I once stood on a Cartesian plane at (0, 0) facing north (along the positive y-axis). I pulled out a rug in the shape of a regular triangle (which I call a 3-gon) and set it on the plane with one vertex at my feet at (0, 0) and with the center in the direction I was facing along the y-axis. I then started walking forward on the rug until I got to the center of the 3-gon, at which point I stopped and turned clockwise until I was facing a vertex, and I walked to that vertex of the 3-gon. Then I pulled out a rug in the shape of a square (which I call a 4-gon), put it with one vertex at my feet and with the center straight ahead of my current view (after that previous clockwise turn). I walked to the center of the 4-gon and then turned counterclockwise (instead of clockwise like on the odd-numbered N-gon rug), and started walking again as soon as I was facing a new vertex of the 4-gon. I kept repeating that for every odd numbered N-gon and every multiple of 4-gon.

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I once stood on a Cartesian plane at (0, 0) facing north (along the positive y-axis). I pulled out a rug in the shape of a regular triangle (which I call a 3-gon) and set it on the plane with one vertex at my feet at (0, 0) and with the center in the direction I was facing along the y-axis. I then started walking forward on the rug until I got to the center of the 3-gon, at which point I stopped and turned clockwise until I was facing a vertex, and I walked to that vertex of the 3-gon. Then I pulled out a rug in the shape of a square (which I call a 4-gon), put it with one vertex at my feet and with the center straight ahead of my current view (after that previous clockwise turn). I walked to the center of the 4-gon and then turned counterclockwise (instead of clockwise like on the odd-numbered N-gon rug), and started walking again as soon as I was facing a new vertex of the 4-gon. I kept repeating that for every odd numbered N-gon and every multiple of 4-gon.

Which way was I facing after I did that forever?

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