Augmenting the dodecadodecahedron
The dodecadodecahedron, vertex figure (5,5/2,5,5/2)
Left: Each pentagonal face of the dodecadodecahedron can be augmented with a pentagonal pyramid. The 5's in the vertex figure are each replaced by (3,3) The original vertex figures become (3,3,5/2,3,3,5/2) with new vertices of (3,3,3,3,3). All vertices lie on the convex hull.
Right: The above figure is shown with the pentagonal pyramids shown in framework style. A framework of this model is linked here.