12. Fibonacci

The Fibonacci sequence of numbers is very famous, it creates something called the golden ratio and it is this ratio that nature uses to create life. The male pattern is found in everything natures does and has been documented else where, if you are interested to learn more just google the Fibonacci sequence.

First we will take a quick look at the male Fibonacci sequence and how it is built but what we will be looking at in great detail is its female pattern.

We start with a 1 and the next number is also a 1 now we add these digit together and get 2 so our sequence starts
1 1 2
To continue we are going to add together the last two digits 1 1+2=3 so our sequence is
1 1 2 3
Again we add the last two digits 1 1 2+3=5 which gives us
1 1 2 3 5
Continuing in this fashion 1 1 2 3+5=8 the next digit is always the sum of the previous two digits
1 1 2 3 5 8
Lets do one more 1 1 2 3 5+8=13 then its into male numbers.

So you see how to build the sequence lets look at the first 24 numbers in the male pattern.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1,597, 2,584, 4,181, 6,765, 10,946, 17,711, 28,657, 46,368

an interesting observation, you can stop the sequence anywhere the last number will always be 1 higher than the sum of all the numbers before less the second to last, not sure that makes sense so lets have a look at it.
lets stop at 5
1 1 2 3 5 lets group them up (1 1 2) 3 (5) add the first group and we get 4 and the last number is 5 which is 1 more.

Another example lets stop at 233.
(1 1 2 3 5 8 13 21 34 55 89) 144 (233) add the 11 numbers in the first group you get 232 which is 1 less than the last number.

Now lets have a look at the golden ratio and how to find it.
We stop the sequence anywhere and divide the last number by the previous number.
lets stop at 610 the number before 610 is 377 so if we divide 610 by 377 we get 1.618037135278515 which is rounded down to 1.618 you can do this at any step and you will always get 1,618 and its this ratio that is found everywhere.

So anyway on to the female pattern, we could implode the 24 numbers above or we can start and implode the female pattern as we go, as we seen it makes no difference if we do male math or female math the female pattern is alway the same.
Here is the Female pattern that is generated by adding the last two 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 we continue the pattern repeats. The male Fibonacci can go on to infinity getting bigger and bigger but this finite 24 digit female pattern will always repeat forever.

At first it looks a bit random so lets start breaking it down, first thing is to draw it in a circle.

fib1

We see straight away that the 9s create a centre axis just like our multiplication ring set, also the 3s and 6s also line up. Notice also that polar pairs are all exactly opposite each other.

lets wrap it around the multiplication ring set to see.

fibtimesoverlay1

When I discovered this I was really excited, this Fibonacci sequence had to be related. I wanted to make a new ring set that contained both the multiplication set and the Fibonacci sequence, but I couldn’t see how.

I was reluctant to change the multiplication ring set as it is complete and any changes could spoil it. I stared along time at the above image trying to break the Fibonacci down even further.

I tried to expand it out into a ring set.

fibspiral1

All that happens is that the exact same number sequence is produced but slightly shifted and if we keep adding rings it spirals. This is interesting because the Fibonacci causes spirals in nature, the spiral on a sea shell or the spiral of a galaxy are just two examples. Maybe I could try and twist the Fibonacci sequence into the multiplication ring set.

When we continue to break down this sequence we find an interesting way to create a repeating ring set but we will get to that later

lets continue.

I noticed that there are too many 1s and 8s they seem to cause an imbalance. So far in all previous sequences there has been complete balance with all digits being represented an equal amount of times. Even though there are too many 1s and 8s they are distributed evenly

fib2

Then I noticed that the 12 numbers left could be split into 2 sets of the doubling sequence. one running clockwise and one running anticlockwise.

fib3

So here we have 2 Patterns of doubling, I am happy to consider the 3 6 and 9 as being invisible, because the 3 and 6 times table are invisible in the multiplication ring set. But I am have problems with the extra 1s and 8s.

If we include the extra 1s and 8s we can implode the three numbers in between the 3s, 6s and 9s. This also creates a doubling pattern clockwise, we can also implode the 3 digits to around the 9 family group and we get doubling going anticlockwise.

fib4

As you see there above the Fibonacci has doubling running forwards and backwards. We have looked at just two of the methods that you can use to create this pattern.

We will leave this sequence for now because it was at this point I had an idea of how to change the multiplication ring set without spoiling it. I wanted to try and add negative numbers.

Advertisements