Augmenting the great rhombicuboctahedron
The great rhombicuboctahedron, vertex figure (4,4,3/2,4)
Left: Each triangular face of the great rhombicuboctahedron can be excavated with a triangular prism completing a cycle around the axis. The 3/2 in the vertex figure is replaced by a (4,4) The original vertex figures become (4,4,4,4,4)/2 with new vertices of (4,4,3). The square faces of the prisms are shown in yellow with the prismatic caps in blue. All vertices lie on the convex hull. A framework model of this figure is linked here.
Centre: A second order augmentation can be generated from the above. The prismatic caps in the above polyhedron can be further augmented with triangular prisms or tetrahedra. This leaves the original vertex figures unchanged at (4,4,4,4,4)/2 but the new vertices of (4,4,3) become (4,4,3,3) vertices and further vertices are added of the form (3,3,3). The tetrahedral faces are shown in red. All vertices are on the exterior of the figure.
Right: Another augmentation of the great rhombicuboctahedron has each triangular face excavated with a triangular antiprism or octahedron completing a cycle around the axis. The 3/2 in the vertex figure is replaced by a (3,3,3) The original vertex figures become (4,4,3,3,3,4)/2 with new vertices of (3,3,3,3). The triangular side faces of the antiprisms are shown in yellow with the prismatic caps in blue. All vertices are on the exterior of the figure.
The above images show rare examples which include both the excavation of a retrograde polygon and the augmentation of prograde polygons at each vertex of a polyhedron
Left: Each triangular face of the great rhombicuboctahedron can be excavated with a triangular prism completing a cycle around the axis. The rhombic dodecahedral square faces (the 2nd and 3rd 4's in the original vertex figure) can be augmented with square pyramids. The 3/2 in the vertex figure is replaced by a (4,4) and the 2nd and 3rd 4's by (3,3) The original vertex figures become (4,3,3,4,4,3,3)/2 with new vertices of (4,4,3) and (3,3,3,3) respectively. The faces of the pyramids are in red. The square faces of the prisms are shown in yellow with the prismatic caps in blue. All vertices are on the exterior of the figure.
Centre: A second order augmentation can also be generated from the above. The prismatic caps in the above polyhedron can be further augmented with triangular prisms or tetrahedra. This leaves the original vertex figures unchanged at (4,3,3,4,4,3,3)/2. The (3,3,3,3) vertices are also unchanged but the new vertices of (4,4,3) become (4,4,3,3) vertices and further vertices are added of the form (3,3,3). The tetrahedral faces are shown in purple. All vertices are on the exterior of the figure.
Right: Each triangular face of the great rhombicuboctahedron can be excavated with a triangular antiprism or octahedron completing a cycle around the axis. The rhombic dodecahedral square faces (the 2nd and 3rd 4's in the original vertex figure) can be augmented with square pyramids. The 3/2 in the vertex figure is replaced by a (3,3,3) and the 2nd and 3rd 4's by (3,3) The original vertex figures become (4,3,3,3,3,3,3,3)/2 with two sets of new vertices both of the form (3,3,3,3). The faces of the pyramids are in red. The triangular side faces of the antiprisms are shown in yellow with the prismatic caps in blue. All vertices are on the exterior of the figure.