92 Dodecahedra, Arranged as a Single Rhombic Triacontahedron

Posted on 5 November 2015 by RobertLovesPi

With 92 dodecahedra, if you arrange them just right, you can make a model of a rhombic triacontahedron:

For purposes of comparison, here is what the rhombic triacontahedron normally looks like:

Also, referring back to the first model shown, here is a picture of just one of the red rhombi-made-of-dodecahedra.

The first polyhedron shown in this post has an interesting dual, as well. Here it is, colored by face-type (position within the overall shape):

Here is another view of the dual, colored by number of edges per face.

Here’s one more view of the dual, in “rainbow color mode.”

Returning to the original model, at the top of this post, here’s what it looks like, if colored by face type:

Here’s one more view, in “rainbow color mode.”

All of these images were created using Stella 4d, a program you can buy, or try for free, right here.

Building a Rhombic Enneacontahedron, Using Icosahedra and Elongated Octahedra

Posted on 5 November 2015 by RobertLovesPi

With four icosahedra, and four octahedra, it is possible to attach them to form this figure:

This figure is actually a rhombus, but the gap between the two central icosahedra is so small that this is hard to see. To remedy this problem, I elongated the octahedra, thereby creating this narrow rhombus:

It is also possible to use the same collection of polyhedra to make a wider rhombus, as seen below.

These aren’t just any rhombi, either, but the exact rhombi found in the polyhedron below, the rhombic enneacontahedron. It has ninety rhombi as faces: sixty wide ones, and thirty narrow ones.

As a result, it is possible to use the icosahedra-and-elongated-octahedra rhombi, above, to construct a rhombic enneacontahedron made of these other two polyhedra. The next several images show it under construction (I “built” it using Stella 4d, available at this website), culminating with the complete figure.

Lastly, I made one more image — the same completed shape, but in “rainbow color mode.”

A Chiral Polyhedron with 120 Pentagonal Faces, Together with Its Dual

Posted on 4 November 2015 by RobertLovesPi

In this chiral polyhedron, sixty faces are the small, purple pentagons, while the other sixty are the larger, orange pentagons. The next image shows its dual.

Both images were created with Stella 4d, a program you can buy, or try for free, at this website: http://www.software3d.com/Stella.php.

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.

A Forgotten Mandala, from 2010

Posted on 27 October 2015 by RobertLovesPi

Someone found this, and “liked” it, in my old Facebook pictures. I had forgotten all about it, until this happened. It is a mandala, made of rhombi, with nine-fold symmetry, made in 2010 with Geometer’s Sketchpad — two years before I started this blog.

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.