Augmenting the small retrosnub icosidodecahedron
The small retrosnub icosidodecahedron, vertex figure (5/3,3,3,3,3,3)/2
Left (77a): Each retrograde pentagrammic face of the small retrosnub icosidodecahedron can be excavated with a pentagrammic pyramid completing a cycle around the axis. This has the effect of uncrossing the original vertex figure. The 5/3 in the vertex figure is replaced by (3,3) The original vertex figure becomes (3,3,3,3,3,3,3)/3 or (37)/3 with new vertices of (3,3,3,3,3)/2. The faces of the pentagrammic pyramids are shown in yellow. The original triangular faces are blue. All vertices lie on the convex hull.
Centre (77b): The middle image above shows a rare example where all of the prograde polygons at each vertex of a polyhedron with crossed vertices have been excavated, reversing the polarity of the vertex and turning the originally retrograde polygon prograde. It is easiest to start by considering the vertex is traversed in the opposite direction so that it becomes (5/2,3/2,3/2,3/2,3/2,3/2)/2. Excavate each triangular face with a triangular pyramid or tetrahedron completing a cycle around the axis. This has the effect of uncrossing the reversed vertex figure. Each 3/2 in the vertex figure is replaced by (3,3). The original vertex figure becomes (5/2,3,3,3,3,3,3,3,3,3,3)/3 with new vertices of the form (3,3,3). The tetrahedral faces from the icosahedral triangles are shown in blue, those from the snub triangles in yellow. Whilst all vertices are on the exterior of the polyhedron, the original pentagrammic faces are not visible. The polyhedron is also partially degenerate as the apices of the icosahedral tetrahedra coincide in pairs.
Right (77c): Each retrograde pentagrammic face of the small retrosnub icosidodecahedron can be excavated with a great retrosnub icosidodecahedron - vertex figure (5/2,3,3,3,3)/2 - completing a cycle around the axis. This has the effect of uncrossing the original vertex figure. The 5/3 in the vertex figure is replaced by (3,3,3,3) The original vertex figure becomes (3,3,3,3,3,3,3,3,3)/4 or (39)/4 with new vertices of (5/2,3,3,3,3)/2. The triangular faces of the original small retrosnub icosidodecahedron are shown in yellow. The triangular and pentagrammic faces of the great retrosnub icosidodecahedron are blue and green respectively. All vertices lie on the convex hull.
Left (77d): An associated augmentation of the small retrosnub icosidodecahedron can be obtained by excavating the pentagrams as in the first example above (77a) and also augmenting one set of icosahedral triangular faces (blue in the original image of the small retrosnub icosidodecahedron) with triangular pyramids or tetrahedra. The 2nd 3 in the vertex figure becomes (3,3), along with the replacement of the 5/3 as above, the original vertex figure becomes (3,3,3,3,3,3,3,3)/3 or (38)/3 with new vertices of (3,3,3,3,3)/2 from the pentagrammic pyramids and (3,3,3) from the tetrahedra. Colours are as per the previous example with the tetrahedra in blue. All vertices are on the exterior of the polyhedron.
Right (77e): Another associated augmentation of the small retrosnub icosidodecahedron can be obtained by excavating the pentagrams as in the first example above (77a) and also augmenting both sets of icosahedral triangular faces (blue in the original image of the small retrosnub icosidodecahedron) with triangular pyramids or tetrahedra. The 2nd and 3rd 3 in the vertex figure becomes (3,3), along with the replacement of the 5/3 as above, the original vertex figure becomes (3,3,3,3,3,3,3,3,3)/3 or (39)/3 with new vertices of (3,3,3,3,3)/2 from the pentagrammic pyramids and (3,3,3) from the tetrahedra. The resulting polyhedron is partially degenerate as the tetrahedral vertices are coincident in pairs. Colours are as per the previous example with the tetrahedra in blue. All vertices are on the exterior of the polyhedron.