sdi

You guys tired of geometry yet? No? Great! Neither am I! I found another neat thing I'd like to show you.

Back in 2002, Kurt Hofstetter showed how fundamental the golden ratio is by demonstrating a very simple construction using five circles:

I discovered today that he considerably undersells that construction. Not only does it contain the golden ratio, but it also contains the other two fundamental ratios of sacred geometry, √2 and √3. Consider this pentacle-like diagram:

In this diagram, notice the the dark circle in the bottom-center. If it has a radius of length 1, then the blue lines are of length √3, the green lines are of length √2, and the red line is of length ϕ. Thus it is almost trivial to construct any of the classical regular polygons—the triangle from √3, the square from √2, the pentagon from ϕ, or their multiples—within the dark circle. Indeed, the triangle is already present; the square and hexagon take a single additional circle each, while the pentagon requires two additional circles: