“Expansion” on polyhedra is the process of moving all faces outward from the center of polyhedron, and fill the gaps with new faces.
An expanded polyhedron has all the faces of the original polyhedron, all the faces of its dual, and new square faces in place of the original edges.
Truncation is the process of cutting the vertex.
In truncation, when a original face becomes a regular polygon again, it's called “uniform truncation”.
In truncation, when original edge disappears (it becomes a point), the truncation is called “complete truncation” or “rectified”.
The word “truncation” by itself without qualification usually means “uniform truncation”.
Dual of Truncated Icosahedron is “Pentakis dodecahedron”.
Icosahedron under rectified truncation results in “Icosidodecahedron”. Icosidodecahedron has 20 triangular faces, 12 pentagonal faces, 30 identical vertices.
Dual of Icosidodecahedron is “Rhombic triacontahedron”
Both Truncated icosahedron and Icosidodecahedron are “Archimedean solids”.
“Archimedean solid” is any convex polyhedron with regular polygon faces meeting in identical vertices, excluding the 5 Platonic solids.
“Identical vertices” means that for any two vertices, there is a global isometry of the solid that takes one vertex to the other.
There are 13 Archimedean solids.
If we do not require a global symmetry that maps one vertice config to another, merely that all vertices have the same configuration, then, there are 14 Archimedean solids. The new one is called
“Elongated square gyrobicupola”