Hexagons and the Golden Ratio

This Demonstrations has to do with Odom's recognition of the relationship between the golden ratio and the equilateral triangle. Construct three triangles by extending the edges of an equilateral triangle.

When the extension is inversely proportional to the golden ratio, two vertices of each triangle are on a circle circumscribing a triangle twice as large as the original triangle.

When the extension is proportional to the golden ratio, the outside vertices of the three triangles determine a hexagon having two different edge lengths whose ratio is equal to the golden ratio. The vertices of the hexagon determine two triangles that can be found in the compound of two icosahedra or the compound of five octahedra.