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.

The Compounds of Five Octahedra and Five Cubes, and Related Polyhedra

Posted on 1 January 2016 by RobertLovesPi

This is the compound of five octahedra, each a different color.

Since the cube is dual to the octahedron, the compound of five cubes, below, is dual to the compound above.

Here are five cubes and five octahedra, compounded together, and shown with the same five colors used above.

This is the same compound, except with all squares/cubes having one color, and triangles/octahedra having another, made by changing the color-scheme used by Stella 4d (the program I use to make these images; it’s available here).

After seeing the two-color version of this ten-part compound, I decided to start stellating it, looking for stellations with an interesting appearance. Here is the 23rd stellation of the ten-part compound, colored by face-type.

Next, the 27th stellation, which is chiral, unlike the stellation showed above.

The 33rd stellation also has an interesting appearance (using, I admit, completely subjective criteria for “interesting”), while still having easily-noticable differences to the stellations shown above.

At the 35th stellation, another interesting chiral polyhedron is found. Unexpectedly, its direction of “twist” appears opposite that seen in the 27th stellation. (It could well be that this “twist-reversal” is a common phenomenon in stellation-series — simply one I have never noticed before.)

Next, the ten-part compound’s 39th stellation.

After the 39th stellation, I entered a sort of “desert,” with many stellations in a row which did not strike me as interesting, often with only tiny differences between one and the next. The 194th stellation, though, I liked.

Although I liked the 194th stellation, I didn’t want to risk trudging through another “desert” like the one which preceded it, so I jumped ahead to the final valid stellation, after which the series “wraps around” to its beginning.

Next, I made another rotating image of this final valid stellation, this time with the color-scheme set to “rainbow color mode.”

I couldn’t resist taking this one stellation further, to see the beginning of the stellation-series, since I knew I might have entered it somewhere in the middle, rather than at the beginning.

What I found, I immediately recognized as the rhombic triacontahedron. In some ways, this was surprising, and in other ways, it was not. The compound of five cubes is, itself, a stellation of the rhombic triacontahedron — but what I started stellating also included the compound of five octahedra, which, so far as I know, is not part of the rhombic triacontahedron’s (very) long stellation-series. Also, I know what the rhombic triacontahedron’s final stellation looks like, and it isn’t the final stellation shown above, but is, instead, this:

To try to better-understand just what was going on here, I went back, and deliberately left out the five-cube part of the ten-part compound (which is a stellation of the rhombic triacontahedron), which left me just with the compound of five octahedra — and then I had Stella produce this compound’s final stellation.

This was another polyhedron I recognized: the final stellation of the icosahedron. To verify that my memory was correct, I stellated it one more time. Sure enough, this is what I got:

This reminded me that the compound of five octahedra is the second stellation of the icosahedron, helping to explain some of this. I also noticed that the five-octahedron compound can be seen as a faceting of the icosidodecahedron. (The icosidodecahedron is dual to the rhombic triacontahedron, and faceting is the reciprocal function of stellation.) However, I have no idea why the final stellation of the ten-part compound above appears as it does.

It is my opinion that a productive polyhedral investigation usually does more than answer questions; it also raises new ones. At least in my mind, that’s exactly what has happened. Therefore, I think this was a perfectly good way to begin the new year.

A Compound of Ten Elongated Octahedra Which Is Also a Particular Faceting of the Rhombicosidodecahedron, Together with Its Dual

Posted on 3 December 2015 by RobertLovesPi

Thinking about the post immediately before this one led me to see if I could connect opposite triangular faces of a rhombicosidodecahedron to form a ten-part compound — and it worked with Stella 4d just as it had when I “previewed” it in my head.

The interesting dual of the above polyhedral compound, also a ten-part compound, I was not able to preview in my head (although that would be a nice ability to have), but creating it was easy with Stella:

It is difficult, in the dual, to tell what the ten components are. To help with this, in the next image, all but one component has been removed. This reveals the components of the dual to be rhombus-faced parallelopipeds which are quite flattened, compared to most parallelopipeds I have seen before. This polyhedron is isomorphic to the cube, just as the elongated octahedra in the first compound were each isomorphic to the Platonic octahedron. Given that the cube and octahedron are duals, this is no surprise.

Stella 4d may be tried for free, as a trial download, at this website: 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.

Both of these polyhedra were created with Stella 4d, software you may try for yourself, right here.

Some Enantiomorphic-Pair Compounds

Posted on 3 November 2015 by RobertLovesPi

In the last post here, three different color-versions of the same cluster-polyhedron were shown. Since this cluster-polyhedron is chiral, it is possible to make a compound of it, and its own enantiomer (or “mirror-image,” if you prefer). This first image shows that, with the face-color chosen by the number of sides of each face.

Shown next is the dual of this figure, also colored by the number of sides of each face.

Next, another image of the first compound shown here, but with the colors chosen by face-type (referring to each face’s position in the overall polyhedron).

Finally, here is the dual, again, also with colors chosen by face-type.

All four of these images were generated with Stella 4d, a computer program available at http://www.software3d.com/Stella.php.

A Twice-Zonohedrified Dodecahedron, Together with Its Dual

Posted on 25 October 2015 by RobertLovesPi

This polyhedron was created by performing vertex-based zonohedrifications of a dodecahedron — twice. The first zonohedrification produced a rhombic enneacontahedron, various version of which I have blogged many times before, but performing a second zonohedrification of the same type was a new experiment. It has 1230 faces, 1532 vertices, and 2760 edges. All of its edges have equal length. I created the models in this post using Stella 4d, a program you can buy, or try for free, right here.

Here is the dual of this zonohedron, which has 1532 faces, 1230 vertices, and 2760 edges. This “flipping” of the numbers of faces and vertices, with the number of edges staying the same, always happens with dual polyhedra. I do not know of a name for the class of polyhedra made of zonohedron-duals, but, if any reader of this post knows of one, please leave this name in a comment.

The Final Stellation of the Rhombic Triacontahedron, Together with Its Dual, a Faceting of the Icosidodecahedron

Posted on 15 October 2015 by RobertLovesPi

Sharp-eyed, regular readers of this blog will notice that this is the same polyhedron shown in the previous post, which was described as the “final stellation of the compound of five cubes,” due to the coloring scheme used in the first image there, which had five colors “inherited” from each of the differently-colored cubes in the five-cube compound. This image, by contrast, is shown in rainbow-color mode.

How can the rhombic triacontahedron and the compound of five cubes have the same final stellation? Simple: the compound of five cubes is, itself, a member of the stellation-series of the rhombic triacontahedron. Because of this, those two solids end up at the same place, after all possible stellations are completed, just as you will reach 1,000, counting by ones, whether you start at one, or start at, say, 170.

I am grateful to Robert Webb for pointing this out to me. He’s the person who wrote Stella 4d, the software I use to make these images of rotating polyhedra. His program may be found at http://www.software3d.com/Stella.php — and there is a free trial version available for download, so you can try Stella before deciding whether or not to purchase the fully-functioning version.

Since faceting is the reciprocal process of stellation, the dual of the polyhedron above is a faceted icosidodecahedron, for the icosidodecahedron is the dual of the rhombic triacontahedron. Here is an image of that particular faceting of the icosidodecahedron, colored, this time, by face-type:

The Great Rhombicosidodecahedron, Built from Rhombic Triacontahedra, and Its Dual

Posted on 8 June 2015 by RobertLovesPi

The great rhombicosidodecahedron is also known as the truncated icosidodecahedron (and, confusingly, several other names). Regardless of what it’s called, these pictures demonstrate that this Archimedean solid can be constructed using rhombic triacontahedra as building-blocks.

First, here’s one in the same color I used for the decagonal ring of rhombic triacontahedra in the last post:

The next one is identical, except I used “rainbow color mode” for it.

Also, just in case you’re curious, here’s the dual of this polyhedron-made-of-polyhedra — this time, colored by face-type.

These virtual models were all built using Stella 4d, software you may buy, or try for free, right here.

There Are Many Faceted Versions of the Dodecahedron. This One Is the Dual of the Third Stellation of the Icosahedron.

Posted on 6 June 2015 by RobertLovesPi

The twelve purple faces of this faceted dodecahedron show up on Stella 4d‘s control interface as {10/4} star decagons, which would make them each have five pairs of two coincident vertices. I’m informally naming this special decagon-that-looks-like-a-pentagram (or “star pentagon,” if you prefer) the “antipentagram,” for reasons which I hope are clear.

Stella 4d, the program I use to make most of my polyhedral images, may be tried for free at http://www.software3d.com/Stella.php.

A Cluster of Twenty Great Icosahedra, Excavated from the Faces of a Central Icosahedron, Along with Its Dual

Posted on 28 May 2015 by RobertLovesPi

These twenty great icosahedra were excavated from the faces of a central icosahedron, which is concealed in the figure’s center. These excavations exceed the limits of the central icosahedron, resulting in each great icosahedron protruding in a direction opposite that of the face from which it is excavated. In a certain sense, then, the figure above has negative volume.

To make this, I used software called Stella 4d: Polyhedron Navigator. It can be researched, bought, or tried for free here.

Also, here is the dual of the polyhedral cluster above, made with the same program.