Augmenting the great dodekicosidodecahedron
The great dodekicosidodecahedron, vertex figure (10/3,5/2,10/3,3)
Left: Each triangular face of the great dodekicosidodecahedron can be augmented with a triangular pyramid or tetrahedron. Each 3 in the vertex figure is replaced by (3,3) The original vertex figure becomes (10/3,5/2,10/3,3,3) 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 dodekicosidodecahedron can be obtained by excavating each decagrammic face with a decagrammic antiprism completing a cycle around the axis, each pentagrammic face with a pentagrammic prism which again completes a cycle around the axis, and each triangular face with a triangular prism which again completes a cycle around the axis. Each 10/3 in the vertex figure is replaced by (3,3,3), the 5/2 and the 3 in the vertex figure are each replaced by (4,4) The original vertex figure becomes (3,3,3,4,4,3,3,3,4,4)/3 with new vertices of (10/3,3,3,3), (5/2,4,4) and (3,4,4) respectively. The square faces of the pentagrammic prisms are shown in yellow and the square faces of the triangular prisms in orange. The triangular faces of the decagrammic antiprisms are shown in blue and the triangular prismatic caps in red. All vertices are on the exterior of the polyhedron.
Right: A second order augmentation can be formed by excavating the pentagrammic prismatic caps from the above polyhedron with 5/3-cupolas. The cupolas themselves are not locally convex but the act of joining them to the pentagrammic caps removes the retrograde polygon. The original vertices of (3,3,3,4,4,3,3,3,4,4)/3 are unaffected, as are the (10/3,3,3,3) and (3,4,4) vertices. The 5/2 in the (5/2,4,4) vertices is replaced by (4,3,4)/2 to give (4,3,4,4,4)/2 vertices. New vertices are added of the form (10/3,3,4). Colours are as per the previous polyhedron with the 5/3-cupolas shown in purple/pink. All vertices are on the exterior of the polyhedron.