A Polyhedral Journey, Starting with the Compound of Five Dodecahedra

This is the compound of five dodecahedra, a shape which is included in the built-in polyhedral library of Stella 4d, a program you can try for yourself, free, right here.

I wanted to see what I could make, starting from this compound. My first modification to it was to create its convex hull, which is shown below.

The next move was to use Stella‘s “Try to Make Faces Regular” function, which produced this:

Next, I augmented this figure’s thirty yellow rhombi with prisms.

I then created the convex hull of this augmented polyhedron.

Next, I used the “Try to Make Faces Regular” function again, producing a solid that looks, to me, like a hybrid of the rhombicosidodecahedron and the rhombic triacontahedron.

This polyhedron has yellow faces that are almost squares. Careful inspection reveals that they are actually isosceles trapezoids. The next thing I did was to augment each of these trapezoids with a tall prism.

The next step was to, again, create the convex hull.

That was the end of this polyhedral journey, but I am confident there will be others.

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.