# 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 