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Fibonacci & Other Integer Sequences – Mod 9 by Anthony Morris

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Fibonacci & Other Integer Sequences – Mod 9 by Anthony Morris Fibonacci & Other Integer Sequences – Mod 9 by Anthony Morris Please see the accompanying spreadsheet 1.618 - The Golden Mean - PHI: The Organising Structure Found Throughout Nature. Evident In Every Living Thing. Having more than a passing acquaintance with Fibonacci and the Golden Mean through my work on financial markets, I immediately looked at the sequence in a new light. For those that do not know, the Fibonacci sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 etc. The interesting thing about this sequence is that by adding the previous 2 terms to get the next term in the series, the numbers adjacent approximate to a Phi / Golden Mean relationship of 1.618. More unusually the numbers once removed are in a relationship of 2.618! Anthony Morris – Market Harmony Ltd – [email protected] The higher up in the sequence, the closer two consecutive numbers of the sequence divided by each other will approach the golden ratio or what is called PHI, which is approximately 1: 1.618 or 0.618: 1. The Golden Ratio is found throughout the human body, even in our DNA. The DNA molecule, the program for all life, measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. I then discovered to my amazement that if you go far enough out you soon realise that the Fibonacci sequence with MOD 9 applied, produces a repeating 24 number sequence: 1,1,2,3,5,8,4,3,7,1,8,9,8,8,7,6,4,1,5,6,2,8,1,9 I couldn’t believe it! I thought, nobody will have found that but research unfortunately revealed that I was not the first to see this recurring pattern but it seemed that it had only recently been discovered. Upon further investigation I noted that if you take the second 12 numbers of the sequence and place them under the first 12 numbers we can see the beautiful symmetry in the MOD 9 sequence where top and bottom numbers are number pairs as described in the Introductory paper - http://archive.org/details/IntroductionToNoCoincidence. 112358437189 887641562819 Adding Totals 1000000000008. Even more interesting is when we write the Fibonacci sequence, Mod 9, like this: 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9 Column 1 shows the Doubling Circuit Numbers of the 1 2 4 8 7 5 Column 2 shows 1 and 8 Number Pair. Column 3 shows the Doubling Circuit Numbers again. Column 4 shows the 3 6 9 Family Number Group (more about these later). Anthony Morris – Market Harmony Ltd – [email protected] Let’s take a quick look at what happens if you start the Fibonacci sequence with 2 2 instead of 1 1 and again using Mod 9 for the results of the sequence, we also find a repeating 24 number pattern that exhibits symmetry. 2 2 4 6 1 7 8 6 5 2 7 9 7 7 5 3 8 2 1 3 4 7 2 9 And then arranged in 4 columns 2 2 4 6 1 7 8 6 5 2 7 9 7 7 5 3 8 2 1 3 4 7 2 9 Column 1 Doubling Circuit. Column 2 2 and 7 Number Pair. Column 3 Doubling Circuit. Column 4 The 3 6 9 Family Number Group. Now, the sequence starting 3 3: Symmetry again. 3 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9 And 3 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9 Interesting to see how these numbers divide themselves Anthony Morris – Market Harmony Ltd – [email protected] Columns 1 2 and 3 show the 3 6 Number Pair Column 4 just the 9. The 4 4 sequence: Symmetry 4 4 8 3 2 5 7 3 1 4 5 9 5 5 1 6 7 4 2 6 8 5 4 9 And 4 4 8 3 2 5 7 3 1 4 5 9 5 5 1 6 7 4 2 6 8 5 4 9 Column 1 Doubling Circuit Numbers Column 2 4 and 5 Number Pair Column 3 Doubling Circuit Numbers Column 4 The 3 6 and 9 Number Group The 5 5 Sequence: Symmetry 5 5 1 6 7 4 2 6 8 5 4 9 4 4 8 3 2 5 7 3 1 4 5 9 And 5 5 1 6 7 4 2 6 8 5 4 9 4 4 8 3 2 5 7 3 1 4 5 9 Anthony Morris – Market Harmony Ltd – [email protected] Column 1 Doubling Circuit. Column 2 The 4 and 5 Number Pair. Column 2 Doubling Circuit. Column 4 The 3 6 and 9 Family Number Group. Now it should be clear that as we progress to the higher numbered sequences they will just be mirrors of their Number Pair as we have just seen with the 4 4 and 5 5 sequence, so I won’t bore you with any more. 4 4 8 3 2 5 7 3 1 4 5 9 5 5 1 6 7 4 2 6 8 5 4 9 5 5 1 6 7 4 2 6 8 5 4 9 4 4 8 3 2 5 7 3 1 4 5 9 There is much more work to do in this area as I feel there is an enormous amount of information to be gleaned from understanding the geometry of the above and investigating the exact proportion of the diameter where there are crossovers directly under the 9, for example. Anthony Morris – Market Harmony Ltd – [email protected] Dec 2013 Update A common mistake is thinking that the ratios of consecutive Fibonacci numbers are the only numbers that can create the Golden Mean. The fact is, ANY sequence of numbers, starting with ANY random two numbers, you will still eventually get to the Phi ratio if you divide the latter into the former. The same is true even if you begin with a negative number. -8 4 -4 0 -4 -4 -8 -12 -20 -32 -52 -84 -136 by now we see that 136 / 84 = 1.619047 The golden mean ratio ALWAYS reasserts itself to control the numbers - it is inherent to any sequence of numbers formed as an addition of previous two numbers after the 9th number in the sequence. However the 1 1 2 3 5 8 13 is the natural unfolding in the universe and is what makes the Fibonacci numbers in this sequence so special. In wondering about this sequence I wondered what terms could come before the zero and leave the positive side of the equation unaffected and saw a rather nice sequence – there is a type of symmetry being displayed here that doesn’t seem very normal because of the alternating positive and negative value numbers as we go beyond 0 -21 13 -8 5 -3 +2 -1 1 0 1 1 2 3 5 8 13 21 The above model contains 17 Fibonacci terms 2 full octaves of eight terms either side of their fulcrum point of balance 0, beginning with -21 and 13 ending with 21. I feel this is pretty deep and essential…. To the left of zero, the total of the terms is -12 and to the right the total is 54 Addition of these terms gives you a total of 42 (2x21) which is rather elegant I would say. So just what did Douglas Adams know? Note Mod 9 -3 4 -8 5 -3 +2 -1 1 0 1 1 2 3 5 8 4 3 Total is 24 21 is the key number in my paper Partition Theory & The Hologram Projector of Physical Reality. Anthony Morris – Market Harmony Ltd – [email protected] Inter Fibonacci Number Effects What follows is a new look at the Fibonacci Sequence and I believe I have turned up some interesting info here using similar simple techniques as before to be able to understand what we are actually looking at. We all know that this sequence approximates to the Golden Mean or Phi but actually I think it is important to examine exactly the numbers as they relate to each other and all other numbers. The best way to do this seems to be by division. Number 1 The first 2 terms of the Fibonacci Sequence, the higher divided by the lower: 1/0 = Infinity Reverse 0/1 = Infinity Number 2 The next two in the sequence, the higher divided by the lower number 1/1 = 1 Reverse 1/1 = 1 Number 3 2/1 = 2 Reverse 1/2 = 0.5 Number 4 3/2 = 1.5 Reverse 2/3 = 0.666 R Number 5 5/3 = 1.6666 R Reverse 3/5 = 0.6 Number 6 8/5 = 1.6 Reverse 5/8 = 0.625 Anthony Morris – Market Harmony Ltd – [email protected] Number 7 13/8 = 1.625 Reverse 8/13 = 0.615384 R 615 384 999 Symmetry Digit Sum of recurring decimal = 27 Number Pairs 0 9 x 0 1 8 x 1 2 7 x 0 3 6 x 1 4 5 x 1 Number 8 21/13 = 1.615384 R 615 384 999 Symmetry Digit Sum of recurring decimal = 27 Anthony Morris – Market Harmony Ltd – [email protected] Number Pairs 0 9 x 0 1 8 x 1 2 7 x 0 3 6 x 1 4 5 x 1 Reverse 13/21 = 0.619047 R Number Pairs 6 1 9 0 4 7 But No Symmetry!! A 6 digit recurring decimal without symmetry is a unique attribute of the number 21 and its multiples.
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