Two Rhombic Triacontahedra, Each Decorated with Birthday Stars

In yesterday’s post, I unveiled my annual birthday star for my new age, 54. Today, I’m placing that 54-pointed star on each of the thirty faces of a rhombic triacontahedron. I use a program called Stella 4d (free trial available right here) to do this, and it allows images on polyhedron-faces to either be placed inside the face, or around the face. Here’s the “inside” version:

And here is the “around each face” version:

Which one do you like better?

A Toroidal Truncated Icosahedron

The components of this toroid are sixty rhombic triacontahedra, as well as ninety rhombic prisms with lateral edges three times as long as their base edges. I made this using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

A Rhombic Triacontahedron, Decorated with Geometric Artwork

To make this rotating .gif, I navigated to the rhombic triacontahedron in Stella 4d, and then loaded images onto its thirty faces, with the image being the one I blogged in the post right before this one. This program, Stella, has a free trial download you can get right here.

Using Rhombic Triacontahedra to Build an Icosidodecahedron

These two polyhedra are the icosidodecahedron (left), and its dual, the rhombic triacontahedron (right).

One nice thing about these two polyhedra is that one of them, the rhombic triacontahedron, can be used repeatedly, as a building-block, to build the other one, the icosidodecahedron. To get this started, I first constructed one edge of the icosidodecahedron, simply by lining up four rhombic triacontahedra.

Three of these lines of rhombic triacontahedra make one of the icosidodecahedron’s triangular faces.

Next, a pentagon is attached to this triangle.

Next, the pentagonal ring is surrounded by triangles.

More triangles and pentagons bring this process to the half-way point. If we were building a pentagonal rotunda (one of the Johnson solids), this would be the finished product.

Adding the other half completes the icosidodecahedron.

All of these images were created using Stella 4d: Polyhedron Navigator. You may try this program yourself, for free, at http://www.software3d.com/Stella.php. The last thing I did with Stella, for this post, was to put the finished model into rainbow color mode.

The Great Rhombicosidodecahedron, Built from Rhombic Triacontahedra, and Its Dual

The great rhombicosidodecahedron is also known as the truncated icosidodecahedron (and, confusingly, several other names). Regardless of what it’s called, these pictures demonstrate that this Archimedean solid can be constructed using rhombic triacontahedra as building-blocks.

First, here’s one in the same color I used for the decagonal ring of rhombic triacontahedra in the last post:

The next one is identical, except I used “rainbow color mode” for it.

Also, just in case you’re curious, here’s the dual of this polyhedron-made-of-polyhedra — this time, colored by face-type.

These virtual models were all built using Stella 4d, software you may buy, or try for free, right here.

Decagonal Ring of Rhombic Triacontahedra

Ten rhombic triacontahedra fit perfectly into a decagonal ring. It’s not a “near-miss” — the fit is exact.