’s introduction to the Hindu-Arab numeral system by an Arab teacher in North Africa, however, that proved to solve the problems that these groups faced. Precisely, in 1202, Fibonacci emphasized the superiority of the Hindu-Arab numeral system over the Roman numeral system in his book called the Liber Abaci, which compared the two systems of arithmetic (O’Conner & Robertson).
Diagram 1
Sourced from Math is Fun (“Fibonacci Sequence”)
Moreover, Fibonacci’s book discussed several mathematical problems, particularly one regarding rabbit breeding in Chapter 12, whose outcome was the Fibonacci Sequence (see Diagram 2) (Gravett). In this problem, a pair of rabbits (one male, one female), who obtain the ability to reproduce at the age of one month, are placed on a vast field. Considering that a female rabbit is able to birth another pair of rabbits (one male, one female) at the end of every month from the second month on and never dies, Fibonacci posed the following challenge: how many pairs of rabbits will be produced by the end of one year? (Gravett).
Diagram 2
Sourced from LibreTexts Mathematics (“Fibonacci’s Rabbits”)
Note: The term “juvenile” refers to a pair of new baby rabbits that have not yet matured to give birth.
Fibonacci used a simple, core equation to solve this problem: Fn = (Fn−1) + (Fn−2), where n denotes the number of rabbits at the end of the month in the field (Fn), Fn-1 represents the total number of rabbits in the previous month, and Fn-2 represents the total number of rabbits two months previously (Gravett). Thus, by adding these two together, you get the total number of rabbits at the end of nth month (see Diagram 3).
Diagram 3
Sourced from Emory Oxford College Math Center (“Fibonacci’s Rabbits”)
Note: Notice the Fibonacci numbers on the right-hand side.
As the months increase, the original pair of rabbits will continue to give birth. After the new pair of rabbits have matured into adulthood, they will have the ability to reproduce too, creating another pair of rabbits (Chasnov). Understanding this, to answer Fibonacci’s puzzle, considering there are 12 months in a year, the number of rabbit pairs at the start of the 13th month must be the outcome. As the rabbit population increases following the Fibonacci Sequence, at the 13th month, F(13) is equal to 233 rabbits (the 13th number in the sequence), which is the solution (Chasnov).
The Liber Abaci not only proved to be extremely beneficial and revolutionary in terms of its contributions to ancient banking and commerce, but also changed the way we look at the basis and growth in nature today. In conjunction with the famous sequence, another process follows closely: the Golden Ratio (“The Golden Ratio: Is It Myth or Math?”).
The Golden Ratio
First recognized by Pythagoras, a famous Greek philosopher and mathematician, though not fully utilized until 447 BC by the Greek sculptor Phidias in order to create a well-known Athenian temple, the Parthenon, the Golden Ratio goes by a special name to commemorate its use: “Phi” (φ) (Dove). Similar to Pi (π), the special number goes on infinitely but is approximately referred to as: 1.618.
Phi is particularly noteworthy because of its correlation with the Fibonacci Sequence. For example, take a golden rectangle with sides “a” and “b”, which is golden because of its proportions (Dove). The ratio between the side lengths of the rectangle matches the value of Phi. You cannot escape from this ratio, even if you were to cut a square from this rectangle, the proportions would still remain the same (see Diagram 4). Ancient philosophers described this as the Golden Ratio. Similarly, as the Fibonacci Sequence increases slowly, the ratio between the two preceding numbers gets closer and closer to the value of Phi. Taking the Fibonacci Numbers, creating squares that have side lengths as long as their respective numbers, grouping them together to be adjacent to one another (golden rectangle), and creating arcs of a circle throughout the squares from their corners, creates an important Golden Spiral, at which as the spiral increases in size, its growth can be attributed to following the Golden Ratio (see Diagram 5) (“The Golden Ratio: Is It Myth or Math?”).
Diagram 4
Sourced from Pi Day (“Golden Rectangle Calculator”)
Diagram 5
Sourced from Imagination Station (“The Fibonacci Sequence”)
While the Golden Spiral can be found in many patterns of nature, many suggest that the frequency at which these patterns are recognized is wholly exaggerated (“The Golden Ratio: Is It Myth or Math?”). Simply, the pattern-sensing abilities of humans heightens the awareness of these spirals in the natural world, making people believe that they can be found everywhere, ranging from the number of family members in each honeybee colony to hurricane spirals. For example, perhaps one of the most recognized images of the Fibonacci Spiral is the Nautilus shell (see Diagram 6). While cited to follow the ratio of the Fibonacci numbers at each respective curve, the shell actually follows a logarithmic spiral, ranging from a ratio of 4 to 3 (providing a value of 1.333, rather than 1.618, the value of Phi). Despite maybe being exaggerated in significance among the mathematical field and popular culture, the fibonacci sequence makes many frequent appearances throughout our daily lives. For example, considering plants thrive on light, they take on certain growth patterns to ensure they receive the maximum amount of sunlight to be absorbed. In contrast to the leaves of plants growing at every 1⁄2, 1⁄4, 1⁄5, or any rational number amount of turns that prompts the leaves to overlap each other eventually, many evolved plants today have a leaf growth pattern turning at the rate of Phi. Specifically, 1/φ multiplied by a fraction of a circle, 360, resulting in a 137.5-degree turn (the Golden Angle) around each time, eventually creating a Fibonacci Spiral. Through this growth pattern, plants are able to reach the maximum amount of sunlight possible because of the position of their leaves (see Diagram 7) (“The Golden Ratio: Is It Myth or Math?”).
Diagram 6
Sourced from The Shallow Sky (“The Fibonacci Spiral and the Nautilus”)
Diagram 7
Sourced from Wikipedia (“Golden angle”)
Sourced from Go Figure Math (“The Golden Angle”)
Conclusion
There is no doubt that the Fibonacci Sequence and the Golden Ratio hold major significance, importance, and a certain sort of mystery to them. Understanding both of these patterns makes it clear that mathematics does not simply exist to find the value of x, or calculate the number of apples Tom will have after t years, but also extends to finding the beauty and order in the universe around us. Perhaps discovering properties like these brings us one step closer to understanding the universe in its complexity and entirety.
Works Cited
Chasnov, Jeffrey R. “Fibonacci’s Rabbits.” LibreTexts Mathematics, LibreTexts, 18 July 2022, math.libretexts.org/Bookshelves/Applied_Mathematics/ Mathematical_Biology_(Chasnov)/02%3A_Age-structured_Populations/2.01%3A_Fibonacci’s_Rabbits#:~:text=In%201202%2C%20Fibonacci%20proposed%20the,pai r%20and%20then%20mate%20again.
Dove, Laura L. “Does the Parthenon really follow the golden ratio?” HowStuffWorks, 22 Apr. 2015, history.howstuffworks.com/ history-vs-myth/parthenon-golden-ratio.htm.
“The Golden Ratio: Is It Myth or Math?” YouTube, uploaded by Be Smart, 10 Mar.
2021, http://www.youtube.com/watch?v=1Jj-sJ78O6M.
Gravett, Emily. “The rabbit problem.” NZ Maths, MacMillan, nzmaths.co.nz/
resource/rabbit-problem#:~:text=Summary%3A,is%20illustrated%20through%20a%20calendar.
Lakshmi, B. “Acharya Pingala’s Maathra Meru.” Prayoga, http://www.prayoga.org.in/post/
acharya-pingala-s-maathra-meru.
O’Connor, JJ, and E F Robertson. “Leonardo Pisano Fibonacci.” MacTutor, University of St Andrews, Oct. 1998, mathshistory.st-andrews.ac.uk/Biographies/Fibonacci/.
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