Making Art with the Golden Ratio
You can do some pretty cool stuff with the golden ratio. The image above is made from taking each quarter-circle in the golden spiral and expanding it into a full circle. In the second image, the spiral and the golden rectangles are overlaid on the the first image, showing how it works.
The length of the radius of the nth circle is 1/phi^(n-1).
The image above is made by connecting the the center of one circle to the center of the circle in the next iteration (or the corner of one square to the corner of the next), using it for the side of a pentagon which is then turned into a star. This shows how the golden ratio is present in pentagrams (the length of the side of one of the triangles to the length of the side of pentagon); notice in the second image below how the length of the side of a pentagon plus the length of the side of a triangle is the same as the radius of the circle (or the length of the side of the square) and how the length of the side one of the triangles is the same as the radius of the circle (or the length of the square side) in the next iteration. And the ratio of each radius (or side) to the next is, of course, phi.
Anyway, there's some really awesome stuff you can do with the Golden Ratio and these are just a couple of them.