# 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.

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.

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.

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

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.

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.

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.

# 13. Negative Numbers

lets take another look at the Rodin Symbol.

In the Marko Rodin presentation he says that in his symbol there are really 18 digits, each digit has a negative version of itself  hiding behind it. There is a negative 9 behind the 9, negative 1 behind the 1 and so on.

I decided to include negative numbers in my multiplication ring set to see what happened but instead of them hiding I wanted to be able to see them.

I started with a ring with the 1 times table.

This time when dividing the ring I want to leave space to include another 9 digits, so I drew the lines in triangles.

Now I can include negative numbers. The obvious place to put negative 9 is straight at the bottom to keep it in line with the other 9s. Then because we are dealing with negative numbers I decided to run anticlockwise, this keeps all the 3s and 6s in line.

We can then expand to the next ring using the same method of triangles to divide it.

Now we can create our next ring of positive numbers using the same method we used when we built our multiplication table.

If we add the positive 1 and 9 we have a place to insert our answer.

We then continue doing the same for the rest of the ring.

Lets repeat the process for the negative numbers.

And repeat for the remaining numbers.

When we complete the ring set and round off the sharp triangle edges into curves we get something like this.

This created spirals

There are 12 of them, but remember if we were to expand the rings out we would only repeat the first ring so as the spiral comes out of the outer ring we can continue its sequence back on the first ring.

That gives us three different  number sequences one of them is shown above but there is also one each side.

Also we have three more number sequences found spinning anticlockwise.

If we expand the ring set to repeat two times so we have a total of 18 rings we can see the full number sequences a little easier. This spiral does a full 360 degree twist in 18 rings.

So lets look at all the spirals remember there are 12 spirals but only 3 different numbers sequences.

I have coloured the same sequence the same colour. and the same anticlockwise.

The numbers in the spirals can be drawn in a circle because it will always repeat the same sequence no matter how big we expand the ring set.

So lets copy the numbers from the spirals and draw them in a circles. this is not a ring set as the rings are not creating each other.
The inner three rings are the number sequences from the spirals that are going anticlockwise and the outer three rings are the spirals going clockwise.

Now if we ignore the 9 family group and look at the numbers in between we find the Fibonacci doubling going forwards and backwards.

By including negative numbers in the multiplication ring set we find that the Fibonacci was there all along I just needed to see the spirals.

lets have a look at how these doubling sequences work.

All the numbers in yellow are doubling clockwise starting from the orange 1 and the grey numbers are running anticlockwise starting from the purple 1.

What does all this mean?

After I had discovered all this I found the work of Daniel Nunez who has been winding electrical coils. I find it extremely fascinating when comparing the above ring set with its 12 spirals going both clockwise and anticlockwise and this coil made by Daniel Nunez

Anyway back to the numbers and in the ring set above we have three Fibonacci type number sequences. I felt like something was missing so Lets return to the Fibonacci and see what we can find.

# 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.

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

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.

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.

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.

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.

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.

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.

# 6. Sequences

In vortex math we find the underlying repeating female patterns in our male number sequences.

It is essential to understand that as we build these number sequences it does not matter if we build these sequences using female or male numbers.

lets look at this

55+42=97

now lets look at this using the female numbers

55>)1

42>)6

97>)7

so we get 1+6=7

Think of it like this any male number that has a female value of 1 added to any male number with a female value of 6 will always equal a male number with a female value of 7

Its exactly the same when we are multiplying numbers together.

look at this male multiplication.

17×21=357

now lets look at this using the female numbers

17>)8

21>)3

357>)6

we get 8×3=24>)6

I cant stress how important this understanding is as when we start looking at some really big number sequences we can just work with the female numbers and it will stop our male math going into the stratosphere.

There are a number of ways we can create an infinite number sequence using our male math but when we look at the female counterparts of these male numbers we find a absolute finite pattern of repeating female numbers.

Once we unlock these female patterns I find the best way to study the pattern is to place the numbers around a circle just like Marko Rodin did in his symbol. The circle is the by far the best method of studying these patterns for many reasons, you can see the symmetry of the numbers and because the pattern is a repeating pattern you can just go round and round the circle.

One last important point before we move on.

we only have 9 digits in vortex math 1 2 3 4 5 6 7 8 9 is all we’ve got we do not use zero however 9 and 0 are very similar in the way they behave. lets have a look.

We can add nine to any number and the female number will remain the same.

We can add zero to any number and the female number will remain the same.

Anything we multiply by nine will always equal nine

Anything we multiply by zero will always equal zero

# 7. Multiplications

So we come to our first number sequence and our first female pattern. If we simply start counting, always increasing the last digit in the sequence by 1 we create our 1 times table.
1 2 3 4 5 6 7 8 9 10
but when we get to 10 we implode the number to 1 and if we continue from 1 we are only going to repeat what has already gone 10 becomes a 1, 11 implodes to a 2, 12 implodes to a 3 and so on.

This can be drawn in a circle similar to the Marko Rodin symbol.

We put the 9 at the top as this acts similar to zero and is where every thing starts. we continue counting clockwise.

Notice that the numbers either side of the 9 add to 9 (1 and 8) then (2 and 7) then (3 and 6) and at the bottom (4 and 5). There are other points worthy of note but we will come back to them.

Now lets look at the 2 times table (adding 2 to the last number in the sequence) we will be imploding the sequence as we go so when we get to 8 add 2 we will only be using the female answer.
2 4 6 8 1 3 5 7 9
After the 9 the whole sequence will repeat, this sequence we can also draw in a circle.

Again we notice that if we draw an imaginary line vertically down the centre all opposite pairs of numbers total 9.

Now we are going to look at the 3 times table, so starting with 3 we are going to continuously add 3 we get a male pattern of
3 6 9 12 15 18 21 24 27
If we implode the sequence we get
3 6 9 3 6 9 3 6 9
This pattern repeats every three numbers
3 6 9
Where as the two previous patterns repeated after nine numbers, for now just note that its pattern is 3 times shorter.

The four times table has a female pattern that looks like this
4 8 3 7 2 6 1 5 9
This can also be drawn in a circle

Once again we notice that if we draw an imaginary line vertically down the centre all opposite pairs of numbers total 9.

Lets look at the female pattern of the 5 times table
5 1 6 2 7 3 8 4 9
This pattern is just a repeat of the previous 4 times table but in reverse , lets draw it in a circle and it should be clear,

If I go clockwise I have my 5 times table but if I go anticlockwise I get my 4 times table. both the 4 and 5 times table are just mirrors of each other.

The female pattern for the 6 times table is
6 3 9
just like the 3 times table it repeats every 3 numbers in fact its just the revers of the 3 times table. Again we will be coming back to take a closer look at the 3 and 6 but for now just take note that its pattern is 3 times shorter

Just like the 5 was a mirror of 4 and 6 was a mirror of 3, the 7 times table is just a mirror of the 2 times table, its female pattern is
7 5 3 1 8 6 4 2 9
Its easier to see in a circle

You are probably seeing a pattern and if you are expecting the 8 times table to be the reverse a of a previous pattern you are right.
The 8 times table is the mirror  of the 1 times table, its female pattern looks like this.
8 7 6 5 4 3 2 1 9
and in a circle

The 9 times table has no pattern it just is 9 forever, it can be seen as the mirror of zero but because we do not use a zero in vortex maths the 9 stands alone if we were to count in 9s we would just get
9 9 9 9 9 9 9 9 9
it is a constant forever.

Lets summarise what we’ve got
We have 6 sequences that use all nine female digits and 2 sequence that are three times shorter for a total of 8 sequences, and 9 that is just a constant. We can break the 8 sequences into two halves with one have being an exact mirror of the other.

That is all there is, if we looked at the female pattern for the 10 times table we would find it is exactly the same as the 1 times table. In fact there is not a times table in existence that does not fall into one of these patterns, lets check this.
65>)2 so if we keep adding 65 we should get the 2 times table pattern 65×2=130 and 130>)4 the next would be a 6 and then an 8 and so on. All multiplication fall into one of these patterns all the way to infinity.

# 8. Polar Pairs

Throughout vortex math we come across polar pairs. these are pairs of numbers that total nine.
If we take 9 as being the whole, there are four ways to split 9 into two whole numbers.
1 and 8
2 and 7
3 and 6
4 and 5
These are the polar pairs and keeping track of them as we go will reveal some fascinating symmetries.

lets have a look at the male 9 times table
9
18
27
36
45
54
63
72
81
90
Notice the 9 times table produces nothing but polar pairs and if we continue
99
108 we can take the first two digits and implode them 10>)1 that gives us 1 and 8 a polar pair
117 we can take the first two digits and implode them 11>)2 that gives us 2 and 7 a polar pair
126  we can take the first two digits and implode them 12>)3 that gives us 3 and 6 a polar pair
We can do this to infinity always leaving the last digit and imploding the remaining digits will always produce a polar pair.

When we made circles out of our multiplication tables we found that polar pair were always opposite.

Lets compare the 1 and 8 times table
1 2 3 4 5 6 7 8 9
8 7 6 5 4 3 2 1 9
Every step creates polar pairs, its the same if we compare the 2 and 7 times table.
2 4 6 8 1 3 5 7 9
7 5 3 1 8 6 4 2 9
Polar pairs again and if we compare 4 and 5 times table we get the same again.
4 8 3 7 2 6 1 5 9
5 1 6 2 7 3 8 4 9

Now lets look at the 3 and 6 times table.
3 6 9
6 3 9
The 3 and 6 are unique for many reasons, one of the reasons is that they are the only Polar Pair to be in the same family member group.

What’s a family member? lets take a look.

# 9. Family Members

There are three family member groups each containing three digits.

We have already seen one of these family groups and that was the 3 times table.
3 6 9
This sequence is a key to unlocking the other two groups, when we were looking at the the 3 times table we started the sequence with a 3, well we can count in 3s but instead of starting with a 3 lets start with a 1 then we will constantly keep adding 3 to the total

Three Times Table (adding 3) Starting With a 1
1 4 7
If we continue by 7 + 3 = 10 would implode back to a 1 so it would repeat, so now we have our second family member group and that leaves three digits left to make up the final group. Lets start a new sequence using the same method of adding 3 but this time we start with a 2.

Three Times Table (adding 3) Starting With a 2
2 5 8
That completes the set and all female digits have been used.

Just for completeness lets have a quick look at the 6 times table, we will create three sequences of numbers starting with a 1 a 2 and a 3 each time adding 6 and we get.
1 7 4
2 8 5
3 9 6

If we implode the digits in a family group get

1 4 7 >)3  so I will be referring to this group as the 3 family group (even though the 3 is not in this group)

2 5 8 >)6  so I will be referring to this group as the 6 family group (even though the 6 is not in this group)

3 6 9 >)9  so I will be referring to this group as the 9 family group.

Notice that the nine family group is the only group that contains a polar pair, the 3 and the 6, also this group contains the almighty 9 as such this group is complete and is different to the other two groups. The 3 and 6 family groups are two halves that make up one whole. A closer look at how these family groups interact should explain what I mean.

1 4 7 and 1 4 7
Notice that when any instance of the 3 family group interacts with any other instance of the same group the result is one of the 6 family member group
3+3>)6

2 5 8 and 2 5 8
Notice that when any instance of the 6 family group interacts with any other instance of the same group the result is one of the 3 family member group.
6+6>)3

1 4 7 and 2 5 8
When member of the 3 and 6 family groups interact the 9 family group is created.
3+6>)9

1 4 7 and 3 6 9 | 2 5 8 and 3 6 9 | 3 6 9 and 3 6 9
Now if any family member group interacts with the 9 group the same group is created.
This is just like adding any single digit to 9 the same digit is created.
3+9>)3
6+9>)6
9+9>)9

So as you see the members of each group act just like their female number.

We will be looking at family groups in all the following sequences but before we move on,  lets have a look at the family members in our times tables

Note that in all times tables and in all family groups each member in each group are perfectly equal distance apart.

Here is something I find interesting and a easy way to remember what digits belong in which family group.

There is a good reason why if one family member group is the right way up then the other must be upside down and that we will see in the next part.