# Zonohedra, Zonish Polyhedra, and Another Puzzle

In a recent post, I showed many images of zonohedra, then challenged readers to figure out, from the images, what zonohedra are: polyhedra with only zonogons as faces. Zonogons, I then explained, are polygons with (A) even numbers of edges, and with opposite edges always (B) congruent and (C) parallel. Here is another collection of zonohedra. (Individual images may be enlarged with a click.)

The next set of polyhedra shown, below, are not true zonohedra (as all the ones above are), but merely “zonish polyhedra.” From examination of the pictures above and below, can you figure out the difference between zonohedra and zonish polyhedra?

When you are ready to see the solution to the puzzle, simply scroll down.

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While zonohedra have only zonogons as faces, this restriction is “loosened” for zonish polyhedra. Such solids are formed by zonohedrifying non-zonohedral polyhedra, and letting at least some of the faces of the resulting polyhedra remain non-zonogonal. Zonish polyhedra  are called “zonish” because many (usually most) of their faces are zonogons, but not all of them — in each case, some non-zonogonal polygons (such as triangles and/or pentagons, with their odd numbers of edges) do appear. Non-zonogonal polygons are not required to have odd numbers of edges, of course: simply having opposite edges be parallel, but of different lengths, is enough to prevent a polygon (such as a hexagon, octagon, or decagon) from being a zonogon.

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Software credit: I used Stella 4d to make these images. This program may be tried for free at this website.

# A Zonish Polyhedron with 522 Faces, Together with Its 920-Faced Dual The polyhedron above is a 522-faced zonish polyhedron, which resembles, but is not identical to, a zonohedron. True zonohedra are recognizable as that type of polyhedron by their exclusively zonogonal faces. Zonogons are polygons with even numbers of sites, and with opposite sides both congruent and parallel. If you examine the polyhedron above carefully, you’ll find it does not follow these rules. Stella 4d, the polyhedral-manipulation software I use to make these images, allows one to create either a true zonohedron, or a mere “zonish” polyhedron, as the user chooses, starting from another polyhedron (which may, itself, be zonish, a true zonohedron, or neither).

The next polyhedron is the dual of the polyhedron above. This dual has 920 faces. The duals of both zonohedra and zonish polyhedra have a distincive appearance, but, to my knowledge, no one has yet given either set of polyhedra a single-word name. In my opinion, such names are both needed, and deserved. If you would like to try Stella 4d for yourself, there is a free trial download available at http://www.software3d.com/Stella.php.

# The Seven Zonish Dodecahedra with Zones Added Based on Faces, Edges, and/or Vertices

If a zonish dodecahedron is created with zones based on the dodecahedron’s vertices, here is the result. If the same thing is done with edges, this is the result — an edge-distorted version of the great rhombicosidodecahedron. Another option is faces-only. Although I haven’t checked the bond-lengths, this one does have the general shape of the most-symmetrical 80-carbon-atom fullerene molecule. Also, this shape is sometimes called the “pseudo-truncated-icosahedron.” The next zonish dodecahedron has had zones added based on the dodecahedron’s faces and edges, both. Here’s the one for vertices and edges. Here’s the one for faces and vertices. Finally, the last of this set of seven has had zones added based on all three: faces, vertices, and edges. All seven of these were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.

# The Seven Zonish Icosahedra with Zones Added Based on Faces, Edges, and/or Vertices

If a zonish icosahedron is created with zones based on the icosahedron’s vertices, here is the result. If the same thing is done with edges, this is the result. Another option is faces-only. The next zonish icosahedron has had zones added based on the icosahedron’s faces and edges, both. Here’s the one for vertices and edges. Here’s the one for faces and vertices. Finally, the last of this set of seven has had zones added based on all three: faces, vertices, and edges. All seven of these were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.

# A Polyhedral Journey, Beginning with Face-Based Zonohedrification of an Icosahedron

To begin this, I took an icosahedron, and made a zonish polyhedron with it, with the new faces based on the zones of the existing faces. Here’s the result. Next, I started stellating the polyhedron above. At the sixth stellation, I found this. It’s a true zonohedron, and the first polyhedron shown here is merely “zonish,” because one has triangles, and the other does not. (One of the requirements for a polyhedron to be a zonohedron is that all its faces must have an even number of sides.) After that, I kept stellating, finding this as the 18th stellation of the first polyhedron shown here. With this polyhedron, I then made its convex hull. At this point, the irregular hexagons were bothering me, so I used Stella 4d‘s “try to make faces regular” option. (Stella 4d is polyhedron-manipulation software you can try for free, or purchase, right here.) The next step I chose was to augment all the yellow trapezoids with prisms, each with a height 1.6 times the trapezoids average edge length. The next step was, again, to make the convex hull. At this point, I tried “try to make faces regular” again, and was pleased with the result. The green rectangles became so thin, however, that I had to stop displaying the edges and vertices, in order for then to be seen. Next, I augmented both the blue faces (decagons) and the yellow faces (dodecagons) with antiprisms, again using a height 1.6 times that of the augmented faces’ average edge-lengths. Next, I made the convex hull again — a step I often take immediately after augmenting a polyhedron. This one surprised me, as it is more complicated than I expected. To clean things up a bit, I augmented only the trapezoids (dark pink) with prisms, and dodecagons (green) with antiprisms, again using the factor 1.6 for the augmentation-height. The next step I chose was to take the convex hull, once more. I had not yet noticed that the greater height of the trapezoidal prisms would cause the dodecagonal antiprisms to be “lost” by this step, though. Next, “try to make faces regular” was used again. This last result had me feeling my polyhedral journey was going in circles, so I tried augmentation again, but in a different way. I augmented this polyhedron, using prisms, on only the red trapezoids (height factor, 1.6 again) and the blue rectangles (new height factor, 2.3 times average edge length). After that, it was time to make another convex hull — and that showed me that I had, indeed, taken a new path. I found the most interesting faces of this polyhedron to be the long, isosceles trapezoids, so I augmented them with prisms, ignoring the other faces, using the new height-factor of 2.3 times average edge length this time. Of course, I wanted to see the convex hull of this. Who wouldn’t? I then started to stellate this figure, choosing the 14th stellation as a good place to stop, and making the edges and vertices visible once more. # A Zonish Icosahedron, and Some of Its “Relatives”

To begin this, I used Stella 4d (available here) to create a zonish polyhedron from the icosahedron, by adding zones along the x-, y-, and z-axes. The result has less symmetry than the original, but it is symmetry of a type I find particularly interesting. After making that figure, I began stellating it, and found a number of interesting polyhedra in this polyhedron’s stellation-series. This is the second such stellation: This is the 18th stellation: The next one, the 20th stellation, is simply a distorted version of the Platonic dodecahedron. This one is the 22nd stellation: This is the 30th stellation: The next really interesting stellation I found was the 69th: At this point, I returned to the original polyhedron at the top of this post, and examined its dual. It has 24 faces, all of which are quadrilaterals. This is the third stellation of this dual — and another distorted Platonic dodecahedron. This is the dual’s 7th stellation: And this one is the dual’s 18th stellation: At this point, I took the convex hull of this 18th stellation of the original polyhedron’s dual, and here’s what appeared: Here is this convex hull’s dual: Stella 4d, the program I use to make these (available here), has a built-in “try to make faces regular” function. When possible, it works quite well, but making the faces of a polyhedron regular, or even close to regular, is not always possible. I tried it on the polyhedron immediately above, and obtained this interesting result: While interesting, this also struck me as a dead end, so I returned to the red-and-yellow convex hull which is the third image above, from right here, and started stellating it. At the 19th stellation of this convex hull, I found this: I also found an interesting polyhedron as the 19th stellation of the dual which is three images above: 