An Icosahedron Augmented with Twenty Great Icosahedra, together with the Dual of this Cluster-Polyhedron

icosa Augmented by great Icosas.gif

The cluster-polyhedron above was formed by augmenting a central isocahedron with twenty great icosahedra. The dual of this cluster is shown below.

icosa Augmented by great Icosas dual.gif

Both these images were created using Stella 4d, which you may try for free at http://www.software3d.com/Stella.php.

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.

A Cluster of Twenty Great Icosahedra, Excavated from the Faces of a Central Icosahedron, Along with Its Dual

Augmented Icosa its excavated with great icosas

These twenty great icosahedra were excavated from the faces of a central icosahedron, which is concealed in the figure’s center. These excavations exceed the limits of the central icosahedron, resulting in each great icosahedron protruding in a direction opposite that of the face from which it is excavated. In a certain sense, then, the figure above has negative volume.

To make this, I used software called Stella 4d: Polyhedron Navigator. It can be researched, bought, or tried for free here.

Also, here is the dual of the polyhedral cluster above, made with the same program.

Augmented Icosa its excavated with great icosas the dual

A Great Icosahedron, Augmented with Twenty Icosahedra

Augmented Great Icosa augmented with icosas

The polyhedral clusters above and below use different coloring-schemes, but are otherwise identical. Invisible, in the center, is a great icosahedron. Each of its faces has been augmented by a (Platonic) icosahedron.

Augmented Great Icosa augmented with icosas colored by face typeBoth images were created using Stella 4d, software you can try here.

The Greatly Augmented Icosidodecahedron, and Its Dual

Augmented Icosidodeca

If a central polyhedron’s pentagonal and triangular faces are augmented by great dodecahedra and great icosahedra, I refer to it as a “greatly augmented” polyhedron. Here, this has been done with an icosidodecahedron. The same figure appears below, but in “rainbow color” mode.

Augmented Icosidodeca colored rainbow

In the next image, “color by face type,” based on symmetry, was used.

Augmented Icosidodeca colored by face type

The next image shows the dual of this polyhedral cluster, with face color chosen on the basis of number of sides.

Augmented Icosidodeca colored by whether sides have 5 or 16 sides

Here is another version of the dual, this one in “rainbow color” mode.

Augmented Icosidodeca colored rainbow DUAL

Finally, this image of the dual is colored based on face type.

Augmented Icosidodeca colored by face type DUAL

These six images were made with Stella 4d, which may be found here.

The Greatly Augmented Rhombicosidodecahedron

Greatly Augmented Rhombicosidodeca

I call this variant on the rhombicosidocahedron “greatly augmented” because it was formed by augmenting each pentagonal face of a central rhombicosidodecahedron with great dodecahedra, while each triangular face is augmented with great icosahedra. It was made using Stella 4d, which may be found here.

An Octahedron, Augmented with Eight Great Icosahedra, and the Dual of this Augmented Polyhedral Cluster

Image

An Octahedron, Augmented with Eight Great Icosahedra, and the Dual of this Augmented Polyhedral Cluster

I made these using Stella 4d, which you can try at http://www.software3d.com/Stella.php. Here is its dual, also:

GrIcosadual-Augmented Octa