**Augmenting the small retrosnub icosidodecahedron**

The small retrosnub icosidodecahedron, vertex figure
(^{5}/_{3},3,3,3,3,3)/2

Left
(77a): Each
retrograde pentagrammic face of the small retrosnub icosidodecahedron can be
excavated with a pentagrammic pyramid 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) The original vertex figure becomes (__3,3__,3,3,3,3,3)/3 or
(3^{7})/3 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 lie on the convex hull.

Centre (77b): The
middle image above shows a rare example where all of the prograde polygons at each vertex of a polyhedron with
crossed vertices have been excavated, reversing the polarity of the vertex and
turning the originally retrograde polygon prograde. It is easiest to start by considering the vertex is
traversed in the opposite direction so that it becomes
(^{5}/_{2},^{3}/_{2},^{3}/_{2},^{3}/_{2},^{3}/_{2},^{3}/_{2})/2.
Excavate each triangular face with a triangular pyramid
or tetrahedron completing a cycle around the axis. This has the effect of uncrossing the
reversed vertex figure. Each ^{3}/_{2} in the vertex figure is replaced by
(3,3). The original vertex figure becomes
(^{5}/_{2},__3,3__,__3,3__,__3,3__,__3,3__,__3,3__)/3
with new vertices of the form (3,3,3).
The tetrahedral faces from the icosahedral triangles are shown in blue, those
from the snub triangles in yellow. Whilst all vertices are on the exterior of the
polyhedron, the original pentagrammic faces are not visible. The polyhedron is
also partially degenerate as the apices of the icosahedral tetrahedra coincide
in pairs.

Right
(77c): Each
retrograde pentagrammic face of the small retrosnub icosidodecahedron can be
excavated with a great retrosnub icosidodecahedron -
vertex figure (^{5}/_{2},3,3,3,3)/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
(3,3,3,3) The original vertex figure becomes (__3,3,3,3__,3,3,3,3,3)/4 or
(3^{9})/4 with new vertices of (^{5}/_{2},3,3,3,3)/2. The
triangular faces of the original small retrosnub icosidodecahedron are shown in yellow. The
triangular and pentagrammic faces of the great retrosnub icosidodecahedron are
blue and green respectively. All
vertices lie on the convex hull.

Left (77d): An associated
augmentation of the small retrosnub
icosidodecahedron can be obtained by excavating the pentagrams as in the first example
above (77a) and also augmenting **one** set of icosahedral triangular faces (blue in
the original image of the small retrosnub
icosidodecahedron) with triangular pyramids or tetrahedra.
The 2^{nd} 3 in the vertex figure becomes (3,3), along with the
replacement of the ^{5}/_{3 }as above, the original vertex
figure becomes (__3,3__,3,__3,3__,3,3,3)/3 or
(3^{8})/3 with new vertices of (3,3,3,3,3)/2 from the pentagrammic
pyramids 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.

Right
(77e): Another associated
augmentation of the small retrosnub
icosidodecahedron can be obtained by excavating the pentagrams as in the first example
above (77a) and also augmenting **both** sets of icosahedral triangular faces (blue in
the original image of the small retrosnub
icosidodecahedron) with triangular pyramids or tetrahedra.
The 2^{nd} and 3^{rd} 3 in the vertex figure becomes (3,3), along with the
replacement of the ^{5}/_{3 }as above, the original vertex
figure becomes (__3,3__,3,__3,3__,__3,3__,3,3)/3 or
(3^{9})/3 with new vertices of (3,3,3,3,3)/2 from the pentagrammic
pyramids and (3,3,3) from the tetrahedra. The resulting
polyhedron is partially degenerate as the tetrahedral vertices are coincident in
pairs. Colours are as per the previous example
with the tetrahedra in blue. All
vertices are on the exterior of the polyhedron.