Three Views of a Rotating Cluster of 33 Icosidodecahedra

33-icosidodeca

To make these three rotating cluster-polyhedra, I started with one icosidodecahedron in the center, then augmented each of its 32 faces with overlapping, additional icosidodecahedra, for a total of 33 icosidodecahedra per cluster. In the first image, only two colors are used: one for the triangular faces, and another for the pentagons. The second version, however, has the colors assigned by face-type, which is determined by each face’s placement in the overall cluster.

33-icosidodeca-ft

For the third version, I simply put Stella 4d (the program I use to make these images) into “rainbow color mode.” If you’d like to give Stella 4d a try, you can do so for free at this website.

33-icosidodeca-rc

 

An Open Cluster of Polyhedra

augmented-icosa

From the center to the outside, this cluster contains one icosahedron, twenty octahedra, twenty icosidodecahedra, twenty more octahedra, and, finally, twenty rhombicosidodecahedra.

augmented-icosa-dc

All three of the images here were created using Stella 4d, a program you may try, free, at this website.

augmented-icosa-rc

 

A Festive Cluster of Polyhedra

augmented-icosa-with-tet-then-octahedra

This is what you get if you start with an icosahedron, augment each of its faces with tetrahedra, and then augment the tetrahedral faces with octahedra. I made it using Stella 4d, a program you may try for free at http://www.software3d.com/Stella.php.

Icosahedral Cluster

augmented-great-icosa

The great icosahedron, one of the Kepler-Poinsot solids, is hidden from view at the center of this cluster. Each of its faces is augmented with a Platonic icosahedron, producing what you see here. Stella 4d is the software I used; more information about that program may be found here.

Two Different Cluster-Polyhedra

Augmented Icosa with RIDs

An icosahedron is hidden from view in the center of this cluster-polyhedron. To create the cluster, each of the icosahedron’s triangles was augmented with a rhombicosidodecahedron. The resulting cluster has the overall shape of a dodecahedron.

To create the next cluster-polyhedron, I started with the one above, and then augmented each of its triangular faces with icosidodecahedra. 

large cluster os icosidodecahedrons.gif

I used a program named Stella 4d: Polyhedron Navigator to create these cluster-polyhedra. This software may be bought (or tried for free) at this website

Two Views of an Icosahedron, Augmented with Great Icosahedra

If colored by face-type, based on face-position in the overall solid, this “cluster” polyhedron looks like this:

Augmented Icosa using grt icosas

There is another interesting view of this polyhedral cluster I like marginally better, though, and that is to separate the faces into color-groups in which all faces of the same color are either coplanar, or parallel. It looks like this.

Augmented Icosa using grt icosas parallel faces colored together

Both versions were created by augmenting each face of a Platonic icosahedron with a great icosahedron, one of the four Kepler-Poinsot solids. I did this using Stella 4d: Polyhedron Navigator, available here.

Unsquashing the Squashed Meta-Great-Rhombcuboctahedron

I noticed that I could arrange eight great rhombcuboctahedra into a ring, but that ring, rather than being regular, resembled an ellipse.

Augmented Trunc Cubocta

I then made a ring of four of these elliptical rings.

Augmented Trunc Cubocta B

After that, I added a few more great rhombcuboctahedra to make a meta-rhombcuboctahedron — that is, a great rhombcuboctahedron made of rhombcuboctahedra. However, it’s squashed. (I believe the official term for this is “oblate,” but “squashed” also works, at least for me.)

Augmented Trunc Cubocta 3

So now I’m wondering if I can make this more regular. In other words, can I “unsquash” it? I notice that even this squashed metapolyhedron has regular rings on two opposite sides, so I make such a ring, and start anew.

Augmented Trunc Cubocta a

I then make a ring of those . . . 

Augmented Trunc Cubocta AA

. . . And, with two more ring-additions, I complete the now-unsquashed meta-great-rhombcuboctahedron. Success!

Augmented Trunc Cubocta AAA

To celebrate my victory, I make one more picture, in “rainbow color mode.”

Augmented Trunc Cubocta AAAR

[All images made using Stella 4d, available here: http://www.software3d.com/Stella.php.]