Augmented Uniform Polyhedra
I wanted to find more polyhedra that looked like the
uniform polyhedra. Norman Johnson had already
enumerated all the convex regular-faced polyhedra, so I thought I'd consider
the nonconvex ones. There are an infinite number of these, so some further
rules are required to find the interesting ones. I considered these two
Various existing polyhedra already follow these rules, eg some nonconvex
cupolae, pyramids, etc. But remembering that none of the Johnson Solids have
higher than dihedral symmetry, I wondered if any of this new class of polyhedra
might. So better-than-dihedral symmetry became a third rule.
After a while I thought of what should have been an obvious course for
investigation. The uniform polyhedra already satisfy the rules, so why not try
augmenting some of their faces with pyramids or cupolae?
This turned out to be fruitful, although initially it only lead to four new
polyhedra in this class: one with octahedral symmetry, and three with
icosahedral symmetry. They are pictured below:
- Must be locally-convex. That is, the faces surrounding each
vertex must loop around the vertex in the same direction (no retrograde
faces spanning back the other way).
- All vertices must be vertices of the convex hull. This is to ensure
that all vertices are visible, and is a condition always met by uniform
|Augmented Great Cubicuboctahedron
||Augmented Great Ditrigonal Dodecicosidodecahedron
||Augmented Snub Dodecadodecahedron
In the three icosahedral cases, the pentagons are augmented with pentagonal
pyramids, in the other case the squares are augmented with square pyramids.
Unfortunately, augmenting with cupolae failed to meet the conditions in all
More interesting new polyhedra could surely be created by changing my rules
Since my initial investigation, Jim McNeill has taken this idea and run with
it. He found many more polyhedra following these rules and variations on
them. His results can be found at