Introduction to the Fibonacci sequence

The Fibonacci sequence was applied to the metrical sciences.  The Fibonacci sequence is a sequence of numbers in which each number equals the sum of the two preceding numbers starting with 0,1. The Fibonacci sequence starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34… etc.  The Fibonacci sequence also helps at doing spirals.  There are different types of spirals like the Archimedean spiral and the logarithmic.

The Fibonacci sequence appears in nature many times. For example an appearance of the Fibonacci Series in seedheads, pincones, pineapples, etc., is that the number of spirals going in each direction is a Fibonacci number. In the picture above there are 13 spirals that turn clockwise and 21 curving counterclockwise. On all other sunflowers, the number of clockwise and counterclockwise spirals will always be consecutive Fibonacci Numbers like 21 and 34 or 55 and 34.

History of the Fibonacci sequence

 The Fibonacci sequence was invented by the Italian Leonardo Pisano Bigollo (1180-1250), who is known in mathematical history by several names: Leonardo of Pisa and Fibonacci.  Fibonacci was the son of an Italian businessman from the city of Pisa. He grew up in a trading colony in North Africa during the middle Ages. Italians were some of the western world's most proficient traders and merchants during the middle Ages, and they needed arithmetic to keep track of their commercial transactions. Mathematical calculations were made using the Roman numeral system (I, II, III, IV, V, VI, etc.), but that system made it hard to do the addition, subtraction, multiplication, and division that merchants needed to keep track of their transactions. While he was growing up in North Africa, Fibonacci learned the more efficient Hindu-Arabic system of arithmetical notation (1, 2, 3, 4...) from an Arab teacher. In 1202, he published his knowledge in a famous book called the Liber Abaci. Liber Abaci means the "book of the abacus.” The Liber Abaci showed how superior the Hindu-Arabic arithmetic system was to the Roman numeral system, and it showed how the Hindu-Arabic system of arithmetic could be applied to benefit Italian merchants. The Fibonacci sequence was the outcome of a mathematical problem about rabbit breeding that was posed in the Liber Abaci. The problem was this: Beginning with a single pair of rabbits, one male and one female, there will be 144 pairs of rabbits born in a year, assuming that every month each male and female rabbit gives birth to a new pair of rabbits, and the new pair of rabbits itself starts giving birth to additional pairs of rabbits after the first month of their birth.

TABLE 1
	Newborns (can't reproduce)		One-month-olds (can't reproduce)		Mature Pairs (can reproduce)		Total Pairs
Month 1	1	+	0	+	0	=	1
Month 2	0	+	1	+	0	=	1
Month 3	1	+	0	+	1	=	2
Month 4	1	+	1	+	1	=	3
Month 5	2	+	1	+	2	=	5
Month 6	3	+	2	+	3	=	8
Month 7	5	+	3	+	5	=	13
Month 8	8	+	5	+	8	=	21
Month 9	13	+	8	+	13	=	34
Month 10	21	+	13	+	21	=	55