The icosahedron has twenty triangular faces, twelve vertices and thirty edges, with five faces meeting at each vertex. Euler's formula gives 12 - 30 + 20 = 2. Of the five regular solids it has the most faces, the greatest volume for a given surface area, and the dihedral angle closest to flat, about 138.19 degrees, all of which amount to saying that it is the most nearly spherical of them.
It is the dual of the dodecahedron. The icosahedron has twenty faces and twelve vertices; the dodecahedron has twelve faces and twenty vertices; both have thirty edges. Join the centres of the icosahedron's twenty triangles and a dodecahedron appears; join the centres of the dodecahedron's twelve pentagons and the icosahedron returns. They share one symmetry group of 120 operations, 60 of them rotations, the largest of any Platonic solid.
The golden ratio is not attached to the icosahedron by analogy. It is in the coordinates. Take three identical rectangles whose sides are in the ratio 1 to phi, set them mutually perpendicular through a common centre, and their twelve corners are exactly the twelve vertices of a regular icosahedron. Nothing is approximated in that construction, and it can be checked with a ruler and three pieces of card.
There is a further consequence that reaches well outside geometry. The 60 rotations of the icosahedron form a group identical to the alternating group on five letters, which is simple: it has no proper structure inside it to break down into. That fact is the reason there is no general formula in radicals for solving equations of the fifth degree, the result of Abel and Galois that Felix Klein later reworked entirely in terms of this solid. The icosahedron is one of the places where geometry and algebra turn out to be the same subject, and that is a considerably stronger claim about its significance than anything the esoteric literature offers.