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.

The Compound of Five Cubes, Augmented with Thirty Snub Cubes: Three Versions

Posted on 3 November 2015 by RobertLovesPi

This cluster-polyhedron was made with Stella 4d, software you can try at this website. Above, it is colored by face-type, referring to each face’s position within the overall cluster. In the image below, the original compound of five cubes contained one cube each, of five colors, and then each snub cube “inherited” its color from the cube to which it was attached.

In the next version, the colors are chosen by the number of sides of each face.

The Final Stellation of the Compound of Five Icosahedra

Posted on 3 November 2015 by RobertLovesPi

This was made using Stella 4d, software available at http://www.software3d.com/Stella.php.

A Cluster of Thirty-One Rhombic Enneacontahedra

Posted on 31 October 2015 by RobertLovesPi

The rhombic enneacontahedron has thirty faces which are narrow rhombi, and sixty faces which are wider rhombi. It is also known as a vertex-based zonohedrified dodecahedron. To create this cluster-polyhedron, I started with one rhombic enneacontahedron in the center, and then augmented its thirty red faces (the narrow rhombi) with additional rhombic enneacontahedra. In the image above, I kept the yellow color for all the wide rhombi, and red for all the narrow ones. In the next image, however, the rhombi are colored by face type, referring to their position in the entire cluster-polyhedron.

Software credit: I created this using Stella 4d, software you can buy, or try for free, at this website.

Eight Selections from the Stellation-Series of the Rhombic Enneacontahedron

Posted on 25 October 2015 by RobertLovesPi

The stellation-series of the rhombic enneacontahedron has many polyhedra which are, to be blunt, not much to look at — but there are some attractive “gems” hidden among this long series of polyhedral stellations. The one above, the 33rd stellation, is the first one attractive one I found — using, of course, my own, purely subjective, esthetic criteria.

The next attractive stellation I found in this series is the 80th stellation. Unlike the 33rd, it is chiral.

And, after that, the 129th stellation, which is also chiral:

Next, the 152nd (and non-chiral) stellation:

I also found the non-chiral 158th stellation worthy of inclusion here:

After that, the chiral 171st stellation was the next one to attract my attention:

The next one to attract my notice was the also-chiral 204th stellation:

Some polyhedral stellation-series are incredibly long, with thousands, or even millions, of stellations possible before one reaches the final stellation, after which stellating the polyehdron one more time causes it to “wrap around” to the original polyhedron. Knowing this, I lost patience, and simply jumped straight to the final stellation of the rhombic triacontahedron — the last image in this post:

All of these images were created using Stella 4d: Polyhedron Navigator, a program available at http://www.software3d.com/Stella.php. For anyone interested in seriously studying polyhedra, I consider this program an indispensable research tool (and, no, I receive no compensation for all this free advertising for Stella which appears on my blog). There’s a free trial version available — why not give it a try?

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.

A Pyritohedral Coloring-Scheme for the Truncated Icosahedron

Posted on 25 October 2015 by RobertLovesPi

While the polyhedron above, informally known as the “soccer ball,” has icosidodecahedral symmetry, its coloring-scheme does not. Instead, I colored the faces in such a way that the coloring-scheme has pyritohedral symmetry — the symmetry of a standard volleyball. This rotating image was made with Stella 4d, a program you can buy, or try for free, right here: http://www.software3d.com/Stella.php.

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 Final Stellation of the Compound of Five Cubes

Posted on 14 October 2015 by RobertLovesPi

The version of the final stellation of the compound of five cubes shown above has its colors derived from the traditional five-color version of the original compound, itself. The one below, by contrast, has its colors selected by face-type, without regard for the original compound.

Both of these virtual models were created with Stella 4d: Polyhedron Navigator, software available at this website. Also, for more about this particular polyhedron, please see the next post.

One of Many Possible Facetings of the Rhombicosidodecahedron

Posted on 12 October 2015 by RobertLovesPi

To make a faceted version of the rhombicosidodecahedron, one first (1) starts with a rhombicosidodecahedron, one of the Archimedean solids, then (2) removes the faces and edges of this polyhedron, leaving all the vertices in place, and then (3) connects these vertices in a different way than they were connected in the original polyhedron, forming new edges and faces. Faceting is the reciprocal operation to polyhedral stellation.

This polyhedron was made using Stella 4d, software available here.