A Zome Model of the Compound of the Icosidodecahedron and Its Dual, the Rhombic Triacontahedron

The polyhedral compound above contains an icosidodecahedron (blue) and a rhombic triacontahedron (red). In this compound, the icosidodecahedron’s edges are bisected, while the rhombic triacontahedron’s edges are split into segments with lengths in the square of the golden ratio (~2.618 to 1).

If you want Zome of your own, the place to buy it is http://www.zometool.com.

Playing with One of Stella’s “Morph Duals” Functions

“Morph Duals By Tilting To Duals” is a Stella 4d feature that I haven’t used much. Here’s what happens if you apply it to an icosahedron, at the 50% morphing level: you get the compound of the icosahedron and its dual, the dodecahedron,

If you apply this same operation a second time, here’s what you get.

This appears to be a three-part compound, with two familiar components: the icosahedron (red) and the dodecahedron (orange). Remove those two components, and you get this:

Since this reminds me of an icosidodecahedron, I colored its faces to better suit that identity.

Little peeks at the edges of the solid above made me suspicious, so I hid these purple and green faces, to see the inner structure. Here’s the result.

I made all of these using Stella 4d, which you can try free at http://www.software3d.com/Stella.php.

Repeated Augmentations of a Dodecahedron With More and More Dodecahedra

Here’s a single dodecahedron.

A new “cluster polyhedron” can be made by augmenting each pentagonal face with another dodecahedron.

If you can do it once, you can do it again, augmenting each pentagon with a new dodecahedron.

Once more.

I made these polyhedral clusters using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.

The Sixth Stellation of the Triakis Octahedron Is a Three-Part Polyhedral Compound

The components of this compound are eight-faced trapezohedra. Here’s what just one of them looks like:

I made these images using Stella 4d, which you can try for free right here.

A Polyhedral Journey, Starting with the Compound of Five Dodecahedra

This is the compound of five dodecahedra, a shape which is included in the built-in polyhedral library of Stella 4d, a program you can try for yourself, free, right here.

I wanted to see what I could make, starting from this compound. My first modification to it was to create its convex hull, which is shown below.

The next move was to use Stella‘s “Try to Make Faces Regular” function, which produced this:

Next, I augmented this figure’s thirty yellow rhombi with prisms.

I then created the convex hull of this augmented polyhedron.

Next, I used the “Try to Make Faces Regular” function again, producing a solid that looks, to me, like a hybrid of the rhombicosidodecahedron and the rhombic triacontahedron.

This polyhedron has yellow faces that are almost squares. Careful inspection reveals that they are actually isosceles trapezoids. The next thing I did was to augment each of these trapezoids with a tall prism.

The next step was to, again, create the convex hull.

That was the end of this polyhedral journey, but I am confident there will be others.

A Compound of the Rhombic Triacontahedron and a Truncation of the Icosahedron

Stellated Dual Morph 50.0%

In the compound above, the yellow hexagons are not quite regular, which is why I’m calling the yellow-and-orange polyhedron a truncation of the icosahedron, rather than simply the truncated icosahedron. I stumbled upon it while playing with Stella 4d, which you may try for free at http://www.software3d.com/Stella.php.

The Compound of Five Rhombic Dodecahedra

This is the compound of five rhombic dodecahedra, with each component shown in a different color. This is one of the few well-known polyhedral compounds which is actually more attractive with the faces hidden, and that’s what’s shown in the next image.

RD 5

I made these images using Stella 4d: Polyehdron Navigator, which you can try for free at http://www.software3d.com/Stella.php.