**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 (3^{6})/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}__,

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
(3^{7})/3 and (3^{5})/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
(3^{9})/4 (as in an equivalent augmentation of the small retrosnub icosidodecahedron)
with the
(3^{7})/3 and (3^{5})/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 (3^{7})/3 with new vertices
from the diminished great icosahedron of (3,3,3,3,3)/2 - or (3^{5})/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 2^{nd} 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 (3^{8})/3 with new vertices
from the diminished great icosahedron of (3,3,3,3,3)/2 - or (3^{5})/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 (3^{8})/3 with new vertices
from the S5/3S5/2 of (3,3,3,3,3,3)/2 - or (3^{6})/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 2^{nd} 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 (3^{8})/3 with new vertices
from the S5/3S5/2 of (3,3,3,3,3,3)/2 - or (3^{6})/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.