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 (39)/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 (36)/2 with new vertices of (3,3,3,3). All vertices are on the exterior of the figure.