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
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
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)
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
Jacques Binet’s Formula (1843)
Golden
Ratio:
allows us to have a geometric
formula so we can get any value
of Fibonacci’s sequence without
the previous two terms.
Fibonacci:
n=1 1
n=2 1
n=3 2
n=4 3
n=5 5
If n = 6, plug into the
n=6 8
calculator and it does equal 8.
9. Fibonacci Numbers Divided
If we divide a Fibonacci number by the
Note that the
previous one, the decimals converge to
Fibonacci
the golden ratio = 1.618…
numbers
increase by
Fib Number Divide Decimal
a factor of
1
the Golden
1 1/1 1.0000
Ratio.
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
10. Fibonacci Spiral
The box below draws squares the
1
size corresponding with the order
1
2
of Fibonacci sequence.
3
5
8
13
21
34
55
89
144
233
377
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
Pythagorean
Theorem:
integers that satisfy the
a² + b² = c²
Pythagorean Theorem.
Some common Triples:
3, 4, 5
5, 12, 13
16, 30, 34
39, 80, 89
14. Use Fibonacci Sequence to create
Pythagorean Triples.
Step 1: Select 4 sequential Fibonacci
1
1 numbers.
2
Step 2 : Multiply the middle two
3
5
numbers and double the product
8
13
Step 3 : Multiply the first and last
21
numbers together.
34
55
Step 4 : Add the squares of the middle
89
144 two numbers.
233
The answers from steps 2 and 3 are
377
the legs of a right triangle and step 4 is
the hypotenuse.
16. Fibonacci in the Arts
Featured in the book The Da Vinci
1
Code by Dan Brown and movie with
1
2 Tom Hanks
3
There is a anagram clue in the
5
8 beginning that is 13 3 2 21 1 1 8 5
13
which turns out to be the Fibonacci
21
Sequence transposed. These
34
55 numbers are the bank account number
89
the characters needed.
144
233 In music we have an example of a
377
series of beats/syllables that follow the
Fibonacci Sequence.
Tool's Lateralus
17. Neat facts about Fibonacci
Sequence
Every third number in the sequence
1, 1, 2, 3, 5,
is even.
8, 13, 21,
34,
All prime numbers in the sequence
55, 89, 144,
have a prime index with the
exception of the 4th term which is 3.
233, 377,
610, 987,
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11
1597, 2584, F12 1 1 2 3 5 8 13 21 34 55 89
144
4181, 6765,
10946…
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)
