**Augmenting the stellatruncated cube**

The stellatruncated cube, vertex figure (^{8}/_{3},^{8}/_{3},3)

Left: Each
octagrammic face of the stellatruncated cube can be excavated with an octagrammic
pyramid
completing a cycle around the axis. The ^{8}/_{3} in the vertex figure is replaced by a
(3,3) The original vertex figures become (__3,3__,__3,3__,3)/2 with new vertices of
(3,3,3,3,3,3,3,3)/3 - or (3^{8})/3. This is a unique case where
the original triangular faces are left in place while all other faces are
excavated. The excavations are deep enough that the original triangular
faces do not disrupt the local convexity of the vertices (diagram).
The faces of the pyramids are shown in
yellow with the original triangles in blue. All vertices lie on the convex
hull.

Centre:
Another
augmentation of the stellatruncated cube has in addition to the octagrammic
pyramids above, each triangular
face also excavated with a triangular pyramid or tetrahedron
completing a cycle around the axis. The 3 in the original vertex figure is replaced by a (3,3)
The original vertex figures become (__3,3__,__3,3__,__3,3__)/2 with new vertices of
(3,3,3,3,3,3,3,3)/3 - or (3^{8})/3 and (3,3,3). The faces of the octagrammic
pyramids are shown in
yellow with the tetrahedral faces in blue. All
vertices are on the exterior of the figure.

Right: A further
augmentation of the stellatruncated cube can be obtained as follows:
Excavate one octagrammic cap of a crossed octagrammic
antiprism with an octagrammic pyramid. For
brevity I refer to the resulting polyhedron as S8/5Y8/3. This polyhedron is not locally convex but the act of joining
it to the stellatruncated cube removes the retrograde polygon . Excavate
each of the octagrammic faces of the stellatruncated cube with an S8/5Y8/3.
The ^{8}/_{3} in the vertex figure is replaced by a (3,3,3)/2 The original vertex figures become (__3,3,3__,__3,3,3__,3)/3 - or (3^{7})/3 with new vertices
from the S8/5Y8/3 of (3,3,3,3,3)/2 - or (3^{5})/2 and
(3,3,3,3,3,3,3,3)/3 - or (3^{8})/3. The original triangular faces
of the stellatruncated cube are shown in blue, those of the anti-prisms in
yellow and the pyramids pink. All vertices lie on the convex
hull.