# Repeating pattern in the Fibonacci Series

## The Fibonacci series 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.

{ 10 comments… read them below or add one }

Hi,

regarding this Repeating pattern in the Fibonacci Series.

I have taken analysis of this a few stages further if you would care to take a look.

The ‘adding up to 9′ thing is just one possible pattern to discover. But there are many more to be found.

You can download some of my analysis here:

http://vbm369.ning.com/forum/topics/fibonexus-and-lucanexus-continued

scroll down and download the files. You might find the

“fib divided into fib mod9 DATA.pdf, 377 KB” file the best way in.

Kind regards,

Tom

I was looking at the Fn sequence and noticed that even when the digits are flipped they still sum to the Fn sequence .By reversing the digits in a mirror image and subtracting there is a “new “number that is the mirror of a Fn,

A few of the sums had more calculations than a mirror flip and subtraction.

For instance 610 flipped is 1006.The digits are intact and “mirrored” like a circle but the sum isn’t flipped.

The other two involve a negative number which resolves by breaking the number up and subtracting.btw…109 also has a unique Fibonacci connection with 89.

With the Fn single digits I paired adjacent numbers to form new numbers and they still formed a Fn sequence.

987,610___789-16 =773 *377*

610.0,377_1006-773 =*233*

377,233___773-332 =441 *144*

233,144___332-441 =-109 *901*_ 90-1=*89*

144,89 ___441-98 =343 343_ 343-*233*=110_110/2=*55*

89,55_____98-55 =43 *34*

55,34_____55-43 =12 *21*

34,21_____43-12 =31 *13*

21,13_____12-31 =-19 91 9-1=*8*

13,8______31-8 =23 *3+2* =5

8,5,3,2____85-32 =53 *3+5* =8

5,3,2,1____53-21 =32 *2+3* =5

3 2,1,1____32-11 =21 *1+2* =3

2,1,1,0____21-10 =11 *1+1* =2

1,1,0_____11-10 =1 *1+0 =1

1,0 ______1-0 =1 1+0 =1

Hello,

Very interesting indeed. I was trying to apply the Fibonacci series in a musical composition and observed the pattern. The conversion from number to musical note was just subtracting octaves (12) until the number is in [1,13] interval. Do you think this reduction is correct, for musical purposes?

Regards

The concept make sense, but you might want to give it a bit more range for a more pleasing, realistic musical interpretation. It simply reduces every note to sit within a single octave while most songs have a range of low to high that covers more than an octave. An alternate approach would be to extend your range of allowable notes to cover two octaves, with a rule that would not allow any note to be more than an octave than the one before it.

That is a good idea! Thank you very much.

Regards

This pattern is a 12 pattern and it’s opposite…

The polyhedron with 12 faces is a DODECAHEDRON

Plato said in his writings that the DODECAHEDRON is the shape of the entire cosmos

N.A.S.A discovered in 2003 that the shape of the Universe is a DODECAHEDRON

Man made time as we know it was “ACCIDENTALLY” created being 2 cycles of 12 representing hours and minor cycles of 5×12 = 60 for minutes and seconds…

Every face of the DODECAHEDRON(12) is a PENTAGON(5)

thank you so much! this is absolutely fascinating! there are 9 major triangles in the sri yantra

My father once ran a number sequence by me that I’ve drifted back to over the past 30 years:

Start with any two, none zero, digits and note them

Sum them (if the value is greater sum those two digits ie 13 becomes 4) not result

Now sum last two digits

Continue until you get your starting numbers

This produces 5 Sequences:

11235843718988764156281911

13472922461786527977538213

14595516742685494483257314

3369663933

9999

Obviously, this is simply a numeric doodle based on reduction and the first sequence is simply reduction Fibonacci but I’ve always felt there was something there.

I’ve stretched this out to 3, 4 and 5 digit reductions (thanks to Excel) and the results show lots of patterns where factors of nine, obviously, run throughout.

Scaling up the listed article’s proposals for these other sequences is head spinning but strangely beautiful

I only really reply here as this is the first time I’ve seen my father’s time-passing exercise listed anywhere.

Fascinating. Here’s another aspect of the Fibonacci series which reveals another repetitive pattern.

I just recently discovered that if you take the Fibonacci squares and translate each number to a pitch in the one octave 7-note scale you end up with palindromatic, infinitely repeating series of pitches.

The first numbers in the Fibonnaci series of squares are 1, 1, 4, 9, 25, 64, 169

This will translate to the following pitches in one octave: 1, 1, 4, 2 (9-7), 4 (25-21), 1 (64-63), 1 (169-168).

If we continue this process, this is the pattern we end up with:

1 1 4 2 4 1 1 7 1 1 4 2 4 1 1 7 etc

Seeing the 7 as the center you end up with a 15 note palindromatic, infinitely repeating pattern. Not only that, but the pitches between the 7s make up another palindromatic, repeating pattern with the 2 as the center made of 7 pitches (1 1 4 2 4 1 1).

Numerif duction in fibonacci series is jst awesome