It's a hot day. You are in the middle of a flat sandy plane that stretches as far as you can see. Your hands are shackled and you are surrounded by soldiers of the mighty Pythagorean Brotherhood. Ten steps from you a man, his hands also in shackles, is standing frozen. His name is Hippasus of Metapontum. A Pythagorean soldier yells at Hippasus. "Right-angle turn and move!"
Hippasus makes a 90° left turn and hesitates. A drop of sweat slowly moves down his front head. He then makes a step followed by a 90° right turn. He again hesitates. Another step, and again a 90° left turn. The soldier raises his sword and shouts "Death to the irrational!" Hippasus' body collapses and his head rolls in your direction, coming to a halt just two steps away from your feet.
The soldier turns to you. "Now you go. Make a right angle turn followed by a nonzero number of straight steps in forward direction, and keep repeating this. Each step needs to be of exactly the same size. Each turn needs to be an exact right angle turn and must happen at a position that is a whole number of straight-line steps away from all previous turns. You need to make a total of 100 right angle turns, 50 to the left and 50 to the right. If you fail, you will be killed on the spot, if you succeed, you are a free man, and a witness of the fact that integer distances rule the world. Go!"
You think for a few seconds, and make a left turn. What's next?