# Breaking down eucliclean algorithm as a series of movements of the numbers across the number line

I was going over a process that tries to show the euclidean algorithm distilling it to a series of movements across the number line. The basic movements are measured by the $$2$$ numbers that we are interested in each time and other movements are built on top of that.
Example for numbers $$133$$ and $$85$$ (I am sorry for the inaccuracy of the diagram, didn’t know of any good tool for such a case)

Now if you notice this process essentially at this point moves around at intervals of $$11$$ steps and if we continue back and forth with moves like that it will eventually reach $$1$$.

Also $$1$$ is the greatest common divisor of both numbers.

If we take two different examples:
E.g. $$8$$ and $$4$$ we would have:

This essentially ends up with an infinite loop as we can see from the diagram

Also for $$91$$ and $$49$$ we would have:

Now from there on the algorithm does not enter a loop but can only move in multiples of $$7$$ which is also the GCD of $$91$$ and $$49$$

So my questions are:

• how do we know when the process stops? In the first case it stops to $$1$$ in the latter goes in an infinite loop but the GCD(8,4) = 4 and in the last it does not go in an infinite loop but the last decrement is $$7$$.
• what would be an intuitive explanation of the process?