Math around Us:Math around Us:Fibonacci Fibonacci NumbersNumbers

John HutchinsonJohn HutchinsonMarch 2005March 2005

Leonardo Pisano FibonacciBorn: 1170 in (probably) Pisa (now in Italy)Died: 1250 in (possibly) Pisa (now in Italy)

What is a Fibonacci What is a Fibonacci Number?Number?

Fibonacci numbers are the Fibonacci numbers are the numbers in the Fibonacci numbers in the Fibonacci sequence sequence

0, 1, 1, 2, 3, 5, 8, 13, 21, . . . , 0, 1, 1, 2, 3, 5, 8, 13, 21, . . . ,

each of which, after the each of which, after the second, is the sum of the two second, is the sum of the two previous ones.previous ones.

The Fibonacci numbers can be considered to be a function with domain the positive integers.

NN 11 22 33 44 55 66 77 88 99 1010

FFNN11 11 22 33 55 88 1313 2121 3434 5555

Note thatFN+2 = FN+1

+ FN

NoteNote

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.

Sums of Fibonacci NumbersSums of Fibonacci Numbers

1 + 1 = 21 + 1 = 2 ????????

1 + 1 + 2 = 41 + 1 + 2 = 4 ????????

1 + 1 + 2 + 3 = 71 + 1 + 2 + 3 = 7 ????????

1 + 1 + 2 + 3 + 5 = 121 + 1 + 2 + 3 + 5 = 12 ????????

1 + 1 + 2 + 3 + 5 + 8 = 201 + 1 + 2 + 3 + 5 + 8 = 20 ????????

Sums of Fibonacci NumbersSums of Fibonacci Numbers

1 + 1 = 21 + 1 = 2 3 - 13 - 1

1 + 1 + 2 = 41 + 1 + 2 = 4 5 - 15 - 1

1 + 1 + 2 + 3 = 71 + 1 + 2 + 3 = 7 8 - 18 - 1

1 + 1 + 2 + 3 + 5 = 121 + 1 + 2 + 3 + 5 = 12 13 - 113 - 1

1 + 1 + 2 + 3 + 5 + 8 = 201 + 1 + 2 + 3 + 5 + 8 = 20 21 - 121 - 1

F1 + F2

+ F3 + … + FN = FN+2 -1

Sums of SquaresSums of Squares

112 2 + 1+ 122 = 2 = 2 ????????

112 2 + 1+ 122 + 2 + 22 2 = 6= 6 ????????

112 2 + 1+ 122 + 2 + 22 2 + 3+ 32 2 = 15= 15 ????????

112 2 + 1+ 122 + 2 + 22 2 + 3+ 32 2 + 5+ 52 2 = 40= 40 ????????

112 2 + 1+ 122 + 2 + 22 2 + 3+ 32 2 + 5+ 52 2 + 8+ 822 = = 104104

????????

Sums of SquaresSums of Squares

112 2 + 1+ 122 = 2 = 2 1 X 21 X 2

112 2 + 1+ 122 + 2 + 22 2 = 6= 6 2 X 32 X 3

112 2 + 1+ 122 + 2 + 22 2 + 3+ 32 2 = 15= 15 3 X 53 X 5

112 2 + 1+ 122 + 2 + 22 2 + 3+ 32 2 + 5+ 52 2 = 40= 40 5 X 85 X 8

112 2 + 1+ 122 + 2 + 22 2 + 3+ 32 2 + 5+ 52 2 + 8+ 822 = = 104104

8 X 138 X 13

The FormulaThe Formula

F12 + F2

2 + F32 + …+ Fn

2 = Fn X FN+1

FN+I = FI-1FN + FIFN+1

Another Formula

Pascal’s TrianglePascal’s Triangle

Sums of RowsSums 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 = 121 = 1+1 = 2

22 = 1+2+1 = 423 = 1+3+3+1 = 8

24 = 1+4+6+4+1 = 16

Add DiagonalsAdd Diagonals

Pascal’s triangle with Pascal’s triangle with odd numbers in odd numbers in redred..

1-White Calla Lily1-White Calla Lily

1-Orchid1-Orchid

2-Euphorbia2-Euphorbia

3-Trillium3-Trillium

3-Douglas Iris3-Douglas Iris

3&5 - Bougainvilla3&5 - Bougainvilla

5-Columbine5-Columbine

5-St. Anthony’s Turnip 5-St. Anthony’s Turnip (buttercup)(buttercup)

5-Unknown5-Unknown

5-Wild Rose5-Wild Rose

8-Bloodroot8-Bloodroot

13-Black-eyed Susan13-Black-eyed Susan

21-Shasta Daisy21-Shasta Daisy

34-Field Daisy34-Field Daisy

Dogwood = 4?????Dogwood = 4?????

Here a sunflower seed illustrates this principal as Here a sunflower seed illustrates this principal as the number of clockwise spirals is 55 (marked in the number of clockwise spirals is 55 (marked in red, with every tenth one in white) and the red, with every tenth one in white) and the number of counterclockwise spirals is 89 (marked number of counterclockwise spirals is 89 (marked in green, with every tenth one in white.)in green, with every tenth one in white.)

SweetwartSweetwart

SweetwartSweetwart

"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 rabbitsLet’s count rabbits

BabiesBabies 11 00 11 11 22 33 55 88 1313 2121 3434 4545

AdultAdult 00 11 11 22 33 55 88 1313 2121 3434 5555 8989

TotalTotal 11 11 22 33 55 88 1313 2121 3434 5555 8989 144144

Let’s count tokensLet’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 themCount 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-HQQQ-HHQ-HQH-QHH

8

89 Measures Total

55 Measures 34 Measures

34 Measures 21 Measures 21 Measures13

First Movement, Music for Strings, Percussion, and Celeste

Bela Bartok

Gets loud here

Strings remove mutes Replace mutes

21 ThemeTexture

13 8

The KeyboardThe Keyboard

                                                  <>            <>                         <>

The handThe hand

Ratios of consecutiveRatios of consecutive

11 11

22 22

33 1.51.5

55 1.666661.66666

88 1.61.6

1313 1.6251.625

2121 1.6153851.615385

3434 1.6190481.619048

5555 1.6176471.617647

8989 1.6181821.618182

144144 1.6179781.617978

233233 1.6180561.618056

377377 1.6180261.618026

610610 1.6180371.618037

987987 1.6180331.618033

etcetc 1.6180341.618034……

The golden ratio is The golden ratio is approximatelyapproximately

1.610833989…

(√5+1)/2 = 2/(√5-1)

Or exactly

Golden SectionGolden Section

S L

S/L = L/(S+L)

If S = 1 then L= 1.610833989…

If L = 1 then S = 1/L = .610833989…

Golden RectangleGolden Rectangle

S

L

Golden TrianglesGolden Triangles

5

3

8

5

L

S

The Parthenon

Holy Family, Michelangelo

Crucifixion - Raphael

Self Portrait - Rembrandt

Seurat

Seurat

FractionsFractions

1/1 = 11/1 = 1 ½ = .5½ = .5 1/3 = .333331/3 = .33333 1/5 = .21/5 = .2 1/8 = .1251/8 = .125 …… 1/89 = ?1/89 = ?

.01.01 1/1001/100 .01.01

.001.001 1/10001/1000 .011.011

.0002.0002 2/100002/10000 .0112.0112

.00003.00003 3/1000003/100000 .01123.01123

.000005.000005 5/10000005/1000000 .011235.011235

.0000008.0000008 8/100000008/10000000 .0112358.0112358

.00000013.00000013 13/10000000013/100000000 .00112393.00112393

.000000021.000000021 21/100000000021/1000000000 .0011235951.0011235951

.0000000034.0000000034 34/100000000034/100000000000

.00112359544.00112359544

.00000000055.00000000055 55/100000000055/10000000000000

.001123595495.0011235954951/89

= .00112359550561798…

Are there negative Are there negative Fibonaccis?Fibonaccis?

Fn = Fn+2 - Fn+1

-1-1 11

-2-2 -1-1

-3-3 22

-4-4 -3-3

-5-5 55

-6-6 -8-8

-7-7 1313

-8-8 -21-21

F-n = (-1)n+1Fn

For any three Fibonacci Numbers For any three Fibonacci Numbers the sum of the cubes of the two the sum of the cubes of the two biggest minus the cube of the biggest minus the cube of the smallest is a Fibonacci number.smallest is a Fibonacci number.

8

5

13

125

512

2197

2709 – 125 = 2584

Fn+23 + Fn+1

3 – Fn3 = F3(n+1)