Augmenting the great retrosnub icosidodecahedron
The small retrosnub icosidodecahedron, vertex figure (5/2,3,3,3,3)/2
Left (79a): The second triangular face at each vertex of the great retrosnub icosidodecahedron (i.e. the icosahedral triangular faces shown in yellow in the image at the top of the page) can be augmented with a triangular pyramid or tetrahedron This 3 in the vertex figure are replaced by (3,3) The original vertex figure becomes ((5/2,3,3,3,3,3)/2 with new vertices of (3,3,3). The tetrahedral faces are shown in yellow. The original triangular faces are blue. All vertices are on the exterior of the polyhedron.
RIght (79b): Another augmentation of the great retrosnub icosidodecahedron can be obtained by augmenting each pentagrammic face with a pentagrammic pyramid. The 5/2 in the vertex figure is replaced by (3,3) The original vertex figure becomes (3,3,3,3,3,3)/2 or (36)/2 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 are on the exterior of the polyhedron.
Left (79c): A further augmentation of the great retrosnub icosidodecahedron can be obtained by excavating each pentagrammic face with a pentagrammic pyramid which completes a cycle around the axis, each triangular face of the great retrosnub icosidodecahedron is then excavated with a stellatruncated cube - vertex figure (8/3,8/3,3) - again completing a cycle around the axis. The 5/2 in the vertex figure is replaced by (3,3), each 3 in the vertex figure is replaced by (8/3,8/3) The original vertex figure becomes (3,3,8/3,8/3,8/3,8/3,8/3,8/3,8/3,8/3)/3 or (32,(8/3)8)/3 with new vertices of (3,3,3,3,3)/2 and (8/3,8/3,3) respectively. The faces of the pentagrammic pyramids are shown in yellow. The octagrammic and triangular faces of the stellatruncated cube are pink and blue respectively. All vertices are on the exterior of the polyhedron.
Right (79d): An unusual augmentation of the great retrosnub icosidodecahedron can be obtained as follows. Excavate all but one retrograde pentagrammic face of the small retrosnub icosidodecahedron with a pentagrammic pyramid completing a cycle around the axis (excavating all faces gives an augmented small retrosnub icosidodecahedron). This forms a polyhedron with one retrograde face. I term this a "modified 77a". All vertices not connected to this face are locally convex and of the forms (37)/3 and (35)/2. The vertices containing the retrograde pentagram are of the form (5/3,3,3,3,3,3)/2. Excavate each pentagrammic face of the great retrosnub icosidodecahedron with this modified 77a. The 5/2 in the vertex figure is replaced by (3,3,3,3,3). The original vertices become (3,3,3,3,3,3,3,3,3)/4 or (39)/4 (as in an equivalent augmentation of the small retrosnub icosidodecahedron) with the (37)/3 and (35)/2 vertices of the modified 77a also present. The original triangles of the great retrosnub icosidodecahedron are shown in yellow. The triangles of the small retrosnub icosidodecahedron are in blue and the triangles of the pentagrammic pyramids in green. All vertices are on the exterior of the polyhedron.
Left (79e): A further augmentation of the great retrosnub 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 (framework). This polyhedron is not locally convex but the act of joining it to the ggreat retrosnub 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 - or (37)/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 great icosidodecahedron are shown in blue and cyan, those of the anti-prisms in yellow and the pyramids red. All vertices are on the exterior of the polyhedron.
Right (79f): An associated augmentation of the great retrosnub icosidodecahedron can be obtained by excavating the pentagrams as in the previous example (79e) and also augmenting the icosahedral triangular faces (blue in the previous model) with triangular pyramids or tetrahedra. The 2nd 3 in the vertex figure becomes (3,3), along with the replacement of the 5/2 as above, the original vertex figure becomes (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 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.
Left (79g): A further variation on form 79e of the great icosidodecahedron can be obtained as follows: Excavate one pentagrammic cap of a crossed pentagrammic antiprism with a pentagrammic antiprism. For brevity I refer to the resulting polyhedron as an S5/3S5/2. 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 an S5/3S5/2. 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 S5/3S5/2 of (3,3,3,3,3,3)/2 - or (36)/2 and (5/2,3,3,3). The original triangular faces of the great icosidodecahedron are shown in blue and cyan, those of the 5/3 anti-prisms yellow and the 5/2 anti-prisms red and green. All vertices are on the exterior of the polyhedron.
Right (79h): An associated augmentation of the great retrosnub icosidodecahedron can be obtained by excavating the pentagrams as in the previous example (79g) and also augmenting the icosahedral triangular faces (blue in the previous model) with triangular pyramids or tetrahedra. The 2nd 3 in the vertex figure becomes (3,3), along with the replacement of the 5/2 as above, the original vertex figure becomes (3,3,3,3,3,3,3,3)/3 - or (38)/3 with new vertices from the S5/3S5/2 of (3,3,3,3,3,3)/2 - or (36)/2 and (5/2,3,3,3), 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.