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# Leo of Pisa

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### Transcript

• 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…