**Augmenting the icosahedron**

The icosahedron, vertex figure (3,3,3,3,3)

Left: Each triangular face of the icosahedron can be excavated with a
triangular cupola completing a cycle around the axis.
The triangular caps of the cupola are coincident to the original triangular
faces. The 3's in the vertex figure
are each replaced by (4,3,4) The original vertex figure becomes (__4,3,4__,__4,3,4__,__4,3,4__,__4,3,4__,__4,3,4__)/4
or ((4,3,4)^{5})/4 with new vertices of (6,4,3).
All
vertices lie on the convex hull.

Centre: The left-hand
polyhedron can be further augmented by augmenting the hexagonal faces of the
triangular cupolas with a further triangular cupola,
such that the square faces of the two sets of cupolas are edge connected.
This is equivalent to excavating a triangular
orthobicupola from the original icosahedron. The original vertex figure
of (4,3,4,4,3,4,4,3,4,4,3,4,4,3,4)/4
or ((4,3,4)^{5})/4 is unchanged. The vertices of (6,4,3) become (__3,4__,4,3).
New vertices are added of the form (3,4,3,4). The square and triangular faces of
the new cupolas are shown in red and yellow respectively.
All
vertices lie on the convex hull.

Right: Another
augmentation of the icosahedron has each face of the icosahedron 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__,__3,3__)/3 - or
(3^{10})/3 with new vertices of (3,3,3). All
vertices are on the exterior of the figure. This figure nearly
escaped me as the (3,3,3) vertices are only barely visible. To see these
vertices click on the linked image above and zoom into one of the triangular
valleys.