14. More Fibonacci

Lets look at all the ways we can create a female Fibonacci pattern. Our original line was.

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

Lets start a new sequence but this time starting 2 2, we will be using the same process as we used to create the original Fibonacci sequence.

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

Everything we did with the original sequence still works but this time we have extra 2s and 7s. we get the same doubling sequence running in both directions.

lets start 4 4 to see what we get.

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 lastly lets start 3 3

3 3 6 9 6 6 3 9

This creates a sequence that is one third the length of the other three sequences just like the multiplication tables.

A point to make is that we can start lines 5 5 or 7 7 and 8 8 but we are only going to create the exact same sequences just starting half way round.

That just leaves 9 9 and this like most things we do with 9 will create a sequence of 9s forever.

doublefibonacci

This is not a ring set but just illustrates how they all line up. Now finally we have balance where all digits are represented an equal amount of times.

We could include negative numbers for completeness and run the 3 3 sequence three times

fibposneg

Notice that all the family groups are lined up with the outside ring.

So we have found all the female patterns that we can produce using the Fibonacci method of adding the last two digits and what we find is that you can start with any two digits, for example lets start with 273 and 9935 add them together and continue the sequence and in a very few steps we create the golden 1.618 ratio, it turns out that its the Fibonacci Process that is important not the numbers we use.

If we take the above example 273>)3 and 9935>)8 we have a 3 8 in that order and if we continue the sequence we are on the sequence starting 2 2, what this means is that there are  no two number that you can pick that will not fall into one of the above sequences.

Lets look at something very interesting we can do to create a ring set.

We will start with the original sequence.

fib2

Now we already know that the white numbers are two interwoven doubling sequences going in opposite directions. Lets take the remaining numbers.

1 3 8 3 1 9 8 6 1 6 8 9

These numbers we draw in a circle then create a ring set, and we get a 12 rings with 12 digits on each ring.

fibringset1

Notice how every other ring creates 12 numbers that make up one half of one of the three Fibonacci sequences. Also the 3s and 6s do not line up this time, they fall into the 3 6 3 6 doubling pattern

If we look at the first ring we find the other half of its Fibonacci pattern is created on the third ring with a 90 degree twist. The first ring pattern is exactly repeated on the seventh ring this time starting from the bottom and the same 90 degree twist can be found on the tenth ring.

fibringset2

Now lets look at the rings that contain the interwoven doubling. Red numbers run clockwise starting from a yellow 1 and blue number run anticlockwise starting from the green 1s.

fibringset3

Notice how the starting points spiral out.

And if all that is not enough we can see every spoke has doubling running from the outside ring to the centre.

fibringset4

Even the rings that contain the 9 family group can be seen to be doubling.

When we look at this we find that the three Fibonacci sequences are interwoven and that they create each other.

When looking at the Fibonacci and its connections to the doubling is just mind boggling.

We can find the Fibonacci sequence in some more sequences we will be looking at.