H3 is a group that consists of 120 elements. The action of these elements can be represented as reflexions and rotations in the three dimensional space. Therefore, this group can transform one single point of space into 120 different points. The result is a geometrical object that consists of 120 vertices.

However, if the point that you start with exactly lies on such  plane of reflexion, it won't be transformed into another point. It turns out that in such a case, exactly half of the 120 elements remain passive and one ends up with a geometrical object that consists of 60 vertices. Similarly, geometrical objects with 30, 20 and 12 vertices can be obtained. Starting from the most general representation with 120 vertices, all other representations are obtained by identifying vertices. This is illustrated here.

There are transformations that allow to generate new polyhedra from these basic polyhedra. The first of such transformations is the duality operation. The center of each face consitutes a vertex to the new polyhedra. The following table shows the duals. The number of faces and vertices interchange, the number of edges stays the same.

Another interesting transformation consists in merging the vertices of a polyhedron with the vertices of its dual and construct from this set of points a new polyhedron. The mergers are visualized in the following animations.

When this merger is applied to the icosahedron, one ends up with dual of the icosidodecahedron, and similarly, the merger applied to the icosidodecahedron yields the dual of the edge-truncated icosahedron.