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

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things, the majority of which are geometrical. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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2 Responses to Ten Different Facetings of the Rhombicosidodecahedron

  1. Brent says:

    Could these different facetings be interpreted as projections of the same higher dimensional object observed from different perspectives?

    Like

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