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General: The Fibonacci sequence has a pattern that repeats every 24 numbers=24 hours
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From: BARILOCHENSE6999  (Original message) Sent: 28/10/2019 18:25

Fibonacci 24 Repeating Pattern

The Fibonacci sequence has a pattern that repeats every 24 numbers.

Numeric reduction is a technique used in analysis of numbers in which all the digits of a number are added together until only one digit remains.  As an example, the numeric reduction of 256 is 4 because 2+5+6=13 and 1+3=4.

Applying numeric reduction to the Fibonacci series produces an infinite series of 24 repeating digits:

1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9

If you take the first 12 digits and add them to the second twelve digits and apply numeric reduction to the result, you find that they all have a value of 9.

 

1st 12 numbers 1 1 2 3 5 8 4 3 7 1 8 9
2nd 12 numbers 8 8 7 6 4 1 5 6 2 8 1 9
Numeric reduction – Add rows 1 and 2 9 9 9 9 9 9 9 9 9 9 9 18
Final numeric reduction – Add digits of result 9 9 9 9 9 9 9 9 9 9 9 9

This pattern was contributed both by Joseph Turbeville and then again by a mathematician by the name of Jain.

We would expect a pattern to exist in the Fibonacci series since each number in the series encodes the sum of the previous two.  What’s not quite so obvious is why this pattern should repeat every 24 numbers or why the first and last half of the series should all add to 9.

For those of you from the “Show Me” state, this pattern of 24 digits is demonstrated in the numeric reduction of the first 73 numbers of the Fibonacci series, as shown below:

Fibonacci Number

Numeric reduction by adding digits
1st Level 2nd Level Final Level
Example: 2,584 2+5+8+4=19 1+9=10 1+0=1
0 0 0 0
1 1 1 1
1 1 1 1
2 2 2 2
3 3 3 3
5 5 5 5
8 8 8 8
13 4 4 4
21 3 3 3
34 7 7 7
55 10 1 1
89 17 8 8
144 9 9 9
233 8 8 8
377 17 8 8
610 7 7 7
987 24 6 6
1,597 22 4 4
2,584 19 10 1
4,181 14 5 5
6,765 24 6 6
10,946 20 2 2
17,711 17 8 8
28,657 28 10 1
46,368 27 9 9
75,025 19 10 1
121,393 19 10 1
196,418 29 11 2
317,811 21 3 3
514,229 23 5 5
832,040 17 8 8
1,346,269 31 4 4
2,178,309 30 3 3
3,524,578 34 7 7
5,702,887 37 10 1
9,227,465 35 8 8
14,930,352 27 9 9
24,157,817 35 8 8
39,088,169 44 8 8
63,245,986 43 7 7
102,334,155 24 6 6
165,580,141 31 4 4
267,914,296 46 10 1
433,494,437 41 5 5
701,408,733 33 6 6
1,134,903,170 29 11 2
1,836,311,903 35 8 8
2,971,215,073 37 10 1
4,807,526,976 54 9 9
7,778,742,049 55 10 1
12,586,269,025 46 10 1
20,365,011,074 29 11 2
32,951,280,099 48 12 3
53,316,291,173 41 5 5
86,267,571,272 53 8 8
139,583,862,445 58 13 4
225,851,433,717 48 12 3
365,435,296,162 52 7 7
591,286,729,879 73 10 1
956,722,026,041 44 8 8
1,548,008,755,920 54 9 9
2,504,730,781,961 53 8 8
4,052,739,537,881 62 8 8
6,557,470,319,842 61 7 7
10,610,209,857,723 51 6 6
17,167,680,177,565 67 13 4
27,777,890,035,288 73 10 1
44,945,570,212,853 59 14 5
72,723,460,248,141 51 6 6
117,669,030,460,994 65 11 2
190,392,490,709,135 62 8 8
308,061,521,170,129 46 10 1
498,454,011,879,264 72 9 9

 

Thanks to Joseph Turbeville for sending “A Glimmer of Light from the Eye of a Giant” and to Helga Hertsig for bringing Jain’s discovery of this pattern to my attention.

https://www.goldennumber.net/fibonacci-24-pattern/


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De: BARILOCHENSE6999 Enviado: 28/10/2019 14:40

Fibonacci 60 Repeating Pattern

The last digit of the numbers in the Fibonacci Sequence form a pattern that repeats after every 60th number:

0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1

This pattern can be seen in the following list of the first 72 Fibonacci numbers:

0 0
1 1
2 1
3 2
4 3
5 5
6 8
7 13
8 21
9 34
10 55
11 89
12 144
13 233
14 377
15 610
16 987
17 1,597
18 2,584
19 4,181
20 6,765
21 10,946
22 17,711
23 28,657
24 46,368
25 75,025
26 121,393
27 196,418
28 317,811
29 514,229
30 832,040
31 1,346,269
32 2,178,309
33 3,524,578
34 5,702,887
35 9,227,465
36 14,930,352
37 24,157,817
38 39,088,169
39 63,245,986
40 102,334,155
41 165,580,141
42 267,914,296
43 433,494,437
44 701,408,733
45 1,134,903,170
46 1,836,311,903
47 2,971,215,073
48 4,807,526,976
49 7,778,742,049
50 12,586,269,025
51 20,365,011,074
52 32,951,280,099
53 53,316,291,173
54 86,267,571,272
55 139,583,862,445
56 225,851,433,717
57 365,435,296,162
58 591,286,729,879
59 956,722,026,041
60 1,548,008,755,920
61 2,504,730,781,961
62 4,052,739,537,881
63 6,557,470,319,842
64 10,610,209,857,723
65 17,167,680,177,565
66 27,777,890,035,288
67 44,945,570,212,853
68 72,723,460,248,141
69 117,669,030,460,994
70 190,392,490,709,135
71 308,061,521,170,129
72 498,454,011,879,264

Lucien Khan arranged these 60 digits of the pattern in a circle, as shown in illustration below:

fibonacci-60-digit-repeating-pattern-copyright-lucien-khan

Here he found other interesting results:

  • The zeros align with the 4 cardinal points on a compass.
  • The fives align with the 8 other points of the 12 points on a clock.
  • Except for the zeros, the number directly opposite each number adds to 10.

Lucien postulates that ancient knowledge of these relationships contributed to the development of our modern use of 60 minutes in an hour, and presentation of numbers on the face of the clock.

I found too that any group of four numbers that are 90 degrees from each other (15 away from each other in the circle) sum to 20, except again for the zeros. As an example, use 1, 7, 9 and 3, which appear one to the right of each of the compass points.

Additionally, every group of five numbers that define the points of the 12 pentagons on the circle also create a pattern. Four of the pentagons have even-numbered last digits of 0, 2, 4, 6, and 8. The remaining eight pentagons have odd-numbered last digits of 1, 3, 5, 7 and 9.
fibonacci-60-digit-12-pentagon-pattern

Another interesting pattern yet was observed by Lucien Khan: The 216th number is this sequence is 619220451666590135228675387863297874269396512. The sum of all the digits in that number add up to 216, as well. He notes that it is believed that the secret or hidden name of God contains 216 characters. There are many other fascinating relationships and sacred geometries, which are presented by Lucien Khan in more detail at the links below.

https://www.goldennumber.net/fibonacci-60-repeating-pattern/

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De: BARILOCHENSE6999 Enviado: 28/10/2019 14:48
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