Augmenting the great stellatruncated dodecahedron
The great stellatruncated dodecahedron, vertex figure (10/3,10/3,3)
Left: Each triangular face of the great stellatruncated dodecahedron can be augmented with a triangular pyramid or tetrahedron. The 3 in the vertex figure is replaced by (3,3) The original vertex figure becomes (10/3,10/3,3,3) with new vertices of (3,3,3). All vertices are on the exterior of the polyhedron.
Right: The above figure is shown with the tetrahedra shown in framework style. A framework of this model is linked here.
Left: Another augmentation of the great stellatruncated dodecahedron can be obtained by excavating each decagrammic face with a decagrammic pyramid completing a cycle around the axis, and each triangular face with a triangular antiprism or octahedron again completing a cycle around the axis. The 10/3's in the vertex figure are each replaced by (3,3) and the 3 with (3,3,3). The original vertex figure becomes (3,3,3,3,3,3,3)/2 or (37)/2 with new vertices of (3,3,3,3,3,3,3,3,3,3)/3 or (310)/3 and (3,3,3,3). The faces of the decagrammic pyramids are shown in red. The triangular side faces of the antiprisms are blue and the prismatic caps yellow. All vertices are on the exterior of the polyhedron.
Right: An unusual augmentation of the great stellatruncated dodecahedron can be obtained as follows. The 5/3-cupola is not itself a locally convex polyhedron. However, local convexity can be obtained if the pentagram is excavated with a pentagrammic prism. For brevity, I refer to the resulting polyhedron as a Q5/3(P5/2). This contains vertices of the form (10/3,4,3), (4,4,4,3,4)/2 (from the (5/3,4,3,4) vertices of the 5/3-cupola) and (5/2,4,4) (from the prismatic caps). We can then excavate each decagrammic face of the great stellatruncated dodecahedron with a Q5/3(P(5/2) completing a cycle around the axis, and each triangular face with a triangular antiprism or octahedron again completing a cycle around the axis. The 10/3's in the vertex figure are each replaced by (4,3) and the 3 with (3,3,3). The original vertex figure becomes (4,3,4,3,3,3,3)/2 with new vertices of (4,4,4,3,4)/2, (5/2,4,4) and (3,3,3,3). The triangular side faces of the anti-prisms are red. The square and triangular faces of the cupola are orange and blue respectively, the pentagrammic and square faces of the prisms are green and yellow respectively. All vertices are on the exterior of the polyhedron.