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 (310)/3 with new vertices of (3,3,3). All vertices lie on the convex hull.