A Zome Rhombicosidodecahedron, and Some of Its Variations

Here’s a Zome rhombicosidodecahedron, made up of white Zomeballs and short blue struts.

If you replace each of the rhombicosidodecahedron’s thirty squares with golden rectangles in a certain way, by replacing certain short struts with medium struts along the triangle/rectangle edges, you get a Zomeball made of Zome — what some have called a “metazomeball.” It has enlarged triangles, compared to the pentagons.

It’s almost possible to augment each of the Zomeball’s 62 faces with all-blue pyramids. Here’s the attempt.

The Zomeball’s twelve pentagons are augmented with pyramids, using medium struts, so that each of these pyramids’ lateral faces is a golden isosceles triangle. The twenty triangles are also augmented by pyramids, with golden isosceles triangles as lateral faces. This requires the use of long blue struts. As for the augmentation performed on the thirty golden rectangles, that’s a bit more complicated.

Here’s what the golden rectangles are augmented with, with the new pieces all being short blue struts. It isn’t quite a pyramid, nor is it quite a prism. Its lateral faces are two equilateral triangles and two golden isosceles trapezoids. It’s a five-faced polyhedron in need of a good name. Until someone comes up with a better name, then, I’m going to call it a golden pentahedron.

Here’s a close-up of a golden isosceles trapezoid, by itself. It’s made of three short struts and one medium strut.

It’s also possible to do the first change in another way. When the rhombicosidodecahedron has its squares replaced with golden rectangles, rotate the golden rectangles 90 degrees, such that it increases the size of the pentagons, rather than the triangles. This has been called an “antizomeball.”

With the antizomeball, augmentation with pyramids is possible with all 62 faces.

Here’s what the three types of pyramid used in the augmentation of the antizomeball look like, separated from the main structure.

With the antizomeball, the pentagonal pyramids’ bases are made of medium struts, and long struts make the five lateral faces into golden isosceles triangles. The triangular faces of the antizomeball are made of short struts, so medium lateral edges for these pyramids make their lateral faces golden isosceles triangles. The golden rectangles in the antizomeball are different from the ones in the Zomeball, and the changes make it possible to augment the antizomeball’s golden rectangles using four medium lateral edges, forming two golden isosceles triangles and two equilateral triangles as the lateral faces of a true pyramid.

If you’d like to try Zome for yourself, which I strongly recommend, the site to visit is http://www.zometool.com.

Modified Rhombicosidodecahedra, as Building Blocks for Larger Structures

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.

The Triple Rhombicosidodecahedron

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.

The Double Rhombicosidodecahedron

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.

A Faceted Rhombicosidodecahedron with 540 Faces

This beauty appeared unexpectedly when I was playing around with polyhedra related to the icosahedron and dodecahedron, using Stella 4d: Polyhedron Navigator. You can try this program for yourself, free, at http://www.software3d.com/Stella.php.

The Thirty Squares of a Rhombicosidodecahedron

I made this by hiding the pentagons and triangles of a rhombicosidodecahedron, then putting the remaining faces, all squares, into “rainbow color mode.” I did this using Stella 4d: Polyhedron Navigator, which you can try for free right here.

A Dodecahedron Made of Lux Blox . . . or Is It a Rhombicosidodecahedron?

This is the third polyhedral model I’ve built with Lux Blox, and the first to use the Lux trigons (the black pieces) which were added to the Lux system in 2017. If you view this polyhedron as having orange pentagonal faces, white edges, and black vertices, it’s a dodecahedron. On the other hand, it can be seen as having orange pentagonal faces, white square faces, and black triangular faces, in which case this is a rhombicosidodecahedron.

Lots of us are stuck inside because of COVID-19, and a set of Lux Blox is the perfect tool (or toy, if you prefer) to avoid boredom while we wait this thing out. You can find Lux for sale at www.luxblox.com, and delivery is fast.