The Dual of a Geodesic Rhombicosidodecahedron

This polyhedron has, as faces, a dozen regular pentagons, thirty rhombi, and sixty irregular heptagons. I made this using Stella 4d, which is available as a free trial download at http://www.software3d.com/Stella.php.

A Rhombicosidodecahedron, with the Square Faces Rendered Invisible

This polyhedron’s square faces are hidden from view, so that you can see both the front and back of the solid as it rotates. I made this using Stella 4d: Polyhedron Navigator, which you can try for yourself, free, at http://www.software3d.com/Stella.php.

Rhombicosidodeca

The Eighteenth Stellation of the Rhombicosidodecahedron Is an Interesting Polyhedral Compound

Rhombicosidodeca 18th stellation and an interesting compound

The 18th stellation of the rhombicosidodecahedron, shown above, is also an interesting compound. The yellow component of this compound is the rhombic triacontahedron, and the blue-and-red component is a “stretched” form of the truncated icosahedron. 

This was made using Stella 4d, which you can try for free right here.

The Equal-Edge-Length, Pyramid-Augmented Rhombicosidodecahedron

Augmented Rhombicosidodeca.gif

The only difference between these two images is that the lower one is in “rainbow color mode.” Both were created using Stella 4d, which you can try for free at this website.

Augmented Rhombicosidodeca rb.gif

I once made a physical model of this thing, when I was still new to the study of polyhedra. I wish I still had it, but it was lost many years ago.

An Expansion of the Rhombicosidodecahedron

An expansion of the RID with 122 faces 30 rhombi 60 almostsquares 12 pentagons and twenty triangles

This expanded version of the rhombicosidodecahedron has, as faces, 30 rhombi, 60 almost-square trapezoids, twelve regular pentagons, and twenty equilateral triangles, for a total of 122 faces. I made it using Stella 4d, software you may try for free at http://www.software3d.com/Stella.php.

I call this an “expansion of the rhombicosidodecahedron” because it is similar in appearance to that Archimedean solid. However, it is formed by augmenting the thirty faces of a rhombic triacontahedron with prisms, taking the convex hull of the result, and then using Stella‘s “try to make faces regular” function on that convex hull.

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