An man starts on 0+0i of the complex plane. During the first walk step he moves a distance 1 to the right and lands on 1+0i. At each walk phase after this initial one the walk length decreases by a factor f<1 and he changes direction by an angle theta (measured from the horizontal and going counterclockwise). On what point does the man end up after N iterations of this? After an infinite number? What are all possible N=infinity locations?
None that I know of. The path seems to form a spiral like pattern from the origin that curves counterclockwise inwards towards a center point.
The Attempt at a Solution
I tried to calculate the the position of the man by adding to the previous steps at each step by using trigonometry and adding the components to get the position at each step but that got very messy fast since the orientation of the angle is always changing and thus I would have to alternate between sines for the x values and cosines for the y values at seemingly irregular places. It looks like a formula is possible using this method but it would be incredibly messy and I doubt this is the way to do it.