It’s quite an informal way to define it, but pyritohedral symmetry is the symmetry-type of a standard volleyball. These images of pyritohedral polyhedra were made using Stella 4d, software available at http://software3d.com/Stella.php.
I made the polyhedron above by performing a faceting of the dodecahedron, and only realized, after the fact, that I had stumbled upon one of the uniform polyhedra, a set of polyhedra I have not yet studied extensively. It is called the small ditrigonal icosidodecahedron, and its faces are twelve star pentagons and twenty equilateral triangles, with the triangles intersecting each other. Below is its fifth stellation, which appears to be a compound of a yellow dodecahedron and a red polyhedron which I do not (yet) recognize, although it does look quite familiar.
Both images were created using Stella 4d, software you can try right here.
Each of these dodecahedra were modified by truncations at exactly four of their three-valent vertices. As a result, each has four equilateral triangles as faces. In the one above, the Platonic dodecahedron’s pentagonal faces are modified into a dozen irregular hexagons by these truncations, while, in the one below, the rhombic dodecahedron’s faces are modified into twelve irregular pentagons.
Both of these polyhedra were created using Stella 4d, software you can try for yourself at this website.
All of these were made using Stella 4d, a program you can try for free at http://www.software3d.com/Stella.php.
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.
Zonohedra are polyhedra made completely of faces which are zonogons. A zonogon is a polygon which:
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.
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.
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.
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.”
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.
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