Stellar Array

A great dodecahedron (red) sits in the middle of this polyhedral cluster. The polyhedra touching the one in the center are blue small stellated dodecahedra. Finally, there are yellow great stellated dodecahedra on the outside.

I assembled this polyhedral cluster using Stella 4d, which you can try for yourself at http://www.software3d.com/Stella.php.

A Compound of the Great Stellated Dodecahedron and the Great Dodecahedron

Compound of Great Stellated Dodeca and Great Dodeca.gif

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.

Compound of Great Stellated Dodeca and Great Dodeca 2

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.

Small Stellated Dodeca and Great Stellated Dodeca.gif

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

Small Stellated Dodeca and Great Stellated Dodeca 2

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

Small Stellated Dodeca and Great Stellated Dodeca 3

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

Augmenting the Dodecahedron with Great Dodecahedra

These two polyhedra are the dodecahedron (left), and the great dodecahedron (right).

Since the faces of both of these polyhedra are regular pentagons, it is possible to augment each of the dodecahedron’s twelve faces with a great dodecahedron. Here is the result.

Augmented Dodeca.gif

I used Stella 4d to make these images. You may try this program for yourself at http://www.software3d.com/Stella.php.

Augmenting the Icosahedron with Great Icosahedra

These two polyhedra are the icosahedron (left), and the great icosahedron (right).

 

Since the faces of both of these polyhedra are equilateral triangles, it is possible to augment each of the icosahedron’s twenty faces with a great icosahedron. Here is the result.

Augmented Icosa with 20 great icosas.gif

I used Stella 4d to make these images. You may try this program for yourself at http://www.software3d.com/Stella.php.

Icosahedral Cluster

augmented-great-icosa

The great icosahedron, one of the Kepler-Poinsot solids, is hidden from view at the center of this cluster. Each of its faces is augmented with a Platonic icosahedron, producing what you see here. Stella 4d is the software I used; more information about that program may be found here.

A Faceted Great Rhombcuboctahedron

faceted-trunc-cubocta

Some prefer to call the great rhombcuboctahedron the “truncated cuboctahedron,” instead. Whichever term you prefer, this is a faceted version of that Archimedean solid. I made it using Stella 4d: Polyhedron Navigator, software you may find here.

Unsquashing the Squashed Meta-Great-Rhombcuboctahedron

I noticed that I could arrange eight great rhombcuboctahedra into a ring, but that ring, rather than being regular, resembled an ellipse.

Augmented Trunc Cubocta

I then made a ring of four of these elliptical rings.

Augmented Trunc Cubocta B

After that, I added a few more great rhombcuboctahedra to make a meta-rhombcuboctahedron — that is, a great rhombcuboctahedron made of rhombcuboctahedra. However, it’s squashed. (I believe the official term for this is “oblate,” but “squashed” also works, at least for me.)

Augmented Trunc Cubocta 3

So now I’m wondering if I can make this more regular. In other words, can I “unsquash” it? I notice that even this squashed metapolyhedron has regular rings on two opposite sides, so I make such a ring, and start anew.

Augmented Trunc Cubocta a

I then make a ring of those . . . 

Augmented Trunc Cubocta AA

. . . And, with two more ring-additions, I complete the now-unsquashed meta-great-rhombcuboctahedron. Success!

Augmented Trunc Cubocta AAA

To celebrate my victory, I make one more picture, in “rainbow color mode.”

Augmented Trunc Cubocta AAAR

[All images made using Stella 4d, available here: http://www.software3d.com/Stella.php.]

Four Different Facetings of the Great Rhombcuboctahedron

faceted GRCO

Faceted Trunc Cubocta 2

Faceted Trunc Cubocta 4

Faceted Trunc Cubocta

All four of these rotating images were created using software called Stella 4d: Polyhedron Navigator. You can buy this program, or try it for free, at this website. Faceting is the inverse function of stellation, and involves connecting the vertices of an already-established polyhedron in new ways, to create different polyhedra from the one with which one started. For each of these, the convex hull is the great rhombcuboctahedron, itself.