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The Octahedron from Curious and Interesting Geometry by David Wells

If the edges of a regular octahedron are divided in the golden section so that the points of division for any face of the octahedron form an equilateral triangle, then the 12 points of division are the vertices of a regular icosahedron.

There are two ways in which the edges can be divided internally in the golden ratio, and two more ways in which they can be divided externally, producing a total of four icosahedrons. For external division, the points of division of the edges of one face are next-but-one vertices of the icosahedron.