Augmenting the Great Cubicuboctahedron
The great cubicuboctahedron, vertex figure (4,8/3,3,8/3)
Left: Each square face of the great cubicuboctahedron can be augmented with an square pyramid, the 4 in the vertex figure is replaced by a (3,3) The original vertex figures become (3,3,8/3,3,8/3) with new vertices of (3,3,3,3). The original triangular faces are in blue, faces of the square pyramids in yellow. All vertices lie on the convex hull.
Right: The above figure is shown with the square pyramids shown in framework style. A framework of this model is linked here.
Left: Another augmentation of the great cubicuboctahedron has each face excavated. The octagrammic faces are excavated with octagrammic antiprisms, the square faces excavated with square pyramids, and the triangular faces with triangular antiprisms or octahedra. Each 8/3 in the vertex figure is replaced by a (3,3,3) completing a cycle around the axis, each 4 in the vertex figure is replaced by a (3,3) completing a cycle around the axis and each 3 in the vertex figure is replaced by a (3,3,3) completing a cycle around the axis. The original vertex figures become (3,3,3,3,3,3,3,3,3,3,3)/3 - or (311)/3 with new vertices of (3,3,3,3,3,3,3,3)/3 - or (38)/3, and (3,3,3,3) from the octagrammic antiprisms, the square pyramids and octahedra respectively. The triangles of the octagrammic antiprisms are shown in purple, the triangles of the square pyramids are blue and the octahedra are yellow. All vertices are on the exterior of the figure.
Right: A further augmentation of the great cubicuboctahedron can be formed by excavating the octagrammic faces with crossed octagrammic cupolas (or 8/5-cupolas, The cupolas themselves are not locally convex but the act of joining them to the octagrams removes the retrograde polygon. The excavation is also shallow enough not to cause the vertex figure to become crossed. Each 8/3 in the vertex figure is replaced by a (4,3,4)/2. The original vertex figures become (4,4,3,4,3,4,3,4)/3. New vertices are added of the form (16/5,3,4). This augmentation is to date unique in that it contains polygons not found in any of the uniform polyhedra. The 16/5-gons are shown in pink. The square and triangular faces of the cupolas are red and yellow respectively. All vertices are on the exterior of the figure.