Math around Us: Fibonacci Numbers
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Math around Us:Math around Us:Fibonacci Fibonacci NumbersNumbersJohn HutchinsonJohn HutchinsonMarch 2005March 2005
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Leonardo Pisano FibonacciBorn: 1170 in (probably) Pisa (now in Italy)Died: 1250 in (possibly) Pisa (now in Italy)
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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.
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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
FFNN 11 11 22 33 55 88 1313 2121 3434 5555
Note thatFN+2 = FN+1
+ FN
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NoteNoteEvery 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.
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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 ????????
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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
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F1 + F2
+ F3 + … + FN = FN+2 -1
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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
????????
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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
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The FormulaThe Formula
F12 + F2
2 + F32 + …+ Fn
2 = Fn X FN+1
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FN+I = FI-1FN + FIFN+1
Another Formula
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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
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Add DiagonalsAdd Diagonals
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Pascal’s triangle with Pascal’s triangle with odd numbers in odd numbers in redred..
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1-Orchid1-Orchid
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3-Douglas Iris3-Douglas Iris
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3&5 - Bougainvilla3&5 - Bougainvilla
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5-St. Anthony’s Turnip 5-St. Anthony’s Turnip (buttercup)(buttercup)
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5-Unknown5-Unknown
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5-Wild Rose5-Wild Rose
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8-Bloodroot8-Bloodroot
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13-Black-eyed Susan13-Black-eyed Susan
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21-Shasta Daisy21-Shasta Daisy
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34-Field Daisy34-Field Daisy
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Dogwood = 4?????Dogwood = 4?????
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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.)
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SweetwartSweetwart
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SweetwartSweetwart
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"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?"
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Let’s count rabbitsLet’s count rabbits
BabiesBabies 11 00 11 11 22 33 55 88 1313 2121 3434 4545AdultAdult 00 11 11 22 33 55 88 1313 2121 3434 5555 8989TotalTotal 11 11 22 33 55 88 1313 2121 3434 5555 8989 144144
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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, …?
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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
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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
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The KeyboardThe Keyboard
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<> <> <>
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The handThe hand
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Ratios of consecutiveRatios of consecutive
11 1122 2233 1.51.555 1.666661.6666688 1.61.61313 1.6251.6252121 1.6153851.6153853434 1.6190481.619048
5555 1.6176471.6176478989 1.6181821.618182144144 1.6179781.617978233233 1.6180561.618056377377 1.6180261.618026610610 1.6180371.618037987987 1.6180331.618033etcetc 1.6180341.618034
……
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The golden ratio is The golden ratio is approximatelyapproximately
1.610833989…
(√5+1)/2 = 2/(√5-1) Or
exactly
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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…
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Golden RectangleGolden Rectangle
S
L
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Golden TrianglesGolden Triangles
5
3
8
5
L
S
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The Parthenon
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Holy Family, Michelangelo
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Crucifixion - Raphael
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Self Portrait - Rembrandt
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Seurat
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Seurat
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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 = ?
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.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…
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Are there negative Are there negative Fibonaccis?Fibonaccis?
Fn = Fn+2 - Fn+1
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-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
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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
21972709 – 125 = 2584
Fn+23 + Fn+1
3 – Fn3 = F3(n+1)