LEONARDO OF PISA

(~1175- 1240)

By: Beth Jarvis MATH 123 Geometry 11/25/08

Fibonacci (1175-1240)

Leonardo of Pisa is the true name for the mathematician many know as Fibonacci.

Fibonacci is a nickname stemming from filiusBonacci, 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

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

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.

“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)

Answer: 144 Pairs of Rabbits

Mont

h

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

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

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.

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…

Fib Number Divide Decimal

1

1 1/1 1.0000

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

Fibonacci Spiral

1123581321345589144233377

The box below draws squares the

size corresponding with the order

of Fibonacci sequence.

Fibonacci Spiral and Shell

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.

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

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.

1123581321345589144233377

Pascal’s Triangle (x + 1)ª

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

1123581321345589144233377

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 4th term which is 3.

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11

F12 1 1 2 3 5 8 13 21 34 55 89

144

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)

Statue of Fibonacci in Pisa

THANK YOU!

Sources upon request

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1

2

3

5

8

13

21…