One of Many Possible Facetings of the Rhombic Triacontahedron

Faceted Rhombic Triaconta

The simplest way I can explain faceting is that it takes a familiar polyhedron’s vertices, and then connects them in unusual ways, so that you obtain different edges and faces. If you take the convex hull of a faceted polyhedron, it returns you to the original polyhedron.

This was created using Stella 4d, software available (including as a free trial download) right 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.

There Are Many Faceted Versions of the Dodecahedron. This One Is the Dual of the Third Stellation of the Icosahedron.

Faceted Dodeca

The twelve purple faces of this faceted dodecahedron show up on Stella 4d‘s control interface as {10/4} star decagons, which would make them each have five pairs of two coincident vertices. I’m informally naming this special decagon-that-looks-like-a-pentagram (or “star pentagon,” if you prefer) the “antipentagram,” for reasons which I hope are clear.

Stella 4d, the program I use to make most of my polyhedral images, may be tried for free at http://www.software3d.com/Stella.php.

One Faceting, Each, of the Snub Cube and Snub Dodecahedron

Faceted snub cube

These are facetings of the snub cube (above) and snub dodecahedron (below). I made both using Stella 4d, software you can try for yourself right here.

faceted Snub Dodeca

A Faceting of the Truncated Dodecahedron, Together with Its Dual

Faceted Trunc Dodeca

This faceting of the truncated dodecahedron, one of many, was made with Stella 4d, software you can buy, or try for free, here. Here is its dual, below.

dual of a faceted trunc dodeca

Another Faceting of the Great Rhombicosidodecahedron

Faceted Trunc Icosidodeca

This could also be called one of many possible faceted truncated icosidodecahedra. I made it using Stella 4d, which you can try and/or buy here. Faceting is the reciprocal operation of stellation, and involves connecting the vertices of a polyhedron into faces which are unlike those of the original polyhedron. At least some, and sometimes all, of the faceted faces intersect each other, inside the polyhedron’s convex hull, as is the case here.

For comparison, here is that convex hull: a (non-faceted) great rhombicosidodecahedron, also made using Stella.

Trunc Icosidodeca

For a different faceting of this polyhedron, just look here: https://robertlovespi.wordpress.com/2013/11/19/a-faceting-of-the-great-rhombicosidodecahedron/

Faceted Snub Dodecahedron

Faceted Snub Dodeca

Facetings are created by joining vertices to other vertices, but not choosing the vertices in the usual manner, which results in new positions for edges and faces. Faceting is also the reciprocal-function for polyhedral stellation. This is one of many possible facetings of the snub dodecahedron, and I created it using Stella 4d, which you can find here.

Creating a Faceting of the Truncated Icosahedron

To make a faceted polyhedron, vertices of the original polyhedron are connected in new ways, to create a different set of faces and edges. To make this particular faceting of the truncated icosahedron, I first connected all vertices in the configuration shown below, to make irregular decagonal faces in the interior of the solid.

fti

In this next pic, the decagonal faces formed above are shown in red, and a new set of pentagonal faces is being created.

fti2

For a polyhedron to be considered mathematically valid, faces must meet in pairs at each edge. To accomplish that, I had to create another set of pentagonal faces, this one smaller than the last, as shown below.

fti3

Here’s the completed polyhedron, with each face-type having its own color.

Faceted Trunc Icosa

This next image is of the same polyhedron, but with a different coloring-scheme. In this second version, each face has its own color — except for faces which are parallel, with those faces given the fame color.

Faceted Trunc Icosa 2

I created these images using Stella 4d: Polyhedron Navigator, a program which is available here.

A Faceting of the Snub Dodecahedron

The snub dodecahedron is chiral, meaning it appears in left- and right-handed forms. This faceted version, where the same set of vertices is connected in different ways (compared to the original), possesses the same property.

Faceted Snub Dodeca

Chiral polyhedra can always be tranformed into interesting polyhedral compounds by combining them with their own mirror-images. If this is done with the polyhedron above, you get this result, presented with a different coloring-scheme.

Compound of enantiomorphic pair

Both of these images were created using Stella 4d:  Polyhedron Navigator, and you may try it at www.software3d.com/Stella.php.