**Augmenting the ditrigonary dodecadodecahedron**

The ditrigonary dodecadodecahedron, vertex figure (^{5}/_{3},5,^{5}/_{3},5,^{5}/_{3},5,)

Left: Each
pentagrammic face of the ditrigonary dodecadodecahedron can be excavated with a pentagrammic
prism completing a cycle around the axis. Each ^{5}/_{3} in the vertex figure is replaced by
(4,4) The original vertex figures become (__4,4__,5,__4,4__,5,__4,4__,5)/4
with new vertices of (^{5}/_{2},4,4). All
vertices lie on the convex hull.

Centre: 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 (4,4,5,4,4,5,4,4,5)/4
are unaffected. 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).
In the model the square faces of the prisms are yellow, the square and
triangular faces of the cupolas are orange and blue respectively, the
decagrammic faces are red. All vertices are on the exterior of the
polyhedron.

Right: Remarkably,
a third order augmentation (the only one to date) can be formed by augmenting
the square faces of the 5/3-cupolas in the
above polyhedron with square pyramids.
As things are getting somewhat complex now, links are given to the overall
excavating polyhedron (P5/2 Q5/3 Y4).
The
original vertices of (4,4,5,4,4,5,4,4,5)/4 are again unaffected. The 4's
originating from the cupolas in the (4,3,4,4,4)/2 vertices are each replaced by
(3,3) to give (__3,3__,3,__3,3__,4,4)/2 vertices, as is the 4 in the (^{10}/_{3},3,4)
vertex to give vertices of the form (^{10}/_{3},3,__3,3__).
New vertices are added of the form (3,3,3,3). Colours are as per the
previous polyhedron with the added square pyramids in orange. All vertices
are on the exterior of the polyhedron.