In the two posts right before this one, I’ve been exploring simple structures made of modified rhombicosidodecahedra, and today I’m going to post a much larger, more complex one. Here’s the rhombicosidodecahedron — the original Archimedean solid which started all of this:
The modified forms of this polyhedron which I’m using as building-blocks are all among the 92 Johnson solids. Here are the two which have already appeared in the last two posts on this blog: the diminished rhombicosidodecahedron (J76) and the parabidiminished rhombicosidodecahedron (J80).
For this new, more ambitious construction, I’m going to need some more pieces, starting with the metabidiminished rhombicosidodecahedron (J81), which will be useful to make angles.
The Johnson solid called the tridiminished rhombicosidodecahedron (J83) can be used to make three-valent vertices.
Finally, here’s the more complex structure for which I needed all these pieces. It could be extended outwards indefinitely, in a manner similar to the tessellation of the plane with regular hexagons.
To make these polyhedral images, I use a program called Stella 4d. If you’d like to give it a try, for free, please visit this website.
This is the rhombicosidodecahedron, one of the thirteen Archimedean solids.
Several of the 92 Johnson solids are modified forms of this polyhedron, such as J76, the diminished rhombicosidodecahedron (shown below). It is formed by removal of a pentagonal cupola from a rhombicosidodecahedron, exposing a decagonal face.
Another variant of this Archimedean solid may be created by removing two pentagonal cupolas, exposing decagons on opposite sides of the figure. This solid, J80, is called the parabidiminished rhombicosidodecahedron.
Two J76s and one J80 can then be joined together, at their decagonal faces, to form this: the triple rhombicosidodecahedron.
I made these using Stella 4d, a program you can try for free at this website.
This is a rhombicosidodecahedron, one of the Archimedean solids.
If one pentagonal cupola is removed from this polyhedron, the result is the diminished rhombicosidodecahedron, which is one of the Johnson solids (J76).
The next step is to take another J76, and attach it to the first one, so that their decagonal faces meet.
I’m calling the result the “double rhombicosidodecahedron.”
I did these manipulations of polyhedra and their images with a program called Stella 4d: Polyhedron Navigator. There’s a free trial download available, if you’d like to try the program for yourself, and it’s at this website.
To make this cluster, start with a cuboctahedron, then augment each of its square faces with rhombicosidodecahedra. Although the cuboctahedron has cuboctahedral symmetry, this cluster does not — rather, it has tetrahedral symmetry. I created this using Stella 4d, which is available here.
This “metarhombicosidodecahedron” took a long time to build, using Stella 4d, which you can find at http://www.software3d.com/Stella.php — so, when I finished it, I made five different versions of it, by altering the coloring settings. I hope you like it.
Since rhombic triacontahedra can form pentagonal rings, triangular rings, and square rings, I wanted to find out if a rhombicosidodecahedron could be built out of these building blocks. As you can see here, the attempt was a success. Each rhombic triacontahedron which appears here is located at the vertex of a rhombicosidodecahedron.