**Augmenting the great stellatruncated
dodecahedron**

The great stellatruncated dodecahedron, vertex figure (^{10}/_{3},^{10}/_{3},3)

Left: Each
triangular face of the great stellatruncated dodecahedron can be augmented with a
triangular pyramid or tetrahedron. The 3 in the vertex figure is replaced by
(3,3) The original vertex figure becomes (^{10}/_{3},^{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 stellatruncated dodecahedron can be obtained by
excavating each decagrammic face with a decagrammic pyramid completing a
cycle around the axis, and each triangular face with a triangular
antiprism or octahedron again completing a
cycle around the axis. The ^{10}/_{3}'s in the vertex figure
are each replaced by
(3,3) and the 3 with (3,3,3). The original vertex figure becomes (__3,3__,__3,3__,__3,3,3__)/2 or
(3^{7})/2 with new vertices of (3,3,3,3,3,3,3,3,3,3)/3 or (3^{10})/3
and (3,3,3,3). The faces of the decagrammic
pyramids are shown in red. The triangular side faces of the antiprisms are
blue and the prismatic caps yellow. All
vertices are on the exterior of the polyhedron.

Right: An unusual
augmentation of the great stellatruncated dodecahedron can be obtained as
follows. The 5/3-cupola is not itself
a locally convex polyhedron. However, local convexity can be obtained if
the pentagram is excavated with a pentagrammic prism.
For brevity, I refer to the resulting polyhedron as a Q5/3(P5/2). This contains
vertices of the form (^{10}/_{3},4,3), (4,4,4,3,4)/2 (from the (^{5}/_{3},4,3,4)
vertices of the 5/3-cupola) and (^{5}/_{2},4,4) (from the
prismatic caps). We can then excavate each decagrammic face of the great stellatruncated dodecahedron
with a Q5/3(P(5/2) completing a
cycle around the axis, and each triangular face with a triangular
antiprism or octahedron again completing a
cycle around the axis. The ^{10}/_{3}'s in the vertex figure
are each replaced by (4,3) and the 3 with (3,3,3). The original vertex figure becomes (__4,3__,__4,3__,__3,3,3__)/2
with new vertices of (4,4,4,3,4)/2, (^{5}/_{2},4,4) and (3,3,3,3).
The triangular side faces of the anti-prisms are red. The square and
triangular faces of the cupola are orange and blue respectively, the
pentagrammic and square faces of the prisms are green and yellow
respectively. All
vertices are on the exterior of the polyhedron.