**Augmenting the great dodekicosidodecahedron**

The great dodekicosidodecahedron, vertex figure (^{10}/_{3},^{5}/_{2},^{10}/_{3},3)

Left: Each
triangular face of the great dodekicosidodecahedron can be augmented with a triangular
pyramid or tetrahedron. Each 3 in the vertex figure is replaced by
(3,3) The original vertex figure becomes (^{10}/_{3},^{5}/_{2},^{10}/_{3},__3,3__)
with new vertices of (3,3,3). All
vertices are on the exterior of the polyhedron.

Right: The above figure is shown with the tetrahedra shown in framework style. A framework of this model is linked here.

Left:
Another augmentation of the great dodekicosidodecahedron can be
obtained by excavating each decagrammic face with a decagrammic
antiprism completing a cycle around the axis, each pentagrammic face with a pentagrammic
prism which again completes a cycle around the axis, and each triangular face with a triangular
prism which again completes a cycle around the axis. Each ^{10}/_{3 }in
the vertex figure is replaced by (3,3,3), the ^{5}/_{2}
and the 3 in the vertex figure are each replaced by (4,4) The original vertex figure becomes (__3,3,3__,__4,4__,__3,3,3__,__4,4__)/3
with new vertices of (^{10}/_{3},3,3,3), (^{5}/_{2},4,4)
and (3,4,4)
respectively. The square faces of the pentagrammic prisms are shown in yellow and
the square faces of the triangular prisms in orange. The triangular faces
of the decagrammic antiprisms are shown in blue and
the triangular prismatic caps in red. All
vertices are on the exterior of the polyhedron.

Right: A second
order augmentation can be formed by excavating the pentagrammic prismatic caps
from the above polyhedron with 5/3-cupolas.
The cupolas themselves are not locally convex but the act of joining them to the
pentagrammic caps removes the retrograde polygon. The original vertices of (3,3,3,4,4,3,3,3,4,4)/3
are unaffected, as are the (^{10}/_{3},3,3,3)
and (3,4,4) vertices. The ^{5}/_{2} in the (^{5}/_{2},4,4)
vertices is replaced by (4,3,4)/2 to give (__4,3,4__,4,4)/2
vertices. New vertices are added of the form (^{10}/_{3},3,4).
Colours are as per the previous polyhedron with the 5/3-cupolas shown in
purple/pink. All
vertices are on the exterior of the polyhedron.