**Augmenting the great rhombicuboctahedron**

The great rhombicuboctahedron, vertex figure (4,4,^{3}/_{2},4)

Left: Each
triangular
face of the great rhombicuboctahedron can be excavated with a triangular
prism
completing a cycle around the axis. The ^{3}/_{2} in the vertex figure is replaced by a
(4,4) The original vertex figures become
(4,4,__4,4__,4)/2 with new vertices of (4,4,3). The square faces of the
prisms are shown in
yellow with the prismatic caps in blue. All vertices lie on the convex
hull. A framework model of this figure is linked here.

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

Right:
Another
augmentation of the great rhombicuboctahedron has each triangular
face excavated with a triangular antiprism or octahedron
completing a cycle around the axis. The ^{3}/_{2} in the vertex figure is replaced by a
(3,3,3) The original vertex figures become
(4,4,__3,3,3__,4)/2 with new vertices of (3,3,3,3). The triangular side
faces of the antiprisms are shown in yellow with the prismatic caps in blue. All
vertices are on the exterior of the figure.

The above images show rare examples which include both the excavation of a retrograde polygon and the augmentation of prograde polygons at each vertex of a polyhedron

Left:
Each
triangular
face of the great rhombicuboctahedron can be excavated with a triangular
prism
completing a cycle around the axis. The rhombic dodecahedral square faces
(the 2^{nd} and 3^{rd} 4's in the original vertex figure) can be
augmented with square pyramids. The ^{3}/_{2} in the vertex figure is replaced by a
(4,4) and the 2^{nd} and 3^{rd} 4's by (3,3) The original vertex figures become
(4,__3,3__,__4,4__,__3,3__)/2 with new vertices of (4,4,3) and
(3,3,3,3) respectively. The faces of the pyramids are in red. The square faces of the
prisms are shown in
yellow with the prismatic caps in blue. All
vertices are on the exterior of the figure.

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

Right:
Each
triangular
face of the great rhombicuboctahedron can be excavated with a triangular antiprism or octahedron
completing a cycle around the axis. The rhombic dodecahedral square faces
(the 2^{nd} and 3^{rd} 4's in the original vertex figure) can be
augmented with square pyramids. The ^{3}/_{2} in the vertex figure is replaced by a
(3,3,3) and the 2^{nd} and 3^{rd} 4's by (3,3) The original vertex figures become
(4,__3,3__,__3,3,3__,__3,3__)/2 with two sets of new vertices both of
the form (3,3,3,3). The faces of the pyramids are in red. The triangular side
faces of the antiprisms are shown in yellow with the prismatic caps in blue. All
vertices are on the exterior of the figure.