The Golden Ratio in Arrangements of Octahedron, Cube, Tetrahedron, and Sphere
The edge of an octahedron of length 1 and the edge of a smaller octahedron of length determine a straight line, one endpoint of which is at the face center of a matching cube, while its other endpoint is on a sphere circumscribing the cube.
This Demonstration is based on the relationship recognized by Odom: suppose that the line connecting the centers and of any two faces of a cube intersects the circumscribing sphere at . Then divides in the ratio , the golden ratio. Odom also noticed the relationship to the two added tetrahedra.