Augmenting the great rhombicosidodecahedron
The great rhombicosidodecahedron, vertex figure (4,5/3,4,3)
Left: Each retrograde pentagrammic face of the great rhombicosidodecahedron can be excavated with a pentagrammic prism 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 (4,4) The original vertex figure becomes (4,4,4,4,3)/2 with new vertices of (5/2,4,4). The square faces of the pentagrammic prisms are shown in yellow. The original square faces are orange. All vertices lie on the convex hull.
Centre: A second order augmentation can be generated from the above. The prismatic caps in the above polyhedron can be further augmented with pentagrammic prisms. This leaves the original vertex figures unchanged at (4,4,4,4,3)/2 but the new vertices of (5/2,4,4) become (3,3,4,4) vertices and further vertices are added of the form (3,3,3,3,3)/2. The faces of the new pyramids are shown in purple. All vertices are on the exterior of the figure.
Right: Another augmentation of the great rhombicosidodecahedron can be obtained by excavating each pentagrammic face with a pentagrammic antiprism 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) The original vertex figure becomes (4,3,3,3,4,3)/2 with new vertices of (5/2,3,3,3). The triangular faces of the pentagrammic antiprisms are shown in yellow. The original triangular faces are blue. All vertices are on the exterior of the polyhedron.
Left: The leftmost 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 (4/3,5/2,4/3,3/2). Excavate each square face with a square pyramid completing a cycle around the axis and each triangular face with a triangular antiprism or octahedron again completing a cycle around the axis. This has the effect of uncrossing the reversed vertex figure. Each 4/3 in the vertex figure is replaced by (3,3) and the 3/2 by (3,3,3) The original vertex figure becomes (3,3,5/2,3,3,3,3,3)/2 with two sets of new vertices both of the form (3,3,3,3). The triangular faces of the square pyramids are shown in yellow. The side faces of the antiprisms are blue with the prismatic caps red. All vertices are on the exterior of the polyhedron.
Centre: The centre image above shows a rare example which includes both the excavation of a retrograde polygon and the augmentation of prograde polygons at each vertex of a polyhedron. Each retrograde pentagrammic face of the great rhombicosidodecahedron can be excavated with a pentagrammic prism completing a cycle around the axis. This has the effect of uncrossing the original vertex figure. The square faces of the great rhombicosidodecahedron are then augmented with square pyramids. The 5/3 in the vertex figure is replaced by (4,4), the 4's in the original vertex figure are each replaced by (3,3). The original vertex figure becomes (3,3,4,4,3,3,3)/2 with new vertices of (5/2,4,4) and (3,3,3,3). The square faces of the pentagrammic prisms are shown in yellow. The original square faces are orange. The triangular faces of the pyramids are in red, the oringinal triangular faces are blue. All vertices lie on the convex hull.
Right: A second order augmentation can be generated from the above. The prismatic caps in the above polyhedron can be further augmented with pentagrammic prisms. This leaves the original vertex figures unchanged at (3,3,4,4,3,3,3)/2. Also unchanged are the (3,3,3,3) vertices but the new vertices of (5/2,4,4) become (3,3,4,4) vertices and further vertices are added of the form (3,3,3,3,3)/2. The faces of the new pyramids are shown in purple. All vertices are on the exterior of the figure.