A Rhombic Triacontahedron, Vertices Surrounded By Smaller Rhombic Triacontahedra, and Its Interesting Dual

The first image shows a central yellow rhombic triacontahedron, with smaller, blue rhombic triacontahedra attached to each of its thirty-two vertices. The second polyhedron shown is the dual of the first one, with colors chosen by the number of sides per face in the second image — pentagons red, and triangles yellow. The convex hull of this second polyhedral complex shown would be an icosidodecahedron, itself the dual of the rhombic triacontahedron.

I use software called Stella 4d: Polyhedron Navigator to make the rotating polyhedral images on this blog. You can try Stella for yourself, for free, at http://www.software3d.com/Stella.php.

Two Images of a Toroidal Rhombic Triacontahedron Made of 212 Dodecahedra

I made these using Stella 4d, a program you can try as a free trial download at http://www.software3d.com/Stella.php.

Two Rhombic Polyhedra with Tessellated Faces

These polyhedra are the rhombic dodecahedron (above), and the rhombic triacontahedron (below).

I made both of these using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php. The tessellation on the faces of these polyhedra first appeared right here on this blog, in the post just before this one.

The Golden Rhombus, the Rhombic Triacontahedron, and the Rhombic Hexecontahedron

There’s a special rhombus which is called the “golden rhombus,” because its diagonals are in the golden ratio. To construct it with compass and straight edge, you first construct a golden rectangle (shown with blue edges and a yellow interior), and then connect the midpoints of its sides to form a rhombus (with edges shown in red).

Several polyhedra can be made which use golden rhombi as their faces. The most well-known of these polyhedra is the rhombic triacontahedron, which has 30 such faces. It is the dual of the icosidodecahedron.

If the rhombic triacontahedron is stellated 26 times, the result is the (non-convex) rhombic hexecontahedron. It has 60 golden rhombi as faces.

Both of these polyhedra can be constructed with Zometools (available at http://www.zometool.com). With white Zomeballs and red Zomestruts, these polyhedra look a lot like this:

The flat image at the top of this post was created using Geometer’s Sketchpad and MS-Paint. The four rotating polyhedral images were created using Stella 4d: Polyhedron Navigator, which you can purchase, or try for free, at http://www.software3d.com/Stella.php.

A Collection of Four Polyhedra Decorated with Mandalas

First, a cuboctahedron.

Rotating Cubocta with rotating mandalasNext, its dual, the rhombic dodecahedron.

Rotating RD with rotating mandalas

And, after that, the icosidodecahedron.

Rotating Icosidodeca with rotating mandalas

And finally, its dual, the rhombic triacontahedron.

Rotating RTC with rotating mandalas

All of these rotating images were assembled using Stella 4d, available at http://www.software3d.com/Stella.php.

A Compound of an Icosahedron and the First Stellation of the Rhombic Triacontahedron

Compound of an icosahedron and the 1st stellation of the RTC

I stumbled across this compound yesterday, an example of exploratory polyhedral manipulation using Stella 4d producing an unexpected result. If you would like to experiment with a free trial download of this program, before deciding whether or not to purchase the fully-functioning version, simply click here:  www.software3d.com/Stella.php.

A Rhombic Triacontahedron, Peeking Through the Faces of an Icosahedron


A Rhombic Triacontahedron, Peeking Through the Faces of an Icosahedron

Software credit: visit http://www.software3d.com/stella.php for more information on the program used to make this rotating image. A free trial download is available.