A Toroidal Truncated Icosahedron

Augmented Rhombic Triaconta

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.

Augmented Rhombic Triaconta rb

A Rhombic Triacontahedron, Covered with Tessellations

The tessellations of the faces of this rhombic triacontahedron first appeared in my last post here. For putting the whole thing together and creating this rotating .gif, I used a program called Stella 4d. If you want to, you can try Stella for free at this website.

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.

A Rhombic Triacontahedron, Constructed From Other Polyhedra

The components of this toroidal polyhedron are 32 rhombicosidodecahedra, 120 pentagonal prisms, and 60 dodecahedra. I assembled it using Stella 4d, a program you can try for free at http://www.software3d.com/Stella.php. Three different coloring-schemes are shown here.

A Rhombic Triacontahedron with Faces Which Are Tessellated

Rhombic Triaconta.gif

I used three programs to make this: Stella 4d, Geometer’s Sketchpad, and MS-Paint. You can try Stella for free at http://www.software3d.com/Stella.php.

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.

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.

ID of RTCs edge

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

ID of RTCs triangle

Next, a pentagon is attached to this triangle.

ID of RTCs pent and triangle

Next, the pentagonal ring is surrounded by triangles.

ID of RTCs star.gif

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.

ID of RTCs pentagonal rotunda.gif

Adding the other half completes the icosidodecahedron.

ID of RTCs complete.gif

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.

ID of RTCs complete rainbow.gif

A Polyhedral Journey, Beginning With an Expansion of the Rhombic Triacontahedron

The blue figure below is the rhombic triacontahedron. It has thirty identical faces, and is one of the Catalan solids, also known as Archimedean duals. This particular Catalan solid’s dual is the icosidodecahedron.

Rhombic Triaconta

I use a program called Stella 4d (available here) to transform polyhedra, and the next step here was to augment each face of this polyhedron with a prism, keeping all edge lengths the same.

Rhombic Triaconta augmented

After that, I created the convex hull of this prism-augmented rhombic triacontahedron, which is the smallest convex figure which can enclose a given polyhedron.

Convex hull

Another ability of Stella is the “try to make faces regular” function. Throwing this function at this four-color polyhedron above produced the altered version below, in which edge lengths are brought as close together as possible. It isn’t possible to do this perfectly, though, and that is most easily seen in the yellow faces. While close to being squares, they are actually trapedoids.

ch after ttmfr

For the next transformation, I looked at the dual of this polyhedron. If I had to name it, I would call it the trikaipentakis icosidodecahedron. It has two face types: sixty of the larger kites, and sixty of the smaller ones, also.

ch after ttmfr dual

Next, I used prisms, again, to augment each face. The height used for these prisms is the length of the edges where orange kites meet purple kites.

aug ch after ttmfr dual

Lastly, I made the convex hull of the polyhedron above. This convex hull appears below.

Convex hull again