Figure 1-Fibonacci. Image: Public domain, via Wikimedia Commons.
Most mathematically inclined people are familiar with the famous and unique Fibonacci sequence. Defined by the recurrence relation (*) Fn=Fn-1+Fn-2 with initial values F1=1 and F2=1 and (or sometimes F0=1 and F1=1), the Fibonacci sequence is an integer sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …) with many remarkable mathematical and real world applications. However, it seems that few are as well informed on the man behind this sequence as they are on the sequence itself. Did you know that Fibonacci didn’t even discover the sequence? Of course not! Predating Fibonacci by almost a century, the so called “Fibonacci sequence” was actually the brainchild of Indian mathematicians interested in poetic forms and meter who, through studying the unique arithmetic properties of certain linguistic sequences and syllable counts, derived a great deal of insight into some of the most fascinating mathematical patterns known today. But with a little bit of time (few hundred years), some historical distortion, inaccurate accreditation[1], and a healthy dose of blind western ethnocentrism and voila! Every high school kid in America now thinks there is a connection between Fibonacci and pizza. Or is it Pisa? (That’s a pun, laugh.) While often given more credit than deserved for the “discovery” of the sequence, Fibonacci was nonetheless an instrumental player in the development of arithmetic sequences, the spread of emerging new ideas, and in the advancement of mathematics as a whole. We thus postpone discussion of Fibonacci’s sequence – don’t worry, we shall return – to examine some of the other significant and often overlooked contributions of the “greatest European mathematician of the middle ages.”[1]
Born around the year 1175 in Pisa, Italy, Leonardo of Pisa (more commonly known as Fibonacci) would have been 840 years old this year! (Can you guess the two indexing numbers between which Fibonacci’s age falls?[2]) The son of a customs officer, Fibonacci was raised in a North African education system under the influence of the Moors.[3] Fibonacci’s fortunate upbringing and educational experience allowed him the opportunity to visit many different places along the Mediterranean coast. It is during these travels that historians believe Fibonacci may have first developed an interest in mathematics and at some point come into contact with alternative arithmetic systems. Among these was the Hindu-Arabic number system – the positional number system most commonly used in mathematics today. It appears that we owe a great deal of respect to Fibonacci for, prior to introducing the Hindu-Arabic system to Europe, the predominant number system relied on the far more cumbersome use of roman numerals. It is interesting to note that while the Hindu-Arabic system may have been introduced to Europe as early as the 10th century in the book Codex Vigilanus, it was Fibonacci who, in conjunction with the invention of printing in 1482, helped to gain support for the new system. In his book Liber abbaci[4], Fibonacci explains how arithmetic operations (i.e., addition, subtraction, multiplication, and division) are to be carried out and the advantages that come with the adoption of such a system.
Figure 2-Golden spiral. Image: Weisstein, Eric W. “Golden Spiral.” From MathWorld–A Wolfram Web Resource.
Whereas the number system most familiar to us uses the relative position of numbers next to each other to represent variable quantities (i.e., the 1’s, 10’s, 100’s, 1000’s, … place), Roman numerals rely on a set of standard measurement symbols which, in combination with others, can be used to express any desired quantity. The obvious problem with this approach is that it severely limits the numbers that can be reasonably represented by the given set of symbols. For example, the concise representation of the number four hundred seventy eight in the Hindu-Arabic system is simply 478 in which “4” is in the hundreds place, “7” is in the tens place, and “8” is in the ones place. In the Roman numeral system, however, this same number takes on the form CDLXXVIII. As numbers increase arbitrarily so does the complexity of their Roman numeral representation. The adoption of the Hindu-Arabic number system was, in large part, the result of Fibonacci’s publications and public support for this new way of thinking. Can you imagine trying to do modern mathematical analysis with numbers as clunky as MMMDCCXXXVIII??? Me either. Thanks, Fibonacci!
Fibonacci’s other works include publications on surveying techniques, area and volume measurement, Diophantine equations, commercial bookkeeping, and various contributions to geometry.[4] But among these works nothing stands out more than that of Fibonacci’s sequence – yes, we have returned! Among the more interesting mathematical properties of Fibonacci’s sequence is undoubtedly its connection to the golden ratio (shall be defined shortly). To illustrate, we look momentarily at the ratios of several successive Fibonacci numbers. Beginning with F1=1 and F2=1 we see that the ratio F2/F1=1. Continuing in this manner using the recurrence relation (*) from above or any suitable Fibonacci table we find that F3/F2=2, F4/F3=3/2, F5/F4=5/3,F6/F5=8/5, F7/F6=13/8, F8/F7=21/13, … As the indexing number tends to infinity, the ratio of successive terms converge to the value 1.6180339887… (the golden ratio) denoted by the Greek letter phi. We may thus concisely represent this convergent value by the expression as the lim n–> infinity (Fn+1/Fn). Studied extensively, the golden ratio is a special value appearing in many areas of mathematics and in everyday life. Intimately connected to the concept of proportion, the golden ratio (sometimes called the golden proportion) is often viewed as the optimal aesthetic proportion of measurable quantities making it an important feature in fields including architecture, finance, geometry, and music. Perhaps surprisingly, the golden ratio has even been documented in nature with pine cones, shells, trees, ferns, crystal structures, and more all appearing to have physical properties related to the value of (e.g., the arrangement of branches around the stems of certain plants seem to follow the Fibonacci pattern). While an interesting number no doubt, we must not forget that mathematics is the business of patterns and all too often we draw conclusions and make big picture claims that are less supported by evidence and facts than we may believe. There is, in fact, a lot of “woo” behind the golden ratio and the informed reader is encouraged to be weary of unsubstantiated claims and grandiose connections to the universe. It is also worth mentioning that, using relatively basic linear algebra techniques, it is possible to derive a closed-form solution of the n-th Fibonacci number.
Figure 3-Computing the 18th Fibonacci Number in Mathematica.
Omitting the details (see link for thorough derivation), the n-th Fibonacci number may be computed directly using the formula Fn=((φ)(n+1)+((-1)(n-1)/(φ)^(n-1))/((φ2)+1).[5] While initially clunky in appearance, this formula is incredibly useful in determining any desired Fibonacci number as a function of the indexing value n. For example, the 18-th Fibonacci number may be calculated using F18=((φ)(18+1)+((-1)(18-1)/(φ)^(18-1))/((φ2)+1)=2584. Comparing this value to a list of Fibonacci numbers and to a Mathematica calculation (see picture above), we see that the 18-th Fibonacci number is, indeed, 2584. Without having to determine all previous numbers in the sequence, the above formula allows us to calculate directly any desired value in the sequence saving substantial amounts of time and processing power.
From the study of syllables and poetic forms in 12th-century India to a closed-form solution for the n-th Fibonacci number via modern linear algebra techniques, our understanding of sequences and the important mathematical properties they possess is continuing to grow. Future study may reveal even greater mathematical truths whose applications we cannot yet conceive. It is thus the beauty of mathematics and the excitement of discovery that push us onward, compel us to dig deeper, and to learn more from the world we inhabit. Who knows, you might even be the next Leonardo of Pizza – errrrr Pisa. What patterns will you find?
[1] French mathematician Edouard Lucas (1842-1891) was the first to attribute Fibonacci’s name to the sequence. After which point little is ever mentioned of the Indian mathematicians who laid the groundwork for Fibonacci’s research.
[2] Answer: n=15 –> 610 and n=16 –> 987.
[3] Medieval Muslim inhabitants of the Maghreb, Iberian Peninsula, Sicily, and Malta.[2]
[4] Translation: Book of Calculation[3]
Bibliography
[1] Knott, Ron. Who Was Fibonacci? N.p., 11 Mar. 1998. Web. 27 Apr. 2015.
[2] “Moors.” Wikipedia. Wikimedia Foundation, n.d. Web. 27 Apr. 2015.
[3] Leonardo Pisano – page 3: “Contributions to number theory”. Encyclopædia Britannica Online, 2006. Retrieved 18 September 2006.
[4] “Famous Mathematicians.” The Greatest Mathematicians of All Time. N.p., n.d. Web. 28 Apr. 2015.
[5] Grinfeld, Pavel. “Linear Algebra 18e: The Eigenvalue Decomposition and Fibonacci Numbers.” YouTube. YouTube, 2 Dec. 2014. Web. 28 Apr. 2015.
Figure 1: Fibonacci. Digital image. Wikimedia Foundation, n.d. Web. 27 Apr. 2015.
Figure 2: Golden Spiral. Digital image. Mathworld. Wolfram, n.d. Web. 1 May 2015.
Figure 3: Ross, Andrew Q. Closed-Form Computation of Fibonacci. Digital image. Mathematica, 28 Apr. 2015. Web. 28 Apr. 2015.
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