A Zoo of Zonohedra

Zonohedra are a subset of polyhedra with all faces in pairs of parallel and congruent zonogons. Zonogons are polygons with sides which occur only as parallel and congruent pairs of line segments. As a consequence of this, the faces of zonohedra must have even numbers of sides.

Considering all the restrictions on zonohedra, it may be surprising that there is so much variety among them. Every polyhedron shown in this post is a zonohedron. The colors are chosen so that all four-sided zonogons have one color, all six-sided zonogons have a second color, and so on.

Octagon-dominated zonohedron.gif

Zonohedrified Cobvjnvex hull.gif

Zonohedrified Conjhvjvvex hull.gif

Zonohedrified Conjhvvex hull.gif

Zonohedrified Connbvj,njkvex hull.gif

Zonohedrified Convb bvvex hull.gif

Zonohedrified Convehckhcx hull.gif

Zonohedrified Convex hull  186 faces.gif

Zonohedrified Convex hull 132 faces

Zonohedrified Convex hull 138 faces.gif

Zonohedrified Convex hull 306 faces colored by number of edges per face.gif

Zonohedrified Convex hull features octadecagons.gif

Zonohedrified Convex hull.gif

Zonohedrified Convex hullygduyd.gif

Zonohedrified Cube 2.gif

Zonohedrified Ochjgta.gif

Zonohedrified Octa 2

Zonohedrified Octa 3.gif

Zonohedrified Octa z.gif

Zonohedrified Octa.gif

Zonohedrified Pmhgcholy.gif

Zonohedrified Polly.gif

Zonohedrified Poly.gif

Zonohedrified Snub Cube.gif

Zonohedrified tet.gif

Zonohedrified Trunc Dodeca featuring octadecagons.gif

Zonohedrified Trunc Dodeca.gif

Zonohedrified Trunc Tetra vef.gif

Zonohedrified Trunc Tetra.gif

Zonohedrified Trunc Tetrahedron.gif

I made all of these using Stella 4d: Polyhedron Navigator. This program may be tried for free at this website.

A Zonohedron with 7802 Faces

7802 faces.gif

Zonohedra with surprisingly large numbers of faces are easy to make with Stella 4d: Polyhedron Navigator. This program is sold at http://www.software3d.com/Stella.php, where there is also a free trial download offered.

Six Zonohedra

Zonohedra are a subset of polyhedra. In a zonohedron, all the faces are zonogons. A zonogon is a polygon with an even number of sides, as well as having opposite sides both congruent and parallel. This small collection of rotating zonohedra was made using Stella 4d, a program you can try for yourself at this website.

Also, if you want to see a larger version of any one of these zonohedra, simply click on it.

 

Zonohedrified Rhombicosidodecahedron with 870 Rhombic Faces

zonohedrified-rhombicosidodeca-870-faces

Manipulating known polyhedra in the effort to find new ones, as I did here, is made easy with Stella 4d, a program available at this website.

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.

Some Zonohedra, and a Puzzle

Every zonohedron is a polyhedron, but not all polyhedra are zonohedra. Examples of zonohedra appear below. If you don’t already know what zonohedra are, can you figure out the definition from the examples shown, before reading the definition below the pictures?

Answer below (scroll down a bit):

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Zonohedra are polyhedra with only zonogons as faces. A zonogon is a polygon with an even number of sides, and also with opposite sides congruent and parallel.

Software credit: I used Stella 4d to make these virtual, rotating zonohedra. This program may be tried for free at http://www.software3d.com/Stella.php.

Three Different Views of a 962-Faced Zonohedron

This zonohedron contains faces which are regular decagons (12 of them), equilateral octagons (30, all of the same type), equilateral hexagons (380 of them, of 7 types, with one of these 7 types, of which there are 20, being regular), squares (60), and non-square rhombi (480 of 8 types, counting reflections as separate types). With each polygon-type, including the reflections, given a different color, this zonohedron looks like this.

Zonohedron with 962 faces colored by face type

If reflected face-types are not counted as separate types, then the coloring-by-face-type uses four fewer colors, and looks like this:

Zonohedron with 962 faces colored by face type 2nd version

Another view simply colors faces by numbers of sides, and is shown below. Each of these rotating images was created with Stella 4d, a program you may buy, or try for free, at http://www.software3d.com/Stella.php.

Zonohedron with 962 faces colored by face number of sides