**Augmenting the ****Cube**

The cube, vertex figure (4,4,4)

Left: Each face of the cube can be excavated with a square
antiprism, each 4 in the vertex figure is replaced by a (3,3,3) completing a
cycle around the axis. The original vertex figures become (__3,3,3__,__3,3,3__,__3,3,3__)/2 - or
(3^{9})/2 with new vertices of (4,3,3,3).
Triangular faces edge connected to the original cubic square faces are in blue,
other triangles in yellow. All vertices lie on the convex hull.

Right: Each face of the cube can be excavated with a
square pyramid, each 4 in the vertex figure is replaced by a
(3,3) completing a
cycle around the axis. The original vertex figures become (__3,3__,__3,3__,__3,3__)/2 - or
(3^{6})/2 with new vertices of (3,3,3,3). All vertices are on the exterior of the
figure.