**Augmenting the great icosahedron**

The great icosahedron, vertex figure (3,3,3,3,3)/2

Each triangular face of the great icosahedron can be excavated with a
triangular pyramid or tetrahedron completing a cycle
around the axis. Each
3 in the vertex figure is replaced by (3,3) The original vertex
figure becomes (__3,3__,__3,3__,__3,3__,__3,3__,__3,3__)/3 or (3^{10})/3
with new vertices of (3,3,3). All
vertices lie on the convex hull.