Augmenting the great vertisnub icosidodecahedron
The great vertisnub icosidodecahedron, vertex figure (5/3,3,3,3,3)
Left: Each retrograde pentagrammic face of the great vertisnub icosidodecahedron can be excavated 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 (3,3,3,3,3,3,3)/2 or (37)/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 lie on the convex hull.
Right: Another augmentation of the great vertisnub icosidodecahedron can be obtained by excavating each pentagrammic face with a great icosidodecahedron, vertex figure (5/2,3,5/2,3) 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 (3,5/2,3) The original vertex figure becomes (4,3,5/2,3,4,3)/2 with new vertices of (5/2,3,5/2,3). The triangular faces of the great icosidodecahedra are shown in yellow. The original triangular faces are blue. All vertices are on the exterior of the polyhedron. Unfortunately I have not yet been able to generate a VRML model of this polyhedron, but clicking on the image above will show a larger image.
Left: 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 centre polyhedron is to excavate each pentagrammic face with such a 'great pentagonal rotunda'. - 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 (3,5/2,3) The original vertex figure becomes (4,3,5/2,3,4,3)/2 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.
Right: A second order augmentation can be generated from the left hand polyhedron. The decagrammic faces of the 'great' pentagonal rotundas can be further augmented with decagrammic pyramids. The original vertex figures of (4,3,5/2,3,4,3)/2 and (5/2,3,5/2,3) are unchanged. The (5/2,10/3,3) vertices become (5/2,3,3,3), and new vertices are added of the form (3,3,3,3,3,3,3,3,3,3)/3 - or (310)/3. The decagrammic pyramids are shown in red. All vertices are on the exterior of the figure.