# A Zonish Rhombicosidodecahedron

This zonish polyhedron is based on the vertices of a rhombicosidodecahedron. I made it using Stella 4d, which you can try here.

# A Zonish Dodecahedron

This zonish polyhedron has 162 faces, and is based on the faces and the vertices of a dodecahedron. I made it using Stella 4d, which you can try for free at this website.

# Two Polyhedra Derived From the Icosahedron

The polyhedron above is a zonish icosahedron, with zones added to that Platonic solid based on its faces and vertices. Its faces are twenty equilateral triangles, thirty equilateral decagons, and sixty rhombi. After making it, I used faceting to truncate the vertices where sets of five rhombi met, creating the polyhedron below. It has twelve regular pentagons as faces, with the sixty rhombi of the polyhedron above turned into sixty isosceles triangles, along with the thirty decagons and twenty triangles from the first of these two polyhedra. This second one could be called either a faceted zonish icosahedron, or a truncated zonish icosahedron.

Both of these polyhedra were created using Stella 4d: Polyhedron Navigator, software you can try for free at http://www.software3d.com/Stella.php.

# 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.