A Faceted Truncated Octahedron with Twenty Faces

Posted on 25 July 2020 by RobertLovesPi

Faceted Trunc Octa

The twenty faces of this polyhedron are six small blue squares, six interpenetrating, larger red squares, and eight irregular, interpenetrating yellow hexagons. I made it using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php. The squares are easy to find, but it can be a challenge to see the yellow hexagons. In the .gif below, all of the yellow faces but one are hidden, which should make it easier to see where the hexagons are positioned, relative to the squares.

A Faceting of the Rhombicosidodecahedron

Posted on 18 January 2020 by RobertLovesPi

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

Three Facetings of the Truncated Octahedron

Posted on 26 November 2019 by RobertLovesPi

I created these faceted polyhedra using Stella 4d, a program you can try for free at http://www.software3d.com/Stella.php.

One of Many Faceted Versions of the Great Rhombicosidodecahedron

Posted on 9 March 2019 by RobertLovesPi

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.

Seven Different Facetings of the Truncated Icosahedron

Posted on 10 June 2018 by RobertLovesPi

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.

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.

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

Posted on 28 April 2018 by RobertLovesPi

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

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.

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.

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.

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

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.

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.

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.

ZigZag: A Faceting of the Great Rhombcuboctahedron

Posted on 22 March 2018 by RobertLovesPi

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

Posted on 27 November 2017 by RobertLovesPi

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.

A Faceting of the Rhombcuboctahedron

Posted on 30 September 2017 by RobertLovesPi

Faceted Rhombicubocta

This particular faceting of the rhombcuboctahedron can also be viewed as a cluster of stella octangulae. I made it using Stella 4d, polyhedron-manipulating software you can try, for yourself, at http://www.software3d.com/Stella.php.

Standard and Faceted Versions, Side by Side, of Each of the Thirteen Archimedean Solids

Posted on 18 February 2016 by RobertLovesPi

These two polyhedra are the truncated tetrahedron on the left, plus at least one faceted version of that same Archimedean solid on the right. As you can see, in each case, the figures have the same set of vertices — but those vertices are connected in a different way in the two solids, giving the polyhedra different faces and edges.

(To see larger images of any picture in this post, simply click on it.)

The next three are the truncated cube, along with two different faceted truncated cubes on the right. The one at the top right was the first one I made — and then, after noticing its chirality, I made the other one, which is the compound of the first faceted truncated cube, plus its mirror-image. Some facetings of non-chiral polyhedra are themselves non-chiral, but, as you can see, chiral facetings of non-chiral polyhedra are also possible.

Faceted Trunc cube Compound of enantiomorphic pair

The next two images show a truncated octahedron, along with a faceted truncated octahedron. As these images show, sometimes faceted polyhedra are also interesting polyhedra compounds, such as this compound of three cuboids.

Faceted Trunc Octa and compound of three cuboids

The next polyhedra shown are a truncated dodecahedron, and a faceted truncated dodecahedron. Although faceted polyhedra do not have to be absurdly complex, this pair demonstrates that they certainly can be.

Next are the truncated icosahedron, along with one of its many facetings — and with this one (below, on the right) considerably less complex than the faceted polyhedron shown immediately above.

The next two shown are the cuboctahedron, along with one of its facetings, each face of which is a congruent isosceles triangle. This faceted polyhedron is also a compound — of six irregular triangular pyramids, each of a different color.

The next pair are the standard version, and a faceted version, of the rhombcuboctahedron, also known as the rhombicuboctahedron.

The great rhombcuboctahedron, along with one of its numerous possible facetings, comes next. This polyhedron is also called the great rhombicuboctahedron, as well as the truncated cuboctahedron.

The next pair are the snub cube, one of two Archimedean solids which is chiral, and one of its facetings, which “inherited” its chirality from the original.

The icosidodecahedron, and one of its facetings, are next.

The next pair are the original, and one of the faceted versions, of the rhombicosidodecahedron.

The next two are the great rhombicosidodecahedron, and one of its facetings. This polyhedron is also called the truncated icosidodecahedron.

Finally, here are the snub dodecahedron (the second chiral Archimedean solid, and the only other one, other than the snub cube, which possesses chirality), along with one of the many facetings of that solid. This faceting is also chiral, as are all snub dodecahedron (and snub cube) facetings.