# The Eleventh Stellation of the Truncated Octahedron Is an Interesting Polyhedral Compound

This compound has three parts: two tetrahedra, plus one smaller cube. I made it using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.

# A Compound of the Great Stellated Dodecahedron and the Great Dodecahedron

In the picture above, each component of this compound has its own color. In the one below, each set of parallel faces is given a color of its own.

These images were made using Stella 4d, software you may try for yourself at this website.

# Three Versions of a Compound of the Great and Small Stellated Dodecahedra

In the first version of this compound shown here, the great stellated dodecahedron is shown in yellow, while the small stellated dodecahedron is shown in red.

In the next version, each face has its own color, except for those in parallel planes, which have the same color.

Finally, the third version is shown in “rainbow color mode.”

All three of these images were created using Stella 4d: Polyhedron Navigator, software you can try for free right here.

# A Pyritohedral, Stellated Polyhedron, and Its Convex Hull

To make this polyhedron using Stella 4d (available here), I began with the dodecahedron, dropped the symmetry of the model from icosahedral to tetrahedral, and then stellated it thirteen times.

This stellated polyhedron has pyritohedral symmetry, but this is easier to see in its convex hull:

The eight blue triangles in this convex hull are equilateral, while the twelve yellow ones are golden isosceles triangles.

# A Second Type of Double Icosahedron, and Related Polyhedra

After seeing my post about what I called the “double icosahedron,” which is two complete icosahedra joined at one common triangular face, my friend Tom Ruen brought my attention to a similar figure he likes. This second type of double icosahedron is made of two icosahedra which meet at an internal pentagon, rather than a triangular face. Tom jokingly referred to this figure as “a double patty pentagonal antiprism in a pentagonal pyramid bun.”

It wasn’t hard to make this figure using Stella 4d, the program I use for polyhedral manipulation and image-creation (you can try it for free here), but I didn’t make it out of icosahedra. It was easier to make this figure from gyroelongated pentagonal pyramids, or “J11s” for short. This polyhedron is one of the 92 Johnson solids.

To make the polyhedron Tom had brought to my attention, I simply augmented one J11 with another J11, joining them at their pentagonal faces. Curious about what the dual of this solid would look like, I generated it with Stella.

The dual of the double J11 appears to be a modification of a dodecahedron, which is no surprise, for the dodecahedron is the dual of the icosahedron.

I next explored the stellation-series of the double J11, and found several attractive polyhedra there. This one is the double J11’s 4th stellation.

The next polyhedron shown is the double J11’s 16th stellation.

Here is the 30th stellation:

I also liked the 43rd:

The next one shown is the double J11’s 55th stellation.

Finally, the 56th stellation is shown below. These stellations, as well as the double J11 itself, and its dual, all have five-fold dihedral symmetry.

Having “mined” the double J11’s stellation-series for interesting polyhedra, I next turned to zonohedrification of this solid. The next image shows the zonohedron based on the double J11’s faces. It has many rhombic faces in two “hemispheres,” separated by a belt of octagonal zonogons. This zonohedron, as well as the others which follow, all have ten-fold dihedral symmetry.

Zonohedrification based on vertices produced this result:

The next zonohedron shown was formed based on the edges of the double J11.

Next, I tried zonohedrification based on vertices and edges, both.

Next, vertices and faces:

The next zonohedrification-combination I tried was to add zones based on the double J11’s edges and faces.

Finally, I ended this exploration of the double J11’s “family” by adding zones to build a zonohedron based on all three of these polyhedron characteristics: vertices, edges, and faces.

# Selected Stellations of the Truncated Dodecahedron

This is the truncated dodecahedron. It is one of the Archimedean solids.

This polyhedron has a long stellation-series, from which I selected several on aesthetic grounds. The figure immediately below is the truncated dodecahedron’s 16th stellation.

Here is the 21st stellation.

It’s easy to stellate polyhedra rapidly, and make many other changes to them, with Stella 4d: Polyhedron Navigator. You can try it for free at http://www.software3d.com/Stella.php.

The stellation shown immediately above is the 25th, and the one shown immediately below is the 27th.

Here is the next stellation: the 28th. Unlike the ones shown above, it is chiral.

This is the truncated dodecahedron’s 31st stellation.

This one is the 38th stellation.

This one is the 44th.

The last one shown here is called the truncated dodecahedron’s final stellation because, if it is stellated once more, it returns to the original truncated dodecahedron.

# Some Tetrahedral Stellations of the Truncated Cube

I created these with Stella 4d, which you may try for free at this website. To make a given polyhedral stellation appear larger, simply click on it.