Rodin Fibonacci Wheel Symmetries

I would like to take a slightly deeper look at the Fibonacci/Rodin number wheel.  But first, a quick review of Marko Rodin's vortex based mathematics for those that aren't so familiar.  It is based on reducing all numbers to whole numbers, for example 25 = 2+5 = 7 or 1.156 = 1+1+5+6 = 13 = 1+3 = 4.  From this we see very interesting patterns emerge.  It may seem simple at first, but I believe it has far-reaching applications some of which we have seen in the development of the Rodin Coil. Take a look at this simple multiplication table - thanks to Chris Te Nyenhuis for this.

Notice first how 9 repeats itself always.  This is the master key of all numbers, we can think of it as zero, the point from which all numbers emerge.  Just as zero divides positive from negative, 9 creates two polarities embodied in 3 and 6.  The 3-6-9 and 6-3-9 cycle can be thought of as clockwise and counter-clockwise, or electricity and magnetism.  We can also see the other pairs which add up to 9: 1 and 8, 2 and 7, 4 and 5, run backwards from each other.  We will call these inverts.  So 8 could be thought of as -1, 7 as -2, etc.

These 6 remaining numbers can also be depicted as a doubling/halving circuit on the lazy infinity shape on this wheel.  Following one way we have doubling or powers of two:  1, 2, 4, 8, 7, 5, 1... equivalent to 1, 2, 4, 8, 16, 32, 64... The other way is halving, or inverse powers of two: 1, 5, 7, 8, 4, 2, 1 ... expressing  1, .5, .25, .125, .625, etc.  Start from any number and this holds true.

If we divide the numbers into three triangles, we get three families of numbers.  Any magic square (http://en.wikipedia.org/wiki/Magic_square ) of nine will give you one of these families diagonally.



Now we turn to the Fibonacci wheel.  The Fibonacci numbers were known in India as far as the 6th century, but Italian mathematician Leonardo Fibonacci introduced them to the West in the 12th century.  They are formed by adding two consecutive numbers in the series to get the next one.  The first few are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...  We see this series  everywhere in nature, including the geneology of bees, the growth of pinecones and sunflowers, and even the relative orbit of Earth and Venus http://www.takayaiwamoto.com/Earth_Moon_Sun/Harmony_Planets.html.  The Fibonacci series can be thought of as a whole number expression of the all-important Golden Ratio, as the ratio between numbers in the series comes closer to approximating phi = 1.61803399...

Watch what happens when we run the Fibonacci series as Rodin numbers.  We get a sequence of 24 numbers, then the sequence repeats!  We can run these numbers round a 24-sided wheel, where we see very interesting symmetries.

First off, we notice that each number is directly opposite its inverted pair.  This allows us to look at the cycle as a sine wave.  When the wave dips down, all the numbers repeat themselves but become inverted, subtracted from 9.  1, 1, 2, 3, 5... is mirrored in 8, 8, 7, 6, 4...

9 is 0, and 1 and 8 are the points of maximum potential.  So the Fibonacci Series maps out a sine wave.  Does this give us an insight into the true spiraling nature of the wave?

This 24 number circle can also be divided into 4 hexagrams.

Note first the 3-6-9 and 6-3-9 triangles in a pair, and also this 1-1-1 and 8-8-8 (the only triangles which do not correspond to the three number families) hexagram at a 90 degree angle to the first hexagram.  Then at the two askew angles, we have 1-4-7 clockwise paired with 2-5-8 counter-clockwise, and vice versa, giving us a doubling hexagram and halving hexagram respectively.  These two sets of hexagrams can also be looked at as two 12-sided objects, one with mirror numbers horizontally, and inverse numbers vertically, the other mirrors itself vertically with invert pairs horizontally.

It is very interesting to me that all these symmetries should come out of the Fibonacci Series, reinforcing the ideas of the invert pairs, three number families, and the six doubling/halving numbers. In my previous post, Template for Universal Mathematics I showed how this 24-numbered circle can also be used to map particle physics, prime numbers, crystallization, the nesting of all platonic solids, and more.  I welcome review/criticism/comments/questions.  Peace to everybody out there and try not to let numbers drive you crazy.