Construction of a Regular Pentagon, Using Only a Compass

[sdi]

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.

Given. Points A and B.

I. Construct a circle from A to B.

II. Construct a circle from B to A.

III. Construct a circle between the two intersection points of circles I and II.

IV. Construct a circle from A to the intersection point of circles II and III not on circle I.

V. Construct a circle from B to the intersection point of circles I and III not on circle II.

VI. Construct a circle from one of the intersection points of circles IV and V to the center of circle III.

VII. Construct a circle from the other intersection point of circles IV and V to the center of circle III.

Result. The five points shown form a regular pentagon.

Comments

[violetcabra]

That is so lovely ---- thank you for sharing this! The method I've used is somewhat different, being the one outlined in Miranda Lundy's little book on sacred geometry. I really love how you made these illustrations look with the SVG :-)

[sdi]

You're welcome! The SVG was a fun little project: computing everything arithmetically really makes the proof of the construction clear. (For a little sightseeing, the ratio of the radii of circle VI to circle III, and of the radii of circle III to circle VII, is ϕ.)

[violetcabra]

That makes a lot of sense given both the ratios of the segments of a pentagram and also how lovely your svg turned out.

Re: beautiful

[sdi]

There's no magic here and I didn't use any tools more sophisticated than a calculator and a text editor! (Indeed, as I understand Plato and Euclid, that's the whole point—the sage transcends the need or desire for complicated tools. As Laozi says, "If a country is governed wisely, its inhabitants will be content. They enjoy the labor of their hands and don't waste time inventing labor-saving machines.")