Six Random Starsplosions of Non-Convex Polyhedra with Cuboctahedral Symmetry

Posted on 7 November 2015 by RobertLovesPi

All of these were made using Stella 4d, a program you can try for free at http://www.software3d.com/Stella.php.

The 21st and 22nd Stellations of the Truncated Dodecahedron

Posted on 7 November 2015 by RobertLovesPi

Stellation of a polyhedron involves extending its faces and/or edges into space to form other polyhedra, often with a star-like appearance, which is where the words “stellation,” “stellate,” and “stellated” originate. (“Stella” is Latin for “star.”)

Since this can be done repeatedly, long stellation-series exist for many polyhedra. In the case of the truncated dodecahedron, it was the 21st and 22nd stellations which I found the most aesthetically pleasing.

Here is the 21st stellation of this polyhedron:

And here is the 22nd:

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

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.

One of Many Possible Facetings of the Rhombic Triacontahedron

Posted on 7 November 2015 by RobertLovesPi

The simplest way I can explain faceting is that it takes a familiar polyhedron’s vertices, and then connects them in unusual ways, so that you obtain different edges and faces. If you take the convex hull of a faceted polyhedron, it returns you to the original polyhedron.

This was created using Stella 4d, software available (including as a free trial download) right here: http://www.software3d.com/Stella.php.

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.