**Augmenting the Octahedron**

The octahedron, vertex figure (3,3,3,3)

Left: Each face of
the octahedron can be excavated with a triangular prism,
each 3 in the vertex figure is replaced by a (4,4) completing a cycle around the
axis. The original vertex figures become (__4,4__,__4,4__,__4,4__,__4,4__)/3 - or
(4^{8})/3
with new vertices of (4,4,3). All vertices lie on the convex hull.

Centre: Each
square face in the first augmentation can be further augmented with a square
pyramid. Each 4 in the vertex figure is replaced by a (3,3). The
original vertex figure becomes (__3,3__,__3,3__,__3,3__,__3,3__,__3,3__,__3,3__,__3,3__,__3,3__)/3 - (or
3^{16})/3. The (4,4,3) vertices in the first augmentation become (__3,3__,__3,3__,3)
vertices. New vertices are of the form (3,3,3,3). In the above figure, triangular
faces in the square pyramids are shown in blue. The caps of the triangular
prisms are shown in yellow. All vertices are on the exterior of the
figure. This is a rare example to date of a second order augmentation.

Right: Another
augmentation of the octahedron has each face of the octahedron excavated with a triangular
pyramid or tetrahedron, each 3 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__,__3,3__)/3 - or
(3^{8})/3 with new vertices of (3,3,3). All
vertices are on the exterior of the figure. With only 24 faces I
believe this to be minimal polyhedron meeting the criteria. A
framework model of this polyhedron is linked here.