A Polyhedral Journey, Beginning with a Near-Miss Johnson Solid Featuring Enneagons

When Norman Johnson first found, and named, all the Johnson solids in the latter 1960s, he came across a number of “near-misses” — polyhedra which are almost Johnson solids. If you aren’t familiar with the Johnson solids, you can find a definition of them here. The “near-miss” which is most well-known features regular enneagons (nine-sided polygons):

ennneagonal-faced near-miss

This is the dual of the above polyhedron:

ennneagonal-faced near-miss dual

As with all polyhedra and their duals, a compound can be made of these two polyhedra, and here it is:

ennneagonal-faced near-miss base=dual compound

Finding this polyhedron interesting, I proceeded to use Stella 4d (polyhedron-manipulation software, available at http://www.software3d.com/Stella.php) to make its convex hull.

Convex hull of near-miss base-dual compound

Here, then, is the dual of this convex hull:

dual of Convex hull of near-miss base-dual compound

Stella 4d has a “try to make faces regular” function, and I next used it on the polyhedron immediately above. If this function cannot work, though — because making the faces regular is mathematically impossible — one sometimes gets completely unexpected, and interesting, results. Such was the case here.

attempt no make latest polyhedron have regular faces

Next, I found the dual of this latest polyhedron.

attempt no make latest polyhedron have regular faces's dual

The above polyhedron’s “wrinkled” appearance completely surprised me. The next thing I did to change it, once more, was to create this wrinkled polyhedron’s convex hull. A convex hull of a non-convex polyhedron is simply the smallest convex polyhedron which can contain the non-convex polyhedron, and this process often has interesting results.

Convex hull of wrinkled dual

Next, I created this latest polyhedron’s dual:

dual of Convex hull of wrinkled dual

I then attempted “try to make faces regular” again, and, once more, had unexpected and interesting results:

dual of latest polyhedron

The next step was to take the convex hull of this latest polyhedron. In the result, below, all of the faces are kites — two sets of twenty-four each.

convex hull of last polyhedron with two sets of two dozen kites each

I next stellated this kite-faced polyhedron 33 times, looking for an interesting result, and found this:

33rd stellation of latest polyhedron

This looked like a compound to me, so I told Stella 4d to color it as a compound, if possible, and, sure enough, it worked.

33rd stellation of latest polyhedron colored as a compound

The components of this compound looked like triakis tetrahedra to me. The triakis tetrahedron, shown below, is the dual of the truncated tetrahedron. However, I checked the angle measurement of a face, and the components of the above compound-dual are only close, but not quite, to being the same as the true triakis tetrahedron, which is shown below.

Triakistetra -- ANGLES AREN'T QUITE A MATCH for last polyhedron

This seemed like a logical place to end my latest journey through the world of polyhedra, so I did.