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Math around Us:
Fibonacci Numbers
John Hutchinson
March 2005
Leonardo Pisano Fibonacci
Born: 1170 in (probably) Pisa (now in Italy)
Died: 1250 in (possibly) Pisa (now in Italy)
What is a Fibonacci Number?
Fibonacci numbers are the
numbers in the Fibonacci
sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, . . . ,
each of which, after the second, is
the sum of the two previous ones.
The Fibonacci numbers can be considered
to be a function with domain the positive
integers.
N 1
2
3
4
5
6
7
FN 1
1
2
3
5
8
13 21 34 55
8
Note that
FN+2 = FN+1 + FN
9
10
Note
Every
Every
Every
Every
Every
Every
Every
3rd Fibonacci number is divisible by 2.
4th Fibonacci number is divisible by 3.
5th Fibonacci number is divisible by 5.
6th Fibonacci number is divisible by 8.
7th Fibonacci number is divisible by 13.
8thFibonacci number is divisible by 21.
9th Fibonacci number is divisible by 34.
Sums of Fibonacci Numbers
1+1=2
????
1+1+2=4
????
1+1+2+3=7
????
1 + 1 + 2 + 3 + 5 = 12
????
1 + 1 + 2 + 3 + 5 + 8 = 20
????
Sums of Fibonacci Numbers
1+1=2
3-1
1+1+2=4
5-1
1+1+2+3=7
8-1
1 + 1 + 2 + 3 + 5 = 12
13 - 1
1 + 1 + 2 + 3 + 5 + 8 = 20
21 - 1
F1 + F2 + F3 + … + FN = FN+2 -1
Sums of Squares
12 + 12 = 2
????
12 + 12 + 22 = 6
????
12 + 12 + 22 + 32 = 15
????
12 + 12 + 22 + 32 + 52 = 40
????
12 + 12 + 22 + 32 + 52 + 82 = 104
????
Sums of Squares
12 + 12 = 2
1X2
12 + 12 + 22 = 6
2X3
12 + 12 + 22 + 32 = 15
3X5
12 + 12 + 22 + 32 + 52 = 40
5X8
12 + 12 + 22 + 32 + 52 + 82 = 104
8 X 13
The Formula
F12 + F22 + F32 + …+ Fn2 = Fn X FN+1
Another Formula
FN+I = FI-1FN + FIFN+1
Pascal’s Triangle
Sums of Rows
The sum of the numbers in any row is equal
to 2 to the nth power or 2n, when n is the
number of the row. For example:
20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16
Add Diagonals
Pascal’s triangle with odd
numbers in red.
1-White Calla Lily
1-Orchid
2-Euphorbia
3-Trillium
3-Douglas Iris
3&5 - Bougainvilla
5-Columbine
5-St. Anthony’s Turnip
(buttercup)
5-Unknown
5-Wild Rose
8-Bloodroot
13-Black-eyed Susan
21-Shasta Daisy
34-Field Daisy
Dogwood = 4?????
Here a sunflower seed illustrates this principal as the
number of clockwise spirals is 55 (marked in red, with
every tenth one in white) and the number of
counterclockwise spirals is 89 (marked in green, with
every tenth one in white.)
Sweetwart
Sweetwart
"Start with a pair of rabbits, (one male and one
female). Assume that all months are of equal
length and that :
1. rabbits begin to produce young two months
after their own birth;
2. after reaching the age of two months, each
pair produces a mixed pair, (one male, one
female), and then another mixed pair each
month thereafter; and
3. no rabbit dies.
How many pairs of rabbits will there be after
each month?"
Let’s count rabbits
Babies
Adult
Total
1 0 1 1 2 3 5 8 13 21 34
0 1 1 2 3 5 8 13 21 34 55
1 1 2 3 5 8 13 21 34 55 89
45
89
144
Let’s count tokens
A token machine dispenses 25-cent
tokens. The machine only accepts
quarters and half-dollars. How many
ways can a person purchase 1 token,
2 tokens, 3 tokens, …?
Count them
25C
Q
1
50C
QQ-H
2
75C
QQQ-HQ-QH
3
100C
QQQQ-QQH-QHQ-HQQ-HH
5
125C
QQQQQ-QQQH-QQHQ-QHQQ-HQQQHHQ-HQH-QHH
8
89 Measures Total
Gets loud here
55 Measures
34 Measures
Strings remove mutes
34 Measures
Replace mutes
21 Measures
13
21 Measures
13
21 Theme
8
Texture
First Movement, Music for Strings, Percussion, and Celeste
Bela Bartok
The Keyboard
<>
<>
<>
The hand
Ratios of consecutive
1
2
3
5
8
13
21
34
1
2
1.5
1.66666
1.6
1.625
1.615385
1.619048
55
89
144
233
377
610
987
etc
1.617647
1.618182
1.617978
1.618056
1.618026
1.618037
1.618033
1.618034…
The golden ratio is approximately
1.610833989…
Or exactly
(√5+1)/2 = 2/(√5-1)
Golden Section
S
L
S/L = L/(S+L)
If S = 1 then L= 1.610833989…
If L = 1 then S = 1/L = .610833989…
Golden Rectangle
L
S
Golden Triangles
8
5
3
L
5
S
The Parthenon
Holy Family, Michelangelo
Crucifixion - Raphael
Self Portrait - Rembrandt
Seurat
Seurat
Fractions







1/1 = 1
½ = .5
1/3 = .33333
1/5 = .2
1/8 = .125
…
1/89 = ?
.01
.001
.0002
.00003
.000005
.0000008
.00000013
.000000021
.0000000034
.00000000055
1/100
1/1000
2/10000
3/100000
5/1000000
8/10000000
13/100000000
21/1000000000
34/10000000000
55/100000000000
.01
.011
.0112
.01123
.011235
.0112358
.00112393
.0011235951
.00112359544
.001123595495
1/89 = .00112359550561798…
Are there negative Fibonaccis?
Fn = Fn+2 - Fn+1
-1
-2
-3
-4
-5
-6
-7
-8
1
-1
2
-3
5
-8
13
-21
F-n = (-1)n+1Fn
For any three Fibonacci Numbers the
sum of the cubes of the two biggest
minus the cube of the smallest is a
Fibonacci number.
Fn+23 + Fn+13 – Fn3 = F3(n+1)
5
125
8
512
13
2197
2709 – 125 = 2584