**Augmenting the great rhombicosidodecahedron**

The great rhombicosidodecahedron, vertex figure
(4,^{5}/_{3},4,3)

Left: Each
retrograde pentagrammic face of the great rhombicosidodecahedron can be
excavated with a pentagrammic prism 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
(4,4) The original vertex figure becomes (4,__4,4__,4,3)/2
with new vertices of (^{5}/_{2},4,4). The square faces of
the pentagrammic prisms are shown in yellow. The original square
faces are orange. All
vertices lie on the convex hull.

Centre: A second
order augmentation can be generated from the above. The prismatic caps in
the above polyhedron can be further augmented with pentagrammic
prisms. This leaves the original vertex figures unchanged at (4,4,4,4,3)/2
but the new vertices of (^{5}/_{2},4,4) become (__3,3__,4,4)
vertices and further vertices are added of the form (3,3,3,3,3)/2. The faces of
the new pyramids are shown in purple. All
vertices are on the exterior of the figure.

Right:
Another augmentation of the great
rhombicosidodecahedron can be
obtained by excavating each pentagrammic face with a pentagrammic
antiprism
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) The original vertex figure becomes (4,__3,3,3__,4,3)/2 with new vertices of (^{5}/_{2},3,3,3).
The triangular faces of the pentagrammic antiprisms are shown in yellow. The
original triangular faces are blue. All
vertices are on the exterior of the polyhedron.

Left: The
leftmost 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 (^{4}/_{3},^{5}/_{2},^{4}/_{3},^{3}/_{2}).
Excavate each square face with a square pyramid
completing a cycle around the axis and each triangular face with a triangular
antiprism or octahedron again completing a cycle around the axis. This has the effect of uncrossing the
reversed vertex figure. Each ^{4}/_{3} in the vertex figure is replaced by
(3,3) and the ^{3}/_{2 } by (3,3,3) The original vertex figure becomes
(__3,3__,^{5}/_{2},__3,3__,__3,3,3__)/2 with two sets
of new vertices both of the form (3,3,3,3).
The triangular faces of the square pyramids are shown in yellow. The side
faces of the antiprisms are blue with the prismatic caps red. All
vertices are on the exterior of the polyhedron.

Centre: The centre image above shows a rare example which includes both the
excavation of a retrograde polygon and the augmentation of prograde polygons at each vertex of a polyhedron.
Each
retrograde pentagrammic face of the great rhombicosidodecahedron can be
excavated with a pentagrammic prism completing a cycle
around the axis. This has the effect of uncrossing the original
vertex figure. The square faces of the great
rhombicosidodecahedron are then augmented with square
pyramids. The ^{5}/_{3} in the vertex figure is replaced by
(4,4), the 4's in the original vertex figure are each replaced by
(3,3). The original vertex figure becomes (__3,3__,__4,4__,__3,3__,3)/2
with new vertices of (^{5}/_{2},4,4) and (3,3,3,3). The square faces of
the pentagrammic prisms are shown in yellow. The original square
faces are orange. The triangular faces of the pyramids are in red, the
oringinal triangular faces are blue. All
vertices lie on the convex hull.

Right: A second
order augmentation can be generated from the above. The prismatic caps in
the above polyhedron can be further augmented with pentagrammic prisms. This leaves the original vertex figures unchanged at (3,3,4,4,3,3,3)/2.
Also unchanged are the (3,3,3,3) vertices but the new vertices of (^{5}/_{2},4,4) become
(__3,3__,4,4)
vertices and further vertices are added of the form (3,3,3,3,3)/2. The faces of
the new pyramids are shown in purple. All
vertices are on the exterior of the figure.