Construction of a Regular Pentagon, Using Only a Compass

As a computer programmer, I'm enamored with the notion of economy: that is, doing as much as one can with as little as possible. It is perhaps because of this that I have always found geometry appealing: one endeavors to begin with the simplest tools possible—a collapsing compass and a straightedge—and, using these, build oneself ever-more-powerful tools, allowing one to easily draw figures of tremendous sophistication.

So it was with some amusement that I recently learned that one does not even need a straightedge: everything that can be constructed with a compass and straightedge can be constructed with a compass alone. (We're assuming that one can draw straight lines after the construction is complete to make the construction clear, but one cannot use a straightedge to find the endpoints of any of those lines!)

I haven't studied the proof of that in any depth, yet, but since we have a few fans of geometry here, I thought I might post one of my favorite compass-only constructions, which is due to Kurt Hofstetter. ( Read more... )