The Golden Ratio: Phi, 1.618

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.

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