Mathematical and Statistical Sciences
FIBONACCI SEQUENCE
Melissa Abeyta
May 5, 2005
The Fibonacci sequence is extremely important and appears in many different places. This paper examines many aspects of the sequence. It is interesting to note that the sequence was not that important to Fibonacci however, since it has been discovered there has been a tremendous amount of research done on it (Evansville website). There is a Fibonacci Association in San Jose, California that was established in 1963 and the Fibonacci Quarterly is a journal that is devoted to the study of integers with special properties. It is amazing how one problem that surfaced in 1202 could make such an impact on the world.
HISTORY OF FIBONACCI AND THE FIBONACCI SEQUENCE
Fibonacci’s real name was Leonardo Pisano. He was born in Italy around 1170. Little is known about his life and what we do know is found in a few sentences in the 1228 edition of his famous Liber Abbaci (sometimes spelt Liber Abaci) (Evansville website). We know that his father, Guilielmo, held a diplomatic post in the Republic of Pisa. “His father’s job was to represent the merchants of the Republic of Pisa who were trading in Bugia,” (St Andrews website). Bugia is now called Bejaia and is located in Northeastern Algeria. Fibonacci went there with his father and was taught mathematics and accounting. He traveled with his father and visited Egypt, Syria, Greece, Sicily, and Provence. He was able to recognize “the enormous advantages of the mathematical systems used in the countries he visited,” (St Andrews website).
Around the turn of the century Fibonacci returned to Pisa and for the next twenty-five years he wrote books. The texts that he wrote “played an important role in reviving ancient mathematical skills and he made significant contributions of his own,” (St Andrews website). Only five works by Fibonacci have been preserved:
1. Liber Abbaci (1202, 1228)
2. Practica Geometriae (1220/1221)
3. A writing entitled Flos (1225)
4. An undated letter to Theodorus, the imperial philosopher
5. Liber quadratorum (1225)
We know that he wrote some other books, however they were lost. He wrote a book on commercial arithmetic, Di mino guisa, that was lost as well as his commentary on Book X of Euclid’s Elements “which contained a numerical treatment of irrational numbers which Euclid had approached from a geometric point of view,” (St Andrews website).
Fibonacci was well known for being a great mathematician and caught the attention of the Holy Emperor, Frederick II. The emperor was called “Stuper Mundi” (Wonder of the World). Frederick was also the King of the Two Sicilies and was considered to be one of the most remarkable men on the Middle Ages. He “encouraged learning and scholarship of every kind, having a special interest in mathematics and science,” (Evansville website). [Evansville website]
“Fibonacci corresponded with two of the Emperor’s scholars, Master Theodore and Michael Scott (Scotus) (whom Dante, in his Divina Commedia, consigned to hell as a wizard for perpetuating “magical frauds”),” (Evansville website). Fibonacci and Frederick met at the Emperor’s palazzo in Pisa. Master John of Palemo, who was a scholar, proposed mathematical questions for Fibonacci to solve. Some writers state that a mathematical tournament between Fibonacci and other mathematicians took place, but this is not likely. “At the time of his meeting with Frederick in the 1220’s, Fibonacci was probably at the height of his prowess,” (Evansville website). [Evansville website]
It is not known when Fibonacci died or how he died. The last time he was mentioned in a document was in 1240. He was a remarkable mathematician and has been called “the first great mathematician of the Christian West,” (Dictionary of Scientific Biography, 1971, p. 604).
The famous Fibonacci sequence came out of Liber Abbaci. The first edition was written in 1202 after he returned to Italy from traveling. It was based on arithmetic and algebra that he had accumulated during his travels with his father. He released a revised edition in 1228 and dedicated it to Michael Scott (Scotus), one of Frederick II scholars. It has been stated that the 2nd edition added new material and removed material that was superfluous (Dictionary of Scientific Biography, 1971). The book was divided into 15 chapters, which were analyzed in four sections.
In section 1 (Chapters 1-7), Fibonacci introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe. He began Liber Abacci with the following statement that was simple but profound.
“The nine Indian figures are: 9 8 7 6 5 4 3 2 1 With these nine figures, and with the sign 0…any number may be written, as is demonstrated below” (Evansville website). The new numbers were favorable in comparison to the traditional Roman numerals (Sigler, 2002). He also introduced the fraction bar. In this section he developed rules for the factoring of fractions into sums of unit factors. “Numerous tables (for multiplication, prime numbers, factoring numbers, etc) complete the text,” (Dictionary of Scientific Biography, 1971, p. 606).
Section 2 (Chapters 8-11), contained problems revolving around merchants, such as price of goods, calculation of profits, how to convert between the various currencies used in the Mediterranean countries, and some Chinese mathematical problems (St Andrews website).
Section 4 (Chapter 14 and 15) showed Fibonacci “to be a master in the application of algebraic methods and an outstanding student of Euclid,” (Dictionary of Scientific Biography, 1971, p. 607). Chapter 14 begins with a few formulas of general arithmetic and is devoted to calculations with radicals (Dictionary of Scientific Biography, 1971). Chapter 15 shows how Leonardo had complete control over the geometrical and algebraic methods for solving quadratic equations and had great skill in using them in applied problems (Dictionary of Scientific Biography, 1971).
Section 3 (Chapters 12 and 13) is being taken out of order for a reason. This section gives us the Fibonacci sequence. It is the most extensive section and contains many questions and puzzles. In Chapter 12 Fibonacci states the following problem:
“A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive” (Evansville website).
The problem has been stated a little differently in many sources. Here is a more simple explanation of the problem.
1. Start with a pair of rabbits (one male and one female)
2. Assume that all months are of equal length
3. Rabbits begin to reproduce two months after their own birth
4. After reaching two months of age, each pair produces a mixed pair (one male and one female) and then another mixed pair each month thereafter
5. No rabbit dies
How many pairs of rabbits will there be after one year? [Morris, n.d.]
The following figure is a visual representation of the rabbits being added each month.
Figure 1.1 taken from the Rabbits, Cows, and Bees Family Tree section on the University of Surrey website.
The following is a table representation of the results of the problem.
|Month |Pairs of Rabbit |
|1 |1 |
|2 |1 |
|3 |2 |
|4 |3 |
|5 |5 |
|6 |8 |
|7 |13 |
|… |… |
The solution of this problem is the famous Fibonacci sequence.
I found a discrepancy in my research. One source said, “although it is almost certain that he knew, Fibonacci never wrote that each term was found by adding two previous terms,” (Ballew, n.d.). It states that the first record of such a statement was made almost 400 years after Fibonacci by Kepler (Ballew, n.d.). Contradicting this is the book A History of Mathematics written by Victor J. Katz in 1998 which states that Fibonacci listed the sequence in the margin and “notes that each number is found by adding the two previous numbers and ‘thus you can do it in order for an infinite number of months’,” (p. 309). I believe the Katz book to be more reliable and think that he was well aware of the pattern that the problem produced. Fibonacci was a highly regarded mathematician and it seems unreasonable to believe that he did not see the pattern and note it.
The resulting Fibonacci sequence was the first recurrent series (Dictionary of Scientific Biography, 1971). Each number within the sequence is called a Fibonacci number. The recursive definition for producing Fibonacci numbers is:
Fn = F(n-1) + F(n-2) where n>2
The following is a proof of the formula.
Proof: Let S = {n [pic][pic] Fn = F(n-1) + F(n-2) where n>2}
Note that 3 [pic] S
F3 = F2 + F1 = 1+1 = 2
Suppose that {2,….,m}[pic]S for m>2
Consider Fm+1
In the m+1 stage we need to have all of the rabbits for the previous stage because they are still alive and then we need to add all of the rabbits from two months previous because they are now able to produce children and represent the number of newborns
Therefore Fm+1 = Fm + Fm-1
Thus, m+1 [pic] S, so by the PCI S = [pic]
FIBONACCI SEQUENCE AND THE GOLDEN RATIO
The story of the golden ratio begins with Ptolemy I. Ptolemy I was a successor to Alexander the Great who rose to power in 323 B.C. He established a school known as the ‘Museum’ in Alexandria, Egypt, which was founded around 300 B.C. The “father of formal deductive geometry,” Euclid, was one of the teachers (Livio, 2003, p. 52).
Euclid defined geometry and number theory in his famous book called the Elements. In this book, Euclid stated that a line can be divided into what he called its “extreme and mean ratio” (Livio, 2003, p. 52). In Book IV of the Elements, “a straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segments, so is the greater to the lesser,” (Livio, 2003, p. 52) Simply stated, “Euclid proposed the possibility of dividing a line into two segments such that the ratio of the whole line to the larger segment is the same as the ratio between the larger and the smaller segments. If we call the endpoints of Euclid’s straight line A and C, and its dividing point B, then mathematically the ratio of the line AC to its larger segment, AB, is equal to the ratio of line segments BC to AB,” (Livio, 2003, p. 53) We can write it numerically as:
AB + BC = AC and AB/AC = BC/AB
The following is a visual representation. Notice that we can also say that the line is 1, where the line AB equals b and the line BC equals 1-b.
This line was taken from the ‘What is the golden section (or Phi)?’ section of the University of Surrey website.
From this we get b/1 = (1-b)/b
b2 = 1-b or b2 + b –1 = 0
This result is a quadratic equation, which can be solved using the quadratic formula
x = (-b +- b2 – 4ac)/2a
Using the quadratic equation we get the following two answers for x.
x = (1 + 5 )/2 or (1 – 5 )/2
The values of these two numbers are 1.6180339887 and –0.618033987. The first number is called the golden ratio or Phi. The second value is called – phi. We use the first letter of each to tell us which one is bigger. Phi is 1.618… and – phi –0.618… [Knotts, 2004, What is the golden section (or Phi)?]
For centuries following Euclid’s publication of Elements mathematicians found it difficult to find an appropriate name to describe the ratio. In 1509, Luca Pacioli called the ratio the “divine proportion” in a book he had published (Livio, 2003). The book contains drawings made by Leonardo da Vinci. “It is probably Leonardo who first called it the section aurea (which is Latin for the golden ratio),” (Knotts, 2004, What is the golden section (or Phi)?). Finally, the 20th century mathematician Mark Burr gave the ratio the designation Phi. “Phi is the first letter in the name of the Greek sculptor Phidias (ca 490 – 430 B.C.) who many people believe used the golden ratio in proportioning his sculptures,” (Livio, 2003, p. 53)
The golden ratio comes up in more than mathematics and art. We will see later that the golden ratio, as well as the Fibonacci numbers, appears in many aspects of nature.
We can find the golden ratio in the Fibonacci sequence as well by taking the ratio of two successive numbers in Fibonacci’s sequence. By dividing each Fibonacci number by the previous number we get the following:
1/1 = 1 2/1 = 2 3/2 = 1.5 5/3 = 1.6666 8/5 = 1.6 13/8 = 1.625
21/13 = 1.61538
We can see what is happening if we plot the points on a graph.
[pic]
Figure 1.6 was taken from ‘Fibonacci numbers and the Golden number’ section of the University of Surrey website.
The ratio seems to be settling down to a particular value: the golden ratio. “In Kepler’s words, “The further we advance from the number one, he more perfect the example becomes,” (Livio, 2003, p. 58).
Why does this happen? Let us look at the following equation that represents the Fibonacci sequence.
F(i+2) = F(i+1) + F(i)
Figure 1.6 on the previous page shows that the ratio of F(i+1)/ F(i) will get closer and closer to a particular value which we will call x. If we look at three consecutive Fibonacci numbers, F(i), F(i+1) , and F(i+2) then for very large values of i, the ratio of F(i) and F(i+1) will be almost the same as F(i+1) and F(i+2). We will now see what happens if both of these are the same value: x.
F(i+1) / F(i) = F(i+2) / F(i+1) = x
We can now take our previously stated formula for the Fibonacci sequence and replace F(i+2) with F(i+1) + F(i) as follows:
F(i+2) / F(i+1) = (F(i+1) + F(i) )/ F(i+1)
F(i+2) / F(i+1) = F(i+1) / F(i+1) + F(i) / F(i+1)
F(i+2) / F(i+1) = 1 + F(i) / F(i+1)
So now we have x = F(i+1) / F(i) = 1 + F(i) / F(i+1)
The last fraction F(i) / F(i+1) is the reciprocal of F(i+1) / F(i) which is x. We can replace F(i) / F(i+1) with 1/x.
x = F(i+1) / F(i) = 1 + F(i) / F(i+1) = 1 + 1/x. By multiplying both sides by x, we get x2 = x +1
We have seen this equation before when defining Phi. The result states that x is exactly Phi. [Knotts, 2004, Phi and the Fibonacci numbers]
There are two values that satisfy x2 = x +1. The other value, which is –0.618… can be found if we extend the Fibonacci sequence backwards. We will keep the same Fibonacci relationship (the next number is the sum of two previous numbers), to find the Fibonacci numbers before 0. The following is a table of the new Fibonacci numbers.
|n |-8 |-7 |
| |1 |1 |
|Parents |1 |2 |
|Grandparents |2 |3 |
|Great Grandparents |3 |5 |
|Great-Great Grandparents |5 |8 |
|Great-Great-Great Grandparents |8 |13 |
Figure 1.2 represents the number of parents, grandparents, etc that male and female honeybees have.
As you can see, the family tree of both the male and female honeybee results in the Fibonacci sequence.
We will now look at Fibonacci Rectangles and Shell Spirals. “Fibonacci rectangles are those built to the proportions of consecutive terms in the Fibonacci series. Because of the nature of the series, any Fib(n):Fib(n+1) rectangle can be divided exactly into the all the previous Fibonacci rectangles. For example, the diagram below shows a 34:21 rectangle within which is contained the 21:13, 13:8, 3:5, 5:3, 3:2, 2:1 and 1:1 rectangles. You will also notice that in doing so we have divided each of our rectangles into perfect squares of length of side Fib(k) for k in the natural numbers, less than or equal to n. (Note: the numbers inside the squares show the length of the sides).
[pic]
Figure1.3 was taken from ‘Fibonacci Rectangles and Spirals’ section of the University of Bath website.
If we draw a spiral, or a quarter of a circle, in the squares it will look very similar to the kind of spiral you see in nature (see Figure 1.4). This spiral is not a true mathematical spiral because it is made up of fragments which are parts of circles and does not get smaller and smaller. “The spirals-in-the-squares makes a line from the center of the spiral and increases by a factor of the golden number in each square. So points on the spiral are 1.618 times as far from the center after a quarter turn,” (Surrey website). [Knotts, 2004, Fibonacci Rectangles and Shell Spirals]
Figure 1.4 is taken from the ‘Fibonacci Rectangles and Shell Spirals’ section of the University of Surrey website.
If we keep examining nature and the Fibonacci numbers we see that it shows up in the architecture of plants. The positioning of leaves around a stem and the number of leaves relates back to the Fibonacci numbers and sequence. Leaves are spirally distributed around the stems of plants. “They tend to be separated by an angle of 137.5o. This is the radical equivalent of the golden ratio, 1.618, the ultimate proportional increase between successive Fibonacci numbers,” (Surridge, 2003, p.27). In addition, the number of leaves contained in these spirals form consecutive Fibonacci numbers. The following is a list of plants and the number of petals that are found on them. [Knotts, 2004, Fibonacci numbers, the Golden section, and plants]
|PLANT |NUMBER OF PETALS |
|Lily, iris |3 |
|Buttercups, columbine |5 |
|Delphiniums |8 |
|Ragwort, Corn marigold, some daisies |13 |
|Black-eyed susan, Aster |21 |
|Plantain, pyrethrum |34 |
|Daisies, Michaelmas, the asteraceae family |55,89 |
Figure 1.4 is adapted from the ‘Fibonacci numbers, the Golden section, and plants’ section on the University of Surrey website.
Fibonacci numbers can also be seen in the how seeds are arranged on flower heads and in pinecones. Again we see spirals in these items that curve to the left and to the right. If you count the spirals they almost always are Fibonacci numbers. The picture on the left is of a real sunflower with 89 and 55 spirals at each edge. Figure 1.5 was taken from the ‘Fibonacci numbers, the Golden section, and plants’ section on the University of Surrey website.
Our last example of Fibonacci numbers in nature is with cauliflower. If you look at a cauliflower you should be able to see a center point where the florets are smallest. The florets are organized in spirals in both directions, similar to sunflowers and pinecones. The resulting number of spirals in each direction is a Fibonacci number. [Knotts, 2004, Fibonacci numbers, the Golden section, and plants]
It has not been explained as to why in nature we have these spirals that are consistent with the Fibonacci numbers. “Many hypotheses, ranging from the prosaic to the mystical, have been proposed to explain why leaves should stick out at this angle. Some invoke physical properties of the stem, whereas others propose that growing leaves emit an inhibitory field to prevent new leaves from arising in their vicinity, but none has had direct supporting evidence,” (Surridge, 2003, p. 237). You will not always find the Fibonacci numbers in the number of petals or spirals on seed heads, however they often come close to the Fibonacci numbers. [Knotts, 2004, Fibonacci numbers, the Golden section, and plants]
FIBONACCI NUMBERS AND LUCAS NUMBERS
The French mathematician Edouard Lucas (1842 – 1891) created a similar series to the Fibonacci series. He used the same rule of adding the last two numbers to get the next number but he started from 2 and 1 (in this order) instead of 0 and 1 like the Fibonacci series. The following is a formula for the Lucas numbers. [Knotts, 2004, The Lucas series]
Ln = Ln-1 + Ln-2 for n>1 L0 = 2 L1 = 1
Here is a table of the Fibonacci (Fn) and Lucas (Ln) numbers.
|n |0 |1 |2 |3 |4 |5 |
|1 |1 | | | | | |
|1 |2 |1 | | | | |
|1 |3 |3 |1 | | | |
|1 |4 |6 |4 |1 | | |
|1 |5 |10 |10 |5 |1 | |
|1 |6 |15 |20 |15 |6 |1 |
We will know examine why the diagonal lines in Pascal’s triangle add up to the Fibonacci numbers.
The blue diagonal sums to 5
The pink diagonal sums to 8
The green diagonal sums to 13
Each green number is the sum of a
Blue and pink number on the row above it
Notice that the blue numbers are on one diagonal and the pink ones on the next. The sum of all the blue numbers is 5 and all the pink numbers add up to 8. All the numbers in Pascal’s Triangle are found by adding the two numbers above and to the left on the row above. We can then see that each green number is just the sum of a blue and a pink number. We end up using all the blue and pink numbers to get all the green numbers. Therefore, the sum of all the green numbers is the same as the sum of all the blue and all the pink numbers (5 + 8 = 13). We have just stated that the sum of the numbers on one diagonal is the sum of the numbers on the previous two diagonals. [Knotts, 2004, The Fibonacci numbers in Pascal’s Triangle]
Let Di stand for the sum of the numbers on the diagonal that starts with one of the extra zeros at the beginning of row i then
D(0) = 0 and D(1) = 1
The blue diagonal sum is D(5) = 5 and the pink diagonal sum is D(6) = 8. The green diagonal is D(7) = 13 = D(6) + D(5)
One diagonals sum is the sum of the previous two diagonal sums. We can represent this with the following.
Di = Di-1 + Di-2 where D(0) = 0 and D(1) = 1
This representation of Di is exactly the same as the Fibonacci sequence
(Fn = F(n-1) + F(n-2)). Therefore Di is equivalent to Fi and the sums of the diagonal lines in Pascal’s Triangle are the Fibonacci numbers. [Knotts, 2004, The Fibonacci numbers in Pascal’s Triangle]
In conclusion, the Fibonacci sequence is extremely interesting and relates to other historical and famous mathematical ideas such as the golden ratio, the Lucas numbers, and Pascal’s Triangle. It is amazing how great mathematical ideas branch out into other areas of mathematics. The Fibonacci sequence also branches outside the realm of mathematics into science and nature. I do not think that Leonardo Pisano ever imagined that the sequence would become as popular as it is today. It has been researched by many and continues to be studied. The Fibonacci Association “focuses on Fibonacci numbers and related mathematics, emphasizing new results, research proposals, challenging problems, and new proofs of old ideas,” (The Fibonacci Association website). Hopefully the continued research will yield new results. The Fibonacci sequence also shows that the greatest advancements in mathematics can come out of the simplest of problems.
References
Ballew, P. (n.d.). Math Words. Retrieved April 8, 2005 from Pat Ballew’s website:
Brooker, C. (n.d.). Fibonacci Rectangles and Spirals. Retrieved May 4, 2005 from the University of Bath website:
Dictionary of Scientific Biography. (1971). Vol. IV. New York: Charles Scribner’s Sons.
Fibonacci (c.1175 – c.1240) Mathematician (n.d.). Retrieved April 8, 2005 from the University of Evansville website:
Katz, V. (1998). A History of mathematics: An introduction. (2nd ed.). New York:
Addison-Wesley Longman.
Knotts, R. (2004). Fibonacci numbers and the Golden number. Retrieved March 31, 2005 from the University of Surrey website:
Knotts, R. (2004). Fibonacci numbers, the Golden Section, and plants. Retrieved March 31, 2005 from the University of Surrey website:
Knotts, R. (2004). Fibonacci Rectangles and Shell Spirals. Retrieved March 31, 2005 from the University of Surrey website:
Knotts, R. (2004). Phi and the Fibonacci numbers. Retrieved March 31, 2005 from the University of Surrey website:
Knotts, R. (2004). Rabbits, Cows, and Bees Family Trees. Retrieved March 31, 2005 from the University of Surrey website:
Knotts, R. (2004). The Fibonacci Numbers in Pascal’s Triangle. Retrieved March 31, 2005 from the University of Surrey website:
Knotts, R. (2004). The Lucas Series. Retrieved March 31, 2005 from the University of Surrey website:
Knotts, R. (2004). What is the golden section (or Phi)?. Retrieved March 31, 2005 from the University of Surrey website:
Leonardo Pisano Fibonacci (n.d.). Retrieved March 28, 2005, from the University of St Andrews, Scotland, School of Mathematics and Statistics website:
Livio, M. (2003). Searching for the golden ratio. Astronomy. Vol. 31, Issue 4, p52-58.
Morris, S.J. (n.d.). Fibonacci Numbers. Retrieved March 28, 2005 from the University of Georgia website:
Osler, T.J. (2003). Variations on a theme from Pascal’s triangle. The College Mathematics Journal. Vol 34, No. 3, p.216.
Sigler, L.E. (2002). Fibonacci’s Liber Abaci: A translation into modern English of Leonardo Pisano’s Book of Calculation. New York: Springer.
Surridge, C. (2003). Plant development:Leaves by number. Nature. Vol 426, No. 6964, p237.
The Fibonacci Association website. (2002). Retrieved April 25, 2005 from the The Fibonacci Association website:
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