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Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
Leo of Pisa
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Leo of Pisa

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  • 1. LEONARDO OF PISA (~1175- 1240) By: Beth Jarvis MATH 123 Geometry 11/25/08
  • 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. Fibonacci’s 4 Major Works
    • 1 st 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. 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. “ 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. 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. 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. Geometric Formula
    • Golden Ratio:
    • Fibonacci:
    • n=1 1 n=2 1 n=3 2 n=4 3 n=5 5 n=6 8
    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. Fibonacci Numbers Divided
    • Note that the Fibonacci numbers increase by a factor of the Golden Ratio.
    • If we divide a Fibonacci number by the previous one, the decimals converge to the golden ratio = 1.618…
  • 10. Fibonacci Spiral
    • 1 1 2 3 5 8 13 21 34 55 89 144 233 377
    The box below draws squares the size corresponding with the order of Fibonacci sequence.
  • 11. Fibonacci Spiral and Shell
  • 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. Pythagorean Triples
    • Pythagorean Triples are 3 positive integers that satisfy the Pythagorean Theorem.
    • Some common Triples:
    • 3, 4, 5
    • 5, 12, 13
    • 16, 30, 34
    • 39, 80, 89
    • Pythagorean Theorem:
    • a² + b² = c²
  • 14. Use Fibonacci Sequence to create Pythagorean Triples.
    • Step 1 : Select 4 sequential Fibonacci numbers.
    • Step 2 : Multiply the middle two numbers and double the product
    • Step 3 : Multiply the first and last numbers together.
    • Step 4 : Add the squares of the middle two numbers.
    • The answers from steps 2 and 3 are the legs of a right triangle and step 4 is the hypotenuse.
    • 1 1 2 3 5 8 13 21 34 55 89 144 233 377
  • 15. Pascal’s Triangle (x + 1)ª
  • 16. Fibonacci in the Arts
    • Featured in the book The Da Vinci Code by Dan Brown and movie with Tom Hanks
    • 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.
    • In music we have an example of a series of beats/syllables that follow the Fibonacci Sequence.
    • Tool's Lateralus
    • 1 1 2 3 5 8 13 21 34 55 89 144 233 377
  • 17. Neat facts about Fibonacci Sequence
    • 1, 1, 2 , 3, 5,
    • 8 , 13, 21, 34 ,
    • 55, 89, 144 ,
    • 233, 377,
    • 610 , 987,
    • 1597, 2584 ,
    • 4181, 6765,
    • 10946 …
    • Every third number in the sequence is even.
    • All prime numbers in the sequence have a prime index with the exception of the 4 th term which is 3.
    • 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
  • 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. Statue of Fibonacci in Pisa
  • 20. THANK YOU! Sources upon request 1 1 2 3 5 8 13 21…

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