7. Multiplications

One Times Table (adding 1)
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.

01

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.

Two Times Table (adding 2)
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.

02
Again we notice that if we draw an imaginary line vertically down the centre all opposite pairs of numbers total 9.

Three Times Table (adding 3)
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.

Four Times Table (adding 4)
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

03

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

Five Times Table (adding 5)
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,

04

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.

Six Times Table (adding 6)
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

Seven Times Table (adding 7)
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

05

Eight Times Table (adding 8)
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

06

Nine Times Table (adding 9)
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.

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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.

07

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)
08

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)
09

3 6 9 >)9  so I will be referring to this group as the 9 family group.
10

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

11

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

12

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

13

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

17

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

20

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.

18                 19

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.

10. Ring Sets

When I first started looking at vortex math one of the first things I did was to take Marko Rodin’s symbol and add the numbers that are next to each other and write them in a new ring.

First lets take a brief look at Marko’s symbol

00rodinssymbol

There are a few things going on in the middle but for now we are only looking at the numbers around the edge, these numbers are our 1 times table and by putting them in a circle we capture the finite female pattern of our infinite 1 times table.

I wanted to know how these numbers related to their neighbours. So lets start with our one times table.

01

Now lets add another ring.

set02

Now lets add the numbers that are next to each other and place the result in between them on the next ring out.

set03

Now this I found interesting, if we start at the 9 at the bottom and go clockwise we have the female pattern of our 2 times table created from the 1 times table in the first ring.

Lets continue to the next ring using the same process.

set04

Again starting with the 9 and go clockwise the female pattern of the 4 times table is created all by its self.
I was getting really excited by this and was expecting the 5 times table to be in the next ring, lets see

set05

Well I was wrong and it was the female pattern of the 8 times table, I was still very excited.
lets continue.

set06

This produced the female 7 times table and that just left the female 5 times table and if we continue to the next ring it is created.

set07

Now we could just keep adding rings but what we find is that the next ring out will just repeat the first ring. so in 6 rings we have a complete ring set. Each you can spin around each ring forever adding 1 on the first ring, 2 on the second ring, 4 on the third ring, 8 on the fourth ring, 7 on the fifth ring  or 5 on the sixth ring.

Lets have a look at this ring set in a little more depth.
The first thing I noticed is all the 3s all the 6s and all the 9s all lined up.

set08

Next lets look at how the other family member groups arrange themselves.

set09

We have three rings that start from the 9 at the top and three rings that start from the 9 at the bottom.
We find that the rings that start from the top are the 1,4 and 7 times table and the 2, 5  and 8 are upside down

set10

This is the ring set for all our multiplications we have our 1, 2, 4, 8, 7 and 5 times table contained in each ring and this pattern is a very key pattern that we will be looking at in depth later, but for now notice that all the times table for the 9 family group hidden.

To find the 3 times table we are only going to use the rings of the 3 family group, so that would be the 1 4 and 7 times table or the first, third and fifth rings.

set11

We are starting from 9 and working clockwise just like we did in each ring and we can use the same method to find the 6 times table

set12

It is interesting to overlay the two patterns and we have something very similar to some sacred geometry.

set13

Remember this pattern would keep going if we were to keep adding rings.
That just leaves the 9 times table and that is found vertically down the centre.

So there we have our first ring set and it is complete we cant add anything to it without repeating what is already there. I was very excited.

set08

We shall be making more ring sets as we look at more number sequences.