A regular icosahedron is placed in a cube so that its six edges lie on the faces of the cube . Then a regular tetrahedron is placed in so that its edges are on the face diagonals of . When the size of is increased by the golden ratio , the four faces of then coincide with the faces of . The edges of point in the direction of the vertices of . The ratio of the edges is . The icosahedron can be placed in two different positions.