Leo of Pisa

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Leo of Pisa

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

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