A Compound of Six Pentagonal Trapezohedra, With Related Polyhedra

Here’s a compound I stumbled across tonight, while playing around with Stella 4d, a program you can try for free at this website. Trapezohedra have kites as faces, and each of the six components of this compound has a different color.

After finding the compound above, I used Stella to create this compound’s dual. Since trapezohedra are the duals of antiprisms, I expected to see a compound of six pentagonal antiprisms — but that’s not what I found. Instead, I saw this:

My initial reaction to this polyhedron was puzzlement. It’s pretty, and it’s interesting, but it’s not a dual of six antiprisms, at least as far as I can tell. I found the first polyhedron by using a lot of stellations, as well as other functions, for a long enough time that I couldn’t even remember what I started with. Faceting is the dual process to stellation, so this second polyhedron should be a faceted polyhedron — which it is.

What about the antiprisms I expected, though? Stella has a large built-in library of polyhedra, including compounds, so I looked up the compound of six regular pentagonal antiprisms, which is the next model shown.

Next, I created the dual of this antiprism-compound, and found myself looking at a compound of six trapezohedra which is quite different from the one at the top of this post.

As the dual of the regular-antiprism compound, this fourth image shows the “canonical” compound of six pentagonal trapezohedra, and it has more elongated kites for faces than the first one has. What I originally found with all of my stellations, etc., shown in the first image above, was a compound of six pentagonal trapezohedra, not the compound of six pentagonal trapezohedra. As for the non-compound dual solid shown in the second image above, it is unusual because it had an unusual origin — my long series of stellations and other transformations of polyhedra. Beyond that, I haven’t yet figured it out.

No matter how much you study geometry, there’s always more to learn.

A Compound of Three Trapezohedra

I came across this little beauty while exploring stellations of the triakis octahedron, which is the dual of the truncated cube. Its three components are each eight-faced trapezohedra, and it showed up as the sixth in that stellation-series.

compound of three trapezohedra.gif

Stella 4d: Polyhedron Navigator was used to make this rotating image. You may try it for free right here.

Compound of Three Eight-Faced Trapezohedra

compound of three eight-faced trapezohedra and 6th stellation of triakis octahedron

I made this using Stella 4d, which you can try right here. In addition to being a compound of trapezohedra, it is also the sixth stellation of the triakis octahedron, the dual of the truncated cube.

Infinite Families of Polyhedra

Many polyhedra appear in finite sets. The most well-known example of such a set is the five Platonic solids. Many know them from role-playing games.


Other finite sets include the thirteen Archimedean, four Kepler-Poinsot, thirteen Catalan, and 92 Johnson solids, and the eight convex deltahedra, among others. There is even a finite set, the near-misses, with an unknown number of members, due to the “fuzziness” of its definition. The symmetrohedra is another set with “fuzzy” criteria, but there are still only so many symmetrohedra to be found. We simply haven’t found them all yet, or, if we have, we don’t know that we have, but it would not be reasonable to claim that infinitely many await discovery.

However, not all groups of polyhedra are finite. Some polyhedra appear, instead, in infinite families. What is needed to generate such infinite families (at the cost of some forms of symmetry, compared to, say, Platonic or Archimedean solids) is the use of bases — special polyhedral faces which play a stronger role in the determination of that polyhedron’s shape than do the other faces. For the familar prism, there are two bases.

5 Penta Prism

In a pentagonal prism, the bases are pentagons, as seen above. For a pentagonal prism, n = 5, for n is simply the number of sides of the base. The smallest value of n which is possible, 3, yields a triangular prism. There’s no upper limit for n, either. Here’s a regular hexacontagonal prism, where each base has sixty sides.

60- Prism

Obviously, n can be increased without limit, although for very large values of n, the prism will be hard to distinguish from a cylinder.

Another infinite family may be found by taking the dual of each prism. This is the dual of the pentagonal prism:

5 Penta Prism dual

Taking a dual of a prism produces a dipyramid, with its n-gonal base hidden between the puramids, but with no guarantee that the triangular faces will be regular — and in this case, they are not. It is possible for a pentagonal dipyramid to have only regular faces . . .

5-dipyramid J13

. . . but regular faces will not work if n = 6 (because the dipyramid collapses to zero height), or indeed any number other than 3, 4, or 5. Therefore, dipyramids will not be an infinite family unless non-equilateral triangles are permitted as faces.

Half of a dipyramid, of course, is simply a pyramid, with the single base now visible:

5- Pyramid

Like the dipyramids, and for the same reason, there are only three pyramids (n = 3, 4, or 5) which can have all faces regular. An infinite family of right, regular pyramids do exist, though, if isosceles triangles are permitted as lateral faces.

While pyramids and dipyramids have only one base each, the already-described prisms have two. Prisms also can maintain regularity of all faces, no matter how large n becomes, unlike pyramids and dipyramids. Moreover, prisms can be transformed into another infinite family of regular-faced polyhedra by rotating one base by one half of one-nth of a rotation, relative to the other base, and replacing the n square lateral faces with 2n equilateral triangles. These polyhedra are called antiprisms. The pentagonal antiprism looks like this:

5 Penta Antiprism

Antiprisms with all faces regular do not have the “only 3, 4, and 5” limitation that affected the pyramids and dipyramids. For example, here is one with all faces regular, and dodecagonal bases:

12- Antiprism

One more infinite family of polyhedra may be found, using the antiprisms: their duals. The dual of the pentagonal antiprism looks like this:

5 Penta Antiprism dual

There is no regularity of faces for any members of this family with n > 3, for their faces are kites. (When n = 3, the kites become squares, and the polyhedron formed is simply a cube.) Unfortunately, “kiteohedron” looks ridiculous, and sounds worse, so efforts have been made to find better names for these polyhedra. “Deltohedra” has been used, but is too easily confused with the deltahedra, which are quite different. The best names yet invented for this infinite family of kite-faced polyhedra are, in my opinion, the antidipyramids, and the trapezohedra.

(Note:  the rotating polyhedral images above were generated using Stella 4d:  Polyhedron Navigator, available at http://www.software3d.com/Stella.php.)