Beyond

NUMERACY

John Allen Paulos 1991

 Page 98

“GOLDEN RECTANGLE, FIBONACCI SEQUENCES”

Page 99

“A golden rectangle is defined to be any rectangle whose-length-to width ratio is the same as this golden ratio. ” “…The famous Fibonacci sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,89, 144, 233,… bears an unexpected relation to these golden rectangles…”

“…The sequence is defined by the fact that each term in it (except for the first two) is the sum of its two predecessors

2 =1 + 1; 3 = 2 + 1; 5 = 3 + 2; 8 = 5 + 3, 13 = 8 + 5.

Remembering that the golden ratio is approximately 1.61803 (the number has a non repeating infinite decimal expansion) and performing a little long division, we notice (what can be proved with a little algebra) that the ratio of a term in the fibonacci sequence to its predecessor app roaches this ratio. For the case of 3 by 5 cards, the ratio 5/3 = 1.66666; for 5 by 8 cards, the ratio is 8/5 = 1.6; 13/8 = 1.625; 21/13 = 1.615384; 34/21= 1.61905; and so on."

 

 

GOD'SECRET FORMULA

Peter Plichta 1997

Page 122 (continues)

"The number 81 is the product of 3 x 3 x 3 x 3; 3 4 = 81. The numbers 3, 4, and 81 had been on my mind for years, and suddenly their interrelation appeared as a ' 3- to-the-power-of- law '. If God had simply arranged the 81 elements according to the ordinal numbers 1, 2, 3, …81, researchers would have discovered this fact a long time ago.

Reight, said Zed Aliz to yonder scribe, the first eight sequined sequence.

There has to be give and take in numbers, writ the scribe, having left out zeros occurring after the decimal point.

Better start again from the beginning said Zed Aliz, continue ignoring said zeros, and cluster eight to each twinkle.

Add to reduce, reduce to deduce for example 8 + 1 iz 9 and 9 + 1 iz 10 and 1 + zero iz won.

Reight wah scribe said Zed Aliz Zed, make thee an 8 x 9 ordered grouping of the first 72 distilled Fibonacchi numbers.

Azin add to reduce, reduce to deduce, said the scribe.

Azin add to reduce, reduce to deduce said AlizZed.

And so yon scribe set about patent pattern prospecting.

The scribe noted the sequined sequence change in the 73rd sequence.

 

AZAZAZAZAZAZAZAZAZ

 

 

1.00

=

1

1.00

=

2

2.00

3

3.00

4

5.00

5

8.00

6

13.00
1+3
=
4

7

21.00
2+1
=
3

8

34.00
3+4
=
7

9

55.00
+
=
1

10
+
=
1

89.00
+
=
8

11

144.00
+
=
9

12

233.00
+
=
8

13

377.00
+
=
8

14

610.00
+
=
7

15

987.00
+
=
6

16

1597.00
+
=
4

17

2584.00
+
=
1

18
+
=
9

4181.00
+
=
5

19

6765.00
+
=
6

20

10946.00
+
=
2

21

17711.00
+
=
8

22

28657.00
+
=
1

23

46368.00
+
=
9

24

75025.00
+
=
1

25

121393.00
+
=
1

26

196418.00
+
=
2

27
+
=
9

317811.00
+
=
3

28

514229.00
+
=
5

29

832040.00
+
=
8

30

1346269.00
+
=
4

31

2178309.00
+
=
3

32

3524578.00
+
=
7

33

5702887.00
+
=
1

34

9227465.00
+
=
8

35

14930352.00
+
=
9

36
+
=
9

24157817.00
+
=
8

37

39088169.00
+
=
8

38

63245986.00
+
=
7

39

102334155.00
+
=
6

40

165580141.00
+
=
4

41

267914296.00
+
=
1

42

433494437.00
+
=
5

43

701408733.00
+
=
6

44

1134903170.00
+
=
2

45
+
=
9

1836311903.00
+
=
8

46

2971215073.00
+
=
1

47

4807526976.00
+
=
9

48

7778742049.00
+
=
1

49

12586269025.00
+
=
1

50

20365011074.00
+
=
2

51

32951280099.00
+
=
3

52

53316291173.00
+
=
5

53

86267571272.00
+
=
8

54
+
=
9

139583862445.00
+
=
4

55

225851433717.00
+
=
3

56

365435296162.00
+
=
7

57

591286729879.00
+
=
1

58

956722026041.00
+
=
8

59

1548008755920.00
+
=
9

60

2504730781961.00
+
=
8

61

4052739537881.00
+
=
8

62

6557470319842.00
+
=
7

63
+
=
9

10610209857723.00
+
=
6

64

17167680177565.00
+
=
4

65

27777890035288.00
+
=
1

66

44945570212853.00
+
=
5

67

72723460248141.00
+
=
6

68

117669030460994.00
+
=
2

69

190392490709135.00
+
=
8

70

308061521170129.00
+
=
1

71

498454011879264.00
+
=
9

72
+
=
9

806515533049393.00
+
=
1

73

1304969544928660.00
+
=
4

74

2111485077978050.00
+
=
2

75

3416454622906710.00
+
=
6

76

5527939700884760.00
+
=
8

77

8944394323791460.00
+
=
4

78

14472334024676200.00
+
=
1

79

23416728348467700.00
+
=
9

80

37889062373143900.00
+
=
1

81
+
=
9

N
A
T
U
R
E
S

F
I
B
O
N
A
C
C
H
I

1
2
3
4
5
6
7
8

1
2
3
4
5
6
7
8

1
2
3
4
5
6
7
8

1
1
2
3
5
8
4
3

9
10
11
12
13
14
15
16

7
1
8
9
8
8
7
6

17
18
19
20
21
22
23
24

4
1
5
6
2
8
1
9

25
26

27
28
29
30
31
32

1
1
2
3
5
8
4
3

33
34
35
36
37
38
39
40

7
1
8
9
8
8
7
6

41
42
43
44
45
46
47
48

4
1
5
6

2
8
1
9

49
50
51
52
53
54
55
56

1
1
2
3
5
8
4
3

57
58
59
60
61
62
63
64

7
1
8
9
8
8
7
6

65
66

67
68
69
70
71
72

4
1

5

6

2
8
1
9

N
A
T
U
R
E
S

F
I
B
O
N
A
C
C
H
I

1
2
3
4
5
6
7
8
=
36

1
1
2
3
5
8
4
3
=
27

9
1
2
3
4
5
6
7
=
37

7
1
8
9
8
8
7
6
=
54

8
9
1
2
3
4
5
6
=
38

4
1
5
6
2
8
1
9
=
36

7
8
9
1
2
3
4
5
=
39

1
1
2
3
5
8
4
3
=
27

6
7
8
9
1
2
3
4
=
40

7
1
8
9
8
8
7
6
=
54

5
6
7
8
9
1
2
3
=
41

4
1
5
6
2
8
1
9
=
36

4
5
6
7
8
9
1
2
=
42

1
1
2
3
5
8
4
3
=
27

3
4
5
6
7
8
9
1
=
43

7
1
8
9
8
8
7
6
=
54

2
3
4
5
6
7
8
9
=
44

4
1
5
6
2
8
1
9
=
36

45
45
45
45
45
45
45
45

36
9
45
54
45
72
36
54

N
A
T
U
R
E
S

F
I
B
O
N
A
C
C
H
I

1
2
3
4
5
6
7
8
=
9

1
1
2
3
5
8
4
3
=
9

9
1
2
3
4
5
6
7
=
1

7
1
8
9
8
8
7
6
=
9

8
9
1
2
3
4
5
6
=
2

4
1
5
6
2
8
1
9
=
9

7
8
9
1
2
3
4
5
=
3

1
1
2
3
5
8
4
3
=
9

6
7
8
9
1
2
3
4
=
4

7
1
8
9
8
8
7
6
=
9

5
6
7
8
9
1
2
3
=
5

4
1
5
6
2
8
1
9
=
9

4
5
6
7
8
9
1
2
=
6

1
1
2
3
5
8
4
3
=
9

3
4
5
6
7
8
9
1
=
7

7
1
8
9
8
8
7
6
=
9

2
3
4
5
6
7
8
9
=
8

4
1
5
6
2
8
1
9
=
9

9
9
9
9
9
9
9
9

9
9
9
9
9
9
9
9
=
9

N
A
T
U
R
E
S

F
I
B
O
N
A
C
C
H
I

1
2
3
4
5
6
7
8

1
1
2
3
5
8
4
3

9
1
2
3
4
5
6
7

7
1
8
9
8
8
7
6

8
9
1
2
3
4
5
6

4
1
5
6
2
8
1
9

7
8
9
1
2
3
4
5

1
1
2
3
5
8
4
3

6
7
8
9
1
2
3
4

7
1
8
9
8
8
7
6

5
6
7
8
9
1
2
3

4
1
5
6
2
8
1
9

4
5
6
7
8
9
1
2

1
1
2
3
5
8
4
3

3
4
5
6
7
8
9
1

7
1
8
9
8
8
7
6

2
3
4
5
6
7
8
9

4
1
5
6
2
8
1
9

Alizzed, fast hold of just enough Fibonacci distillations, precipitates a number of numbers.

 

F
=
1741
x
3
=
522

F
=
1741
x
3
=
522

I
=
1
x
9
=
9

1

I

B
=
285
x
3
=
855

B
=
285
x
3
=
855

O
=
396
x
3
=
1188

O
=
396
x
3
=
1188

N
=
582
x
3
=
1746

N
=
582
x
3
=
1746

A
=
8
x
9
=
72

8

A

C
=
471
x
3
=
1413

C
=
471
x
3
=
1413

C
=
369
x
3
=
1107

C
=
369
x
3
=
1107

I
=
3853
x
36
=
6912

I

3844

18

6831

6912

MINUS

6831

81