Leonardo of Pisa
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Transcript of Leonardo of Pisa
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 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
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)