One of Many Faceted Versions of the Great Rhombicosidodecahedron

The great rhombicosidodecahedron is also known as the truncated icosidodecahedron. I created this faceting of it using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.

A Faceted Version of the Truncated Cube

This is the truncated cube, which is one of the Archimedean solids.

Convex hull.gif

To make a faceted version of this solid, one must connect at least some of the vertices in different ways. Doing that creates new faces.

Faceted Trunc Cube 8 hexagons blue and yellow triangles.gif

This faceted version of the truncated cube includes eight blue equilateral triangles, eight larger, yellow equilateral triangles, and eight irregular, red hexagons. It’s easy to spot the yellow and blue triangles, but seeing the red hexagons is harder. In the final picture here, I have hidden all faces except for three of the hexagons, so that their positions can be more easily seen.

Faceted Trunc Cube 8 hexagons blue and yellow triangles some parts hidden.gif

I made all three of these images using Robert Webb’s program called Stella 4d: Polyhedron Navigator. It is available for purchase, or as a free trial download, at http://www.software3d.com/Stella.php.

Seven Different Facetings of the Truncated Icosahedron

Trunc Icosa.gif

The polyhedron above is the truncated icosahedron, widely known as the pattern for most soccer balls. In the image below, the faces and edges have been hidden, leaving only the vertices.

Trunc Icosa vertices only

To make a faceted version of this polyhedron, these vertices must be connected in novel ways, creating new edges and faces. There are many faceted versions of this polyhedron, of which seven are shown below.

Faceted Trunc Icosa

Faceted Trunc Icosa 8

Faceted Trunc Icosa 7

Faceted Trunc Icosa 5.gif

Faceted Trunc Icosa 4.gif

Faceted Trunc Icosa 3

Faceted Trunc Icosa 2.gif

I used Stella 4d to make these polyhedral images, and you’re invited to try the program for yourself at http://www.software3d.com/Stella.php.

Ten Different Facetings of the Rhombicosidodecahedron

This is the rhombicosidodecahedron. It is considered by many people, including me, to be the most attractive Archimedean solid.

Rhombicosidodeca

To create a faceted polyhedron, the first step is to get rid of all the faces and edges, leaving only the vertices, as shown below.

Vertices of a RID

In the case of this polyhedron, there are sixty vertices. To create a faceted version of this polyhedron, these vertices are connected by edges in ways which are different than in the original polyhedron. The new positioning of edges defines new faces, often in the interior of the original polyhedron. Here is one such faceting, with the red hexagonal faces in the interior of the now-removed original polyhedron.

Faceted RID

The rhombicosidodecahedron can be faceted in many different ways. I don’t know how many possible facetings this polyhedron has, but it is a finite number much larger than the ten shown in this post. Here’s another one.

Faceted RID 2

In faceted polyhedra, many faces intersect other faces, as is the case with the red and yellow faces above. The next faceting demonstrates that faceted polyhedra are sometimes incredibly complex.

Faceted Rhombicosidodeca 3

Faceted polyhedra can even contain holes that go all the way through the solid, as seen in the next image.

Faceted Rhombicosidodeca 4.gif

Sometimes, a faceting of a non-chiral polyhedron can be chiral, as seen below. Chiral polyhedra are those which exist in “left-handed” and “right-handed” reflections of each other.

Faceted Rhombicosidodeca 7 chiral

Any chiral polyhedron may be fused with its mirror image to form a compound, and that’s exactly what was done to produce the next image. In addition to being a polyhedral compound, it is also, itself, another faceted version of the rhombicosidodecahedron.

Compound of enantiomorphic pair.gif

All these polyhedral manipulations and gif-creations were performed using a program called Stella 4d: Polyhedron Navigator. If you’d like to try Stella for yourself, please visit http://www.software3d.com/Stella.php, where a free trial download is available.

The rest of the rhombicosidodecahedron-facetings needed to round out this set of ten are shown below, without further comment.

Faceted Rhombicosidodeca 10

Faceted Rhombicosidodeca 9

Faceted Rhombicosidodeca 5

Faceted Rhombicosidodeca 6

ZigZag: A Faceting of the Great Rhombcuboctahedron

Faceted Great Rhombcuboctahedron

I made this using Stella 4d: Polyhedron Navigator. You can try this program for free at http://www.software3d.com/Stella.php.

A Faceted Truncated Icosahedron

Faceted truncated icosahedron

This is one of many possible facetings of the truncated icosahedron. I made it using Stella 4d, which you can try for yourself at this website: http://www.software3d.com/Stella.php.