pentakis dodecahedron

— The pentakis dodecahedron with its full icosahedral symmetry illustrates the fact that it is impossible to form a convex polyhedron with more than sixty identical non-right triangles. If right triangles are permitted then each isosceles triangle of the pentakis dodecahedron can be sub-divided into two coplanar right triangles, giving 120 as the greatest number of like units which can be used to form a convex polyhedron. It is this 120th lowest-common-denominator "icosahedral triangle" from which Fuller has derived his geodesic domes. The pentakis dodecahedron is topologically equivalent to the simplest of Fuller's geodesic domes.