Transcript 1·618034
DUYGU KANDEMİR 200822024 CONTENT • • • • • • • • Who was Fibonnaci? Fibonacci Sequence Fibonacci Rectangle and Spiral Golden Ratio Fibonacci Rabbit Problem Pascal Triangle and Fibonacci Numbers Fibonacci Examples from Nature References 1) Who was Fibonacci? • • • • • • • He was born in 1180 in Pisa Italy. His father was a diplomat in North Africa. He travelled with his father. He received instruction in accounting. He returned to Pisa in 1200. He wrote a book on commercial arithmetic. He introduced the Hindu-Arabic decimal system into Europe. • He taught others how to convert between various currencies for trade between countries. • He died in 1250. 1) Who was Fibonacci? • Fibonacci, a mathematician who lived 800 years ago!!! • These numbers are to be found everywhere in nature. • Applications of Fibonacci series are nearly limitless. • Lots of matematicians added a new piece to the Fibonnaci puzzle. • Fibonacci mathematics is a constantly expanding branch of number theory. 2) Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, ………….. The first two numbers in the series are one and one. To obtain each number of the series, you simply add the two numbers that came before it. In other words, each number of the series is the sum of the two numbers preceding it. Fibonacci Rectangle and Spiral • • • • • If we start with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (=1+1). We can now draw a new square which is 3 units long Then another square is drawn which is 5 units long. We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. (Fibonacci Rectangle) Here is a spiral drawn in the squares, a quarter of a circle in each square Golden Ratio If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, we will find the following series of numbers: 1/1 = 1 2/1 = 2 3/2 = 1·5 5/3 = 1·666... 8/5 = 1·6 13/8 = 1·625 21/13 = 1·61538... Golden ratio which has a value of approximately 1·618034 Fibonacci Examples from Nature Pine cones Pine cones show the Fibonacci Spirals clearly. Here is a picture of an ordinary pine cone seen from its base where the stalk connects it to the tree. 5 fingers, each of which has ... 3 parts separated by ... 2 knuckles • Interesting video for Fibonacci Series • http://www.youtube.com/watch?v=wS7CZI JVxFY REFERENCES • http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fibnat.html • http://www.slideshare.net/merilynhancock2/fibonacci1834195 • http://www.slideshare.net/timsmurphy/golden-meanpresentation01 • http://www.slideshare.net/ameya/fibonacci • http://www.slideshare.net/rmukilan/fibonacci-15739710 • http://www.slideshare.net/arifsulu/game-theory-fibonacciseries • http://www.slideshare.net/timsmurphy/golden-meanpresentation01