**Augmenting the small stellated dodecahedron**

The small stellated dodecahedron, vertex figure (^{5}/_{2},^{5}/_{2},^{5}/_{2},^{5}/_{2},^{5}/_{2})

Left: Each pentagrammic face of the
small stellated dodecahedron can be excavated with a pentagrammic
antiprism completing a cycle around the axis. The ^{5}/_{2}'s
in the vertex figure
are each replaced by (3,3,3) The original vertex figures become (__3,3,3__,__3,3,3__,__3,3,3__,__3,3,3__,__3,3,3__)/4
or (3^{15})/4 with new vertices of (^{5}/_{2},3,3,3).
All
vertices are on the exterior of the figure.

Centre: A second
order augmentation can be formed from the above polyhedron by excavating the pentagrammic
antiprismatic caps 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^{15})/4
are unaffected. The ^{5}/_{2} in the (^{5}/_{2},3,3,3)
vertices is replaced by (4,3,4)/2 to give (__4,3,4__,3,3,3)/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
red/orange/yellow. All
vertices are on the exterior of the polyhedron.

Right: A
further augmentation of the
small stellated dodecahedron can be obtained by excavating each pentagrammic face
with a great retrosnub icosidodecahedron - vertex (^{5}/_{2},3,3,3,3)/2
- completing a cycle around the axis. The ^{5}/_{2}'s_{ }in the vertex figure
are each replaced by (3,3,3,3) The original vertex figures become (__3,3,3,3__,__3,3,3,3__,__3,3,3,3__,__3,3,3,3__,__3,3,3,3__)/9 or
(3^{20})/9 with new vertices of (^{5}/_{2},3,3,3,3)/2. All
vertices are on the exterior of the figure.

Left: Another augmentation of the
small stellated dodecahedron can be obtained by excavating each pentagrammic face
with a great icosidodecahedron - vertex (^{5}/_{2},3,^{5}/_{2},3)
- completing a cycle around the axis. The ^{5}/_{2}'s_{ }in the vertex figure
are each replaced by (3,^{5}/_{2},3) The original vertex figures become (__3, ^{5}/_{2},3__,

Right:
In the same way that an icosidodecahedron is
composed of two pentagonal rotundas, a great
icosidodecahedron can be thought of as being composed of two 'great'
pentagonal rotundas. A variation on the theme of the left hand
polyhedron is to excavate each pentagrammic face
with such a 'great pentagonal rotunda'.
- completing a cycle around the axis. The ^{5}/_{2}'s_{ }in the vertex figure
are each replaced by (3,^{5}/_{2},3) The original vertex figures become (__3, ^{5}/_{2},3__,