Augmenting the great icosidodecahedron
The great icosidodecahedron, vertex figure (5/2,3,5/2,3)
Left: Each triangular face of the great icosidodecahedron can be augmented with a triangular pyramid or tetrahedron. Each 3 in the vertex figure is replaced by (3,3) The original vertex figure becomes (5/2,3,3,5/2,3,3) with new vertices of (3,3,3). All vertices are on the exterior of the polyhedron.
Right: The above figure is shown with the tetrahedra shown in framework style. A framework of this model is linked here.
Left: Another augmentation of the great icosidodecahedron can be obtained by excavating each pentagrammic face with a pentagrammic pyramid completing a cycle around the axis and each triangular face with a triangular pyramid or tetrahedron which again completes a cycle around the axis. Each 5/2 and each 3 in the vertex figure is replaced by (3,3) 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 and (3,3,3) respectively. The faces of the pentagrammic pyramids are shown in yellow and the tetrahedral faces in blue. All vertices lie on the convex hull.
Centre: A further augmentation of the great icosidodecahedron can be obtained by excavating each pentagrammic face with a pentagrammic antiprism which completing a cycle around the axis and each triangular face with a triangular pyramid or tetrahedron which again completes a cycle around the axis. Each 5/2 in the vertex figure is replaced by a (3,3,3) and each 3 in the vertex figure is replaced by (3,3) The original vertex figure becomes (3,3,3,3,3,3,3,3,3,3)/3 or (310)/3 with new vertices of (5/2,3,3,3) and (3,3,3) respectively. The triangular faces of the pentagrammic antiprisms are shown in yellow and the tetrahedral faces in blue. All vertices are on the exterior of the polyhedron.
Right: A further augmentation of the great icosidodecahedron can be obtained as follows: Excavate one pentagrammic cap of a crossed pentagrammic antiprism with a pentagrammic pyramid. The resulting polyhedron is equivalent to a diminished great icosahedron. This polyhedron is not locally convex but the act of joining it to the great icosidodecahedron removes the retrograde polygon . Excavate each of the pentagrammic faces of the great icosidodecahedron with a diminished great icosahedron. The 5/2 in the vertex figure is replaced by a (3,3,3)/2 The original vertex figures become (3,3,3,3,3,3,3,3)/3 - or (38)/3 with new vertices from the diminished great icosahedron of (3,3,3,3,3)/2 - or (35)/2. The original triangular faces of the stellatruncated cube are shown in blue, those of the anti-prisms in yellow and the pyramids red. All vertices are on the exterior of the polyhedron.