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# Leonardo of Pisa

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### Leonardo of Pisa

1. 1. LEONARDO OF PISA (~1175- 1240) By: Beth Jarvis MATH 123 Geometry 11/25/08
2. 2. Fibonacci (1175-1240) Leonardo of Pisa is the  true name for the mathematician many know as Fibonacci. Fibonacci is a nickname  stemming from filius Bonacci, meaning son of Bonacci. He is most well known  for the Fibonacci Sequence and Numbers. Born in Italy and ended  in Italy; he spend his younger years in North Africa
3. 3. Fibonacci’s 4 Major Works 1st and most famous: Liber abaci  (The Book of Calculations), 1202 Practica Geometriae (The Practice  of Geometry), 1220 Flos (The Flower), 1223  Liber quadratorum (Book of  Squares), 1225
4. 4. Liber Abaci One of the first to introduced to Italy  and Europe the Hindu/Arabic value placed decimal system that we use today. 9,8,7,6,5,4,3,2,1 and the symbol 0.  Recall they were using Roman  Numerals. So 1998 was MCMXCVIII, and adding CLXXIV plus XXVIII equals CCII.
5. 5. “How many pairs of rabbits can be bred from a single pair in one year?” This problem states several important factors: rabbits take 1 month to grow up  after they have matured (for 1 month) it  takes a pair of rabbits 1 more month to produce another pair of newly born rabbits. we assume that rabbits never die  we assume that whenever a new pair of  rabbits is produced, it is always a male and a female we assume that these rabbits live in ideal  conditions the problem begins with just 1 pair of  newly born rabbits (1 male, 1 female)
6. 6. Answer: 144 Pairs of Rabbits Mont Rabbit h pairs 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 9 34 10 55 11 89 12 144
7. 7. Fibonacci Sequence is born 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,  144… Each term is created by adding the  two previous terms. 1+1 = 2; 1+2 = 3; 2+3 = 5;…  Recursive formula  
8. 8. Geometric Formula Jacques Binet’s Formula (1843) Golden Ratio: allows us to have a geometric formula so we can get any value of Fibonacci’s sequence without the previous two terms. Fibonacci: n=1 1 n=2 1 n=3 2 n=4 3 n=5 5 If n = 6, plug into the n=6 8 calculator and it does equal 8.
9. 9. Fibonacci Numbers Divided If we divide a Fibonacci number by the  Note that the previous one, the decimals converge to Fibonacci the golden ratio = 1.618… numbers increase by Fib Number Divide Decimal a factor of 1 the Golden 1 1/1 1.0000 Ratio. 2 2/1 2.0000 3 3/2 1.5000 5 5/3 1.6667 8 8/5 1.6000 13 13/8 1.6250 21 21/13 1.6154 34 34/21 1.6190 55 55/34 1.6176
10. 10. Fibonacci Spiral The box below draws squares the 1 size corresponding with the order 1 2 of Fibonacci sequence. 3 5 8 13 21 34 55 89 144 233 377
11. 11. Fibonacci Spiral and Shell
12. 12. Fibonacci Sequence in Nature Male honeybees are produced by  only a female without male fertilization. So they have only a mother and no father. Female honeybees need a male  and female to produce a female honeybee. They have a mother and father. The number of bees in the life of a  male honeybee follows the Fibonacci Sequence.
13. 13. Pythagorean Triples Pythagorean Triples are 3 positive  Pythagorean Theorem: integers that satisfy the a² + b² = c² Pythagorean Theorem. Some common Triples:  3, 4, 5  5, 12, 13  16, 30, 34  39, 80, 89 
14. 14. Use Fibonacci Sequence to create Pythagorean Triples. Step 1: Select 4 sequential Fibonacci  1 1 numbers. 2 Step 2 : Multiply the middle two 3  5 numbers and double the product 8 13 Step 3 : Multiply the first and last  21 numbers together. 34 55 Step 4 : Add the squares of the middle  89 144 two numbers. 233 The answers from steps 2 and 3 are 377  the legs of a right triangle and step 4 is the hypotenuse.
15. 15. Pascal’s Triangle (x + 1)ª
16. 16. Fibonacci in the Arts Featured in the book The Da Vinci  1 Code by Dan Brown and movie with 1 2 Tom Hanks 3 There is a anagram clue in the  5 8 beginning that is 13 3 2 21 1 1 8 5 13 which turns out to be the Fibonacci 21 Sequence transposed. These 34 55 numbers are the bank account number 89 the characters needed. 144 233 In music we have an example of a  377 series of beats/syllables that follow the Fibonacci Sequence. Tool's Lateralus 
17. 17. Neat facts about Fibonacci Sequence Every third number in the sequence  1, 1, 2, 3, 5, is even. 8, 13, 21, 34, All prime numbers in the sequence  55, 89, 144, have a prime index with the exception of the 4th term which is 3. 233, 377, 610, 987, F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11  1597, 2584, F12 1 1 2 3 5 8 13 21 34 55 89 144 4181, 6765, 10946…
18. 18. Fibonacci Groups Fibonacci Association in San  Jose, CA. Started in 1960 Fibonacci Association Fibonacci Quarterly – a journal  published 4 times a year International Conference for the  Applications of Fibonacci Numbers (Winston-Salem, NC hosted in 1990)
19. 19. Statue of Fibonacci in Pisa
20. 20. 1 1 2 3 5 8 13 21… THANK YOU! Sources upon request