During my initial research I was focused on trying to find the point on earth where the sun was directly overhead; since this would be the point you were walking toward always. If I remember correctly, the sun is directly overhead somewhere on earth between the tropic of cancer and the tropic of capricorn. Therefore, it seemed like given enough time to walk, the person would always end up somewhere between these lines of latitude in some sort of cyclical pattern.

It would be great to see this as a mapping plugin which would show the crazy path the walking man would take.