Zonohedra, Zonish Polyhedra, and Another Puzzle

Posted on 11 August 2016 by RobertLovesPi

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.)

Zonohedrified Trunc Cubocta faces are 552 rhombi

2256 rhombic faces Zonohedrified Trunc Cubocta

Zonohedrified Dual of Zonohedrified Triakisocta

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.

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

Posted on 18 January 2016 by RobertLovesPi

zonol

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.

zono 920 faces dual of the one that had 522 faces

If you would like to try Stella 4d for yourself, there is a free trial download available at http://www.software3d.com/Stella.php.

A Zonohedron with 3540 Faces, Together with Its Dual

Posted on 7 November 2015 by RobertLovesPi

Zonohedra are polyhedra made completely of faces which are zonogons. A zonogon is a polygon which:

Has an even number of sides,
Has opposite sides congruent, and
Has opposite sides parallel.

Parallelograms are the simplest zonogons.

Here is the dual of the zonohedron above; it has 3542 faces. Although zonohedra-duals do have distinctive appearances, they do not, as a class, have a name of their own, to the best of my knowledge. They are definitely not zonohedra, themselves.