While I was doodling, I discovered something interesting regarding drawing a walk in the coordinate plane.
First, I selected 4 random lattice points as the vertices of a quadrilateral. Then, I started on a certain vertex. Next, I drew a line segment to the next vertex (going clockwise), then continued the line segment to have the extension be equal in length to the original line segment. Next, I did this again with the next vertex, drawing from where I was to the vertex, then drawing another line segment equal in length to my previous segment.
I noticed that whenever my original quadrilateral was a parallelogram, my path was bounded. However, if it wasn't a parallelogram, then my path became unbounded, growing indefinitely.
I tried the experiment again, except instead I chose a random point on the plane to start. Same results.
What do you guys think? Can you prove this conjecture? I managed to link this with another similar problem that is easy to prove. Can you find that problem?