How to determinate the convergence the start and the finish points?

Question

I have found the Gridpatterns page.

One can apply the next algoritm and obtaine the grid pattern "1-2-3".

1. On square grid paper start in the middle.
2. Draw a line 1-unit long.
3. Turn a right angle clockwise.
4. Draw a line 2-unit long.
5. Turn a right angle clockwise.
6. Draw a line 3-unit long.
7. Repeat steps 1-6 four times.

Result is below on the left figure. As you can see we returned to the start point (red).

Edit. After the lesnik's answer. Your positions after step 6 will be four points forming a square (right figure).

Question. Is there a rule by which it can be determined that the pattern "a-b-c" will return (convergence) to the starting point? Here a, b, c are integer.

Answer 1

Looks like for any $a$, $b$ and $c$ following your algorithm you will return to the starting point.

After steps 1-6 you shift from original position by some vector $\vec{x_1}$ and turn by 90 degrees. Now you repeat steps 1-6, and shift from your current position by some vector $\vec{x_2}$. Which is exactly like $\vec{x_1}$, but turned by 90 degrees. And so on. Sum of vectors $\vec{x_1}$, $\vec{x_2}$, $\vec{x_3}$ and $\vec{x_4}$ is zero because all of them have the same length, and each one is turned by 90 degrees.

Your positions after step 6 will be four points forming a square.