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1. A straight edge

2. The ability to judge any distance to within 20%, or any angle to within 20 degrees.

What can you construct with these tools, with an unlimited? In particular, is there a method for constructing an approximate circle that tends towards perfect accuracy as the number of steps you use tends towards infinity?

For an example of what you can do, here is how to construct an approximate parallel line to a given line m through a point P:

First prepare a length x = 100 light years.

1. draw a line n through P that seems about parallel to your angle judgment. This line is either parallel to m (statistically impossible) or it intersects m at point Q.

2. Does PQ appear, to your 20% accurate length judgment, to be at least 21% greater than x? If not, return to step 1. Otherwise, let x = PQ, and either return to step 1 or decide you are finished and n is your nearly parallel line.

Given enough time, this method will produce an asymptotically accurate parallel line. It will never be exactly accurate. The fact that it may take a huge number of steps to become accurate is not important. Also, please disregard the fact that by some statistical fluke it might take a vast number of steps yet still not be any more accurate than it was to begin with--under reasonable expectations of chance, its accuracy will increase.

So, can you find a similar method for creating a circle? Feel free to share any interesting sub-algorithms you come across. One that I am working on is how to, given an angle, reflect it over one of its rays.