A man named Leonardo Pisano, who was known by his nickname, “Fibonacci”, and named the series after himself, first discovered the Fibonacci sequence around 1200 A.D. The Fibonacci sequence is a sequence in which each term is the sum of the 2 numbers preceding it. The first 10 Fibonacci numbers are: (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89). These numbers are obviously recursive.
Fibonacci was born around 1170 in Italy, and he died around 1240 in Italy, but the exact dates of his birth and death are not known. He played an important role in reviving ancient mathematics and made significant contributions of his own. Even though he was born in Italy, he was educated in North Africa where his father held a diplomatic post. He published a book called Liber abaci, in 1202, after his return to Italy and it was in this book that the Fibonacci numbers were first discussed. It was based on bits of Arithmetic and Algebra that Fibonacci had accumulated during his travels with his father.
Liber abaci introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe. Though people were interested, this book was somewhat controversial because it contradicted some of the foremost Roman and Grecian Mathematicians of the time, and even proved many of their calculations to be false. The Fibonacci sequence is also used in the Pascal triangle. The sum of each diagonal row is a Fibonacci number. They are also in the right sequence.
The Fibonacci sequence has been a big factor in many patterns of things in nature, which is quite fascinating. It’s been discovered that the numbers representing the screw-like arrangements of leaves on flowers and trees are very often numbers in the Fibonacci sequence. On many plants, for instance, the number of petals is a Fibonacci number: buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21, and even daisies can be found with 34, 55 or even 89 petals.
Fibonacci numbers are also used with animals. For example, the first problem Fibonacci had when using the Fibonacci numbers was trying to figure out was how fast rabbits could breed in ideal circumstances. Using the sequence he was able to approximate the answer pretty accurately.
The Fibonacci numbers can also be found in many other patterns. The diagram below is what is known as the Fibonacci spiral, in circle form. We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21,.. in a square form if we start with two small squares of size 1, one on top of the other. Now on the right of these draw a square of size 2 (=1+1). We can now draw a square on top of these, which has sides 3 units long, and another on the left of the picture which as side 5. We can continue adding squares around the picture, each new square having a side that is as long as the sum of the last two squares drawn.
One of the most interesting things about the Fibonacci sequence is the number found by dividing each number by the one preceding it. 1/1 = 1, 2/1 = 2, 3/2 = 15, 5/3 = 1666…, 8/5 = 16, 13/8 = 1625, 21/13 = 161538, and so on. These numbers are obviously settling down to a particular value. This value, approximately 1.618034, is known as the Golden Ratio.
The Golden Ratio is a number that has fascinated mathematicians for generations. It can be expressed succinctly in the ratio of the number “1” to the irrational “l.618034..”, but it has meant so many things to so many people, that a basic investigation of what might is the “Golden Ratio Phenomenon” seems in order.
In modern times there has been much interest in the Golden Ratio, which is also known as the Golden Section or Golden Ratio. Since the Renaissance time it has been used extensively in art and architecture. It was used in the building of Venetian Church of St. Mark built early in the 16th century, and has become a standard proportion for width in relation to height as used in facades of buildings, in window sizing, in proportion of first story to second, and even at times in the dimensions of paintings and picture frames.
In the l930’s the Pratt Institute of New York did a study on various rectangular proportions laid out as vertical frames, and asked several hundred art students to comment on which seemed the most pleasing. Interestingly enough, the ratio of 1: 2 was least liked, while the Golden Ratio was favored by a very large margin. Judging by this response, it seems things constructed according to the Golden Ratio tend to be quite aesthetically pleasing.
Another interesting fact is the way the Golden Ratio appears in the measurements of the human body. The French architect LeCorbusier noted that the human body, when measured from foot to navel, and then again from navel to top of head, showed average numbers very near to the Golden Ratio. He extended this to the height compared with arm-span, and designed doorways consonant with these numbers. Obviously, this is not an exact science, because the numbers were based on averages instead of precise measurements, but they still helped in his design and building techniques.
Studies have shown that the patterns of tree- branching often adhere to the Golden Ratio, although again not exactly, while the dendritic cracking in certain metallic alloys, which occurs as very low temperatures, is basically based on the Golden Ratio. Also, in an entirely different area, Zsolt Lendvai has demonstrated that Bela Bartok used the Golden Ratio extensively while composing music. An interesting question brought up by this is whether the composer uses the Golden Ratio on purpose, or totally unconsciously because of the ubiquitous appearance in the world around us, something we sense by living in a world governed by the Golden Ratio.
In conclusion, I have found that the Golden Ratio is very constructive in many different aspects of life. It was very interesting to do the research for this paper, and find out more about the Golden Ratio and the many uses it has.
- Smith, Karl J. The Nature Of Mathematics: 9th Edition. 2001
- Ruskey, Frank; Lausch, Scott. “Information on Fibonacci Sequence.” http://theory.cs.uvic.ca/cos/amof/e_fiboI.htm.