Augmenting the icositruncated dodecadodecahedron
The icositruncated dodecadodecahedron, vertex figure (10,6,10/3)
Left: Each decagonal face of the icositruncated dodecadodecahedron can be augmented with a pentagonal cupola such that the triangles in the cupola are edge connected to the decagrams. The 10 in the vertex figure is replaced by (3,4) The original vertex figures become (3,4,6,10/3) with new vertices of (5,3,4,3). All vertices are on the exterior of the polyhedron.
Right: The above figure is shown with the pentagonal cupolas shown in framework style, a model with only one decagon augmented is linked here. A framework of the fully augmented model is linked here.