A Faceted Great Rhombcuboctahedron


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.

Compound of the Great and Small Stellated Dodecahedra

Compound of Great Stellated Dodeca and Small Stellated Dodeca color scheme 2

In this compound, as shown above, the small stellated dodecahedron is yellow, while the red polyhedron is the great stellated dodecahedron. Below, the same compound is colored differently; each face has its own color, unless faces are in parallel planes, in which case they have the same color.

Compound of Great Stellated Dodeca and Small Stellated Dodeca color scheme 1

Making a physical model of this compound would have taken most of the day, if I did it using such things as posterboard or card stock, compass, ruler, tape, scissors, and pencils. For the first several years I built models of polyhedra, starting about nineteen years ago, that was how I built such models. The virtual polyhedra shown above, by contrast, took about ten minutes to make, using Stella 4d: Polyhedron Navigator, which you can try for free, or purchase, here.

There’s also a middle path: using Stella to print out nets on cardstock, cutting them out, and then taping or gluing these Stella-generated nets together to make physical models. I haven’t spent much time on this road myself, but I have several friends who have, including the creator of Stella. You can see some of his incredible models here, and some amazing photographs of other Stella users’ paper models, as well as some in other media, at this website.

The Greatly Augmented Rhombicosidodecahedron

Greatly Augmented Rhombicosidodeca

I call this variant on the rhombicosidocahedron “greatly augmented” because it was formed by augmenting each pentagonal face of a central rhombicosidodecahedron with great dodecahedra, while each triangular face is augmented with great icosahedra. It was made using Stella 4d, which may be found here.

Stellating the Great Dodecahedron, by Twentieths, to Beethoven’s Ninth

In this video, the great dodecahedron is stellated, by twentieths, into the great stellated dodecahedron, while a selection from Ludwig van Beethoven’s Ninth Symphony plays. The images for this video were created using Stella 4d, a program you can try for yourself (free trial download available), right here: http://www.software3d.com/Stella.php