Augmenting the snub dodecadodecahedron
The snub dodecadodecahedron, vertex figure (5/2,3,5,3,3)
Left: Each pentagonal face of the snub dodecadodecahedron can be augmented with a pentagonal pyramid. The 5 in the vertex figure is replaced by (3,3) The original vertex figures become (5/2,3,3,3,3,3) with new vertices of (3,3,3,3,3). The triangular facees of the pyramids are shown in yellow, the original triangular faces are in blue. 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.