Truncated Icosahedron

A truncated icosahedron is the only Archimedian solid that does not contain triangles and squares. It is often used to make geodesic structures by subdividing its faces into triangles.

The icosahedral polyhedron can also be created using polyHédronisme v0.2.1 with the polyhedral recipe "dkdI" to create a Geodesic Goldberg polyhedron :

Each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they have rotational icosahedral symmetry.

There are exactly 12 regular planar pentagons. This geodesic projection of a higher order Goldberg polyhedrons does not have planar hexagonal faces, though it is not noticeable at a distance.

Any pentagonal face is m + n steps away from another pentagonal face on a Goldberg polyhedron, where m and n are steps in a straight line. Since each pentagon is 1 step in a straight line, and then another 1 step in a different direction from another pentagon, this polyhedron is also identified as GP(1,1).

The solid has 60 vertices, 90 edges, and 32 faces: 12 pentagonal faces and 20 hexagonal faces.