Math around Us: Fibonacci Numbers

1
Math around UsFibonacci Numbers

• John Hutchinson
• March 2005

2
Leonardo Pisano Fibonacci

• Born 1170 in (probably) Pisa (now in
  Italy) Died 1250 in (possibly) Pisa (now in
  Italy)

3
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.

4
The Fibonacci numbers can be considered to be a
function with domain the positive integers.
N 1 2 3 4 5 6 7 8 9 10
FN 1 1 2 3 5 8 13 21 34 55
Note that FN2 FN1 FN
5
Note
Every 3rd Fibonacci number is divisible by
2. Every 4th Fibonacci number is divisible by
3. Every 5th Fibonacci number is divisible by
5. Every 6th Fibonacci number is divisible by
8. Every 7th Fibonacci number is divisible by
13. Every 8thFibonacci number is divisible by
21. Every 9th Fibonacci number is divisible by 34.
6
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 ????
7
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
8
F1 F2 F3 FN FN2 -1
9
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 ????
10
Sums of Squares
12 12 2 1 X 2
12 12 22 6 2 X 3
12 12 22 32 15 3 X 5
12 12 22 32 52 40 5 X 8
12 12 22 32 52 82 104 8 X 13
11
The Formula
F12 F22 F32 Fn2 Fn X FN1
12
Another Formula
FNI FI-1FN FIFN1
13
Pascals Triangle
14
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 example20 121 11 222
121 423 1331 824 14641 16
15
Add Diagonals
16
Pascals triangle with odd numbers in red.
17
1-White Calla Lily
18
1-Orchid
19
2-Euphorbia
20
3-Trillium
21
3-Douglas Iris
22
35 - Bougainvilla
23
5-Columbine
24
5-St. Anthonys Turnip (buttercup)
25
5-Unknown
26
5-Wild Rose
27
8-Bloodroot
28
13-Black-eyed Susan
29
21-Shasta Daisy
30
34-Field Daisy
31
Dogwood 4?????
32
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.)
33
Sweetwart
34
Sweetwart
35
(No Transcript)
36
"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?"
37
Lets count rabbits
Babies 1 0 1 1 2 3 5 8 13 21 34 45
Adult 0 1 1 2 3 5 8 13 21 34 55 89
Total 1 1 2 3 5 8 13 21 34 55 89 144
38
Lets 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, ?

39
Count them
40
89 Measures Total
Gets loud here
55 Measures
34 Measures
Strings remove mutes
Replace mutes
34 Measures
21 Measures
21 Measures
13
13
8
21 Theme
Texture
First Movement, Music for Strings, Percussion,
and Celeste Bela Bartok
41
(No Transcript)
42
The Keyboard
43
(No Transcript)
44
                        ltgt
                                                
  ltgt
           ltgt
45
(No Transcript)
46
(No Transcript)
47
(No Transcript)
48
(No Transcript)
49
The hand
50
Ratios of consecutive
1 1
2 2
3 1.5
5 1.66666
8 1.6
13 1.625
21 1.615385
34 1.619048
55 1.617647
89 1.618182
144 1.617978
233 1.618056
377 1.618026
610 1.618037
987 1.618033
etc 1.618034
51
The golden ratio is approximately
1.610833989
Or exactly
(v51)/2 2/(v5-1)
52
Golden Section
S
L
S/L L/(SL)
If S 1 then L 1.610833989 If L 1 then S
1/L .610833989
53
Golden Rectangle
L
S
54
Golden Triangles
8
5
L
3
5
S
55
The Parthenon
56
(No Transcript)
57
(No Transcript)
58
(No Transcript)
59
(No Transcript)
60
(No Transcript)
61
(No Transcript)
62
(No Transcript)
63
(No Transcript)
64
(No Transcript)
65
Holy Family, Michelangelo
66
Crucifixion - Raphael
67
Self Portrait - Rembrandt
68
(No Transcript)
69
(No Transcript)
70
Seurat
71
Seurat
72
Fractions

• 1/1 1
• ½ .5
• 1/3 .33333
• 1/5 .2
• 1/8 .125
• 1/89 ?

73
.01 1/100 .01
.001 1/1000 .011
.0002 2/10000 .0112
.00003 3/100000 .01123
.000005 5/1000000 .011235
.0000008 8/10000000 .0112358
.00000013 13/100000000 .00112393
.000000021 21/1000000000 .0011235951
.0000000034 34/10000000000 .00112359544
.00000000055 55/100000000000 .001123595495
1/89 .00112359550561798
74
Are there negative Fibonaccis?
Fn Fn2 - Fn1
75
-1 1
-2 -1
-3 2
-4 -3
-5 5
-6 -8
-7 13
-8 -21
F-n (-1)n1Fn
76
For any three Fibonacci Numbers the sum of the
cubes of the two biggest minus the cube of the
smallest is a Fibonacci number.
Fn23 Fn13 Fn3 F3(n1)
5
125
8
512
2709 125 2584
13
2197