Augmenting the small stellated dodecahedron
The small stellated dodecahedron, vertex figure (5/2,5/2,5/2,5/2,5/2)
Left: Each pentagrammic face of the small stellated dodecahedron can be excavated with a pentagrammic antiprism completing a cycle around the axis. The 5/2's in the vertex figure are each replaced by (3,3,3) The original vertex figures become (3,3,3,3,3,3,3,3,3,3,3,3,3,3,3)/4 or (315)/4 with new vertices of (5/2,3,3,3). All vertices are on the exterior of the figure.
Centre: A second order augmentation can be formed from the above polyhedron by excavating the pentagrammic antiprismatic caps 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 (315)/4 are unaffected. The 5/2 in the (5/2,3,3,3) vertices is replaced by (4,3,4)/2 to give (4,3,4,3,3,3)/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 red/orange/yellow. All vertices are on the exterior of the polyhedron.
Right: A further augmentation of the small stellated dodecahedron can be obtained by excavating each pentagrammic face with a great retrosnub icosidodecahedron - vertex (5/2,3,3,3,3)/2 - completing a cycle around the axis. The 5/2's in the vertex figure are each replaced by (3,3,3,3) The original vertex figures become (3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3)/9 or (320)/9 with new vertices of (5/2,3,3,3,3)/2. All vertices are on the exterior of the figure.
Left: Another augmentation of the small stellated dodecahedron can be obtained by excavating each pentagrammic face with a great icosidodecahedron - vertex (5/2,3,5/2,3) - completing a cycle around the axis. The 5/2's in the vertex figure are each replaced by (3,5/2,3) The original vertex figures become (3,5/2,3,3,5/2,3,3,5/2,3,3,5/2,3,3,5/2,3)/4 or ((3,5/2,3)5)/4 with new vertices of (5/2,3,5/2,3). All vertices are on the exterior of the figure.
Right: In the same way that an icosidodecahedron is composed of two pentagonal rotundas, a great icosidodecahedron can be thought of as being composed of two 'great' pentagonal rotundas. A variation on the theme of the left hand polyhedron is to excavate each pentagrammic face with such a 'great pentagonal rotunda'. - completing a cycle around the axis. The 5/2's in the vertex figure are each replaced by (3,5/2,3) The original vertex figures become (3,5/2,3,3,5/2,3,3,5/2,3,3,5/2,3,3,5/2,3)/4 or ((3,5/2,3)5)/4 with new vertices of (5/2,3,5/2,3) and (5/2,10/3,3). All vertices are on the exterior of the figure.