A Survey of Polyhedra with Pyritohedral Symmetry

The simplest way for many to understand pyritohedral symmetry is simply to realize that it is the symmetry of the seams in a volleyball. The first time I encountered this unusual symmetry-type was in the golden icosahedron I blogged about here, a figure which much resembles this pyritohedral icosahedron, except the dozen isosceles triangles in this one have a leg-to-base ratio which is not the golden ratio.

non golden pyritohedral icosahedron

Earlier today, I went on a search for polyhedra with pyritohedral symmetry. I found several, but the worthwhile findings from the search are far from exhausted. Here are some others I found, exploring and manipulating polyhedra using Stella 4d, which you can try at this website.

another pyritohedral version of an icosahedronIn the version of the pyritohedral icosahedron above, the twelve green triangles have become heptagons which use very short sides to approximate triangles. The one below is of a similar figure, but one in which truncations has happened, so I call it a truncated pyritohedral icosahedron.

pyritohedral version of a truncated icosahedron

There also exist many pyritohedral polyhedra based, more or less, on the cube. These are a few I have found:

pyritohedral cube

pyritohedral cube variant

another pyritohedral cube

Now, is this next one a pyritohedral cube, or a pyritohedral dodecahedron? A case could be made for either, so it inhabits a “gray zone” between varying categories.

pyritohedral dodecahedron

Here is a pyritohedral icosidodecahedron:

pyritohedral icosidoecahedronl

This one could probably be described in multiple ways, also, but it looks, to me, like a rhombic dodecahedron with its six four-valent vertices being double-truncated in a pyritohedral manner, with pairs of isosceles trapezoids appearing where the truncations took place.

Convex hull of icosahedron plus CO

One thing that this one, and the last, have in common is that the largest faces are heptagons. It appears to be a pyritohedral dodecahedron which has been only partially truncated.

12 helptagons and 8 trianlgesl

This survey could not have been performed without a program called Stella 4d, which I rely on heavily for polyhedral investigations. It may be purchased, or tried for free, at http://www.software3d.com/Stella.php.

If You Have Enough Platonic Dodecahedra Around, and Glue Them Together Just Right, You Can Make a Rhombic Triacontahedron.

Aren’t you glad to know that? As soon as I found out isocahedra can form a rhombic dodecahedron (see last post), I knew this would be true as well. Why? Zome explains why, actually. It’s at http://www.zometool.com. Anything buildable with yellow Zome can be built out of icosahedra. Dodecahedra con form anything buildable with red Zome. Finally, if you can make it with blue Zome, it can be built out of rhombic triacontahedra. It follows that rhombicosidoecahedra can build anything Zome-constructable — but one look at a Zomeball makes that easy to believe, since Zomeballs are modified rhombicosidodecahedra.

Anyway, here’s the rhombic triacontahedron, made of dodecahedra:

Augmented Dodeca

[Image created with Stella 4d; see http://www.software3d.com/Stella.php for more info re: this program.]

A Compound of Three Square Dipyramids

The reason I am not calling this a compound of three octahedra is that the faces of the dipyramids aren’t quite equilateral. They are, however, isosceles.

compound of three square dipyramids

This was created with Stella 4d, which you can buy, or try for free, right here.

Happy Second Anniversary of Your Simulated Existence

386145

The world ended on this day in 2012 — December 21 — when the Mayan calendar began a new cycle. We now secretly live in a computer simulation run by highly advanced ancient Mayan aliens. They have authorized me to wish you a happy second anniversary of the end of your previous existence.

[Image credit: within this simulation, you can find this picture at http://wall.alphacoders.com/by_sub_category.php?id=206132.]

A Partially-Invisible Rhombicosidodecahedron, and One of Its Stellations

Rhombicosidodeca

The polyhedron above originally had thirty yellow square faces, but I rendered them invisible so that the interior structure of this polyhedron could be seen.

When stellating such a partially-invisible figure, the new faces “inherited” from the “parent polyhedron” are either visible or invisible, depending on which type of face they are derived from. This makes for a very unusual look for some stellations, such as this, the rhombicosidodecahedron’s 50th:

Rhombicosidodeca w inv squares 50th stellations

I created these images using a program called Stella 4d: Polyhedron Navigator. You may try it for yourself at http://www.software3d.com/Stella.php.

The Seven Zonish Dodecahedra with Zones Added Based on Faces, Edges, and/or Vertices

If a zonish dodecahedron is created with zones based on the dodecahedron’s vertices, here is the result.

zonish dod v

If the same thing is done with edges, this is the result — an edge-distorted version of the great rhombicosidodecahedron.

zonish dodeca edges only

Another option is faces-only. Although I haven’t checked the bond-lengths, this one does have the general shape of the most-symmetrical 80-carbon-atom fullerene molecule. Also, this shape is sometimes called the “pseudo-truncated-icosahedron.”

zonish dodec faces only

The next zonish dodecahedron has had zones added based on the dodecahedron’s faces and edges, both.

zonish dodeca e & f

Here’s the one for vertices and edges.

zonish dodec v & e

Here’s the one for faces and vertices.

zonish dodec v & f

Finally, the last of this set of seven has had zones added based on all three: faces, vertices, and edges.

zonish dodec vfe

All seven of these were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.