Augmenting the vertisnub dodecadodecahedron
The vertisnub dodecadodecahedron, vertex figure (5/3,3,5,3,3)
Left: Each pentagrammic face of the vertisnub dodecadodecahedron can be excavated with a small ditrigonal icosidodecahedron, vertex figure (3,5/2,3,5/2,3,5/2) 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 (3,5/2,3,5/2,3) The original vertex figure becomes (3,5/2,3,5/2,3,3,5,3,3) with new vertices of (3,5/2,3,5/2,3,5/2). Triangular faces of the small ditrigonal icosidodecahedron are shown in yellow. The original triangular faces are in blue. All vertices are on the exterior of the polyhedron.
Right: Dr Richard Klitzing has explored the facetings of the small ditrigonal icosidodecahedron. One faceting, which he terms 'sid-6-10-0-2' has the necessary 5-fold pyramidical symmetry around a pentagrammic face and is also locally convex. As the vertices of the sid-6-10-0-2 are a subset of the vertices of the small ditrigonal icosidodecahedron then excavating each pentagrammic face of the vertisnub dodecadodecahedron with the sid-6-10-0-2 will also generate a valid augmented polyhedron. The 5/3 in the vertex figure is again replaced by (3,5/2,3,5/2,3). The original vertex figure again becomes (3,5/2,3,5/2,3,3,5,3,3). New vertices are now though of the form (5/2,3,5/2,5) and (3,5/2,3,5). Triangular and pentagonal faces of the sid-6-10-0-2 are shown in yellow and red respectively. The original triangular faces are in blue. All vertices lie on the convex hull.