Augmenting, and Then Reaugmenting, the Icosahedron, with Icosahedra

A reader of this blog, in a comment on the last post here, asked what would happen if each face of an icosahedron were augmented by another icosahedron. I was also asked what the convex hull of such an icosahedron-cluster would be. Here are pictures which answer both questions, in order.

Augmented Icosa with more icosas.gif

Convex hull of icosa augmented with icosas.gif

While the icosahedron augmented by twenty icosahedron forms an unusual non-convex shape, its convex hull is simply a slightly “stretched” version of the truncated dodecahedron, one of the Archimedean solids.

The reader who asked these questions did not ask what would happen if the icosahedron-cluster above were to be augmented, on every face, by yet more icosahedra. However, I got curious about this, myself, and created the answer: the following cluster of even-more numerous icosahedra. This could be called, I suppose, the “reaugmented” icosahedron.

Augmented Icosa with more icosas and then yet more icosas.gif

Finally, here is the convex hull of this even-larger cluster. No one asked for it; I simply got curious.

Convex hull of the reaugmented icosahedral cluster

To accomplish the polyhedron-manipulation and image-creation for this post, I used a program called Stella 4d: Polyhedron Navigator, which is available at http://www.software3d.com/Stella.php. A free trial download is available there, so you can try the software before deciding whether or not to purchase it. 

Augmenting the Dodecahedron with Great Dodecahedra

These two polyhedra are the dodecahedron (left), and the great dodecahedron (right).

Since the faces of both of these polyhedra are regular pentagons, it is possible to augment each of the dodecahedron’s twelve faces with a great dodecahedron. Here is the result.

Augmented Dodeca.gif

I used Stella 4d to make these images. You may try this program for yourself at http://www.software3d.com/Stella.php.

Augmenting the Icosahedron with Great Icosahedra

These two polyhedra are the icosahedron (left), and the great icosahedron (right).

 

Since the faces of both of these polyhedra are equilateral triangles, it is possible to augment each of the icosahedron’s twenty faces with a great icosahedron. Here is the result.

Augmented Icosa with 20 great icosas.gif

I used Stella 4d to make these images. You may try this program for yourself at http://www.software3d.com/Stella.php.

Augmenting the Octahedron with Octahedra, Repeatedly.

This is an octahedron.

Octa

If you augment each face of an octahedron with more octahedra, you end up with this.

Augmented Octa

One can then augment each triangular face of this with yet more octahedra.

Aug Augmented Octa

Here’s the next iteration:

Aug Aug Augmented Octa

This could, of course, go on forever, but one more step in the series is all you will see here. I don’t want to get caught in an infinite loop.

next aug Octa

Performing various manipulations of polyhedra is easy with Stella 4d: Polyhedron Navigator, which I used to make all five of these rotating images. If you’d like to try this program for yourself, just check out http://www.software3d.com/Stella.php.

The Compound of Five Cubes, Augmented with Thirty Snub Cubes: Three Versions

Cubes 5 augmented by 30 snub cubes

This cluster-polyhedron was made with Stella 4d, software you can try at this website. Above, it is colored by face-type, referring to each face’s position within the overall cluster. In the image below, the original compound of five cubes contained one cube each, of five colors, and then each snub cube “inherited” its color from the cube to which it was attached.

Cubes 5

In the next version, the colors are chosen by the number of sides of each face.

Cubes 5

A Polyhedral Investigation, Starting with an Augmentation of the Truncated Octahedron

If one starts with a central truncated octahedron, leaves its six square faces untouched, and augments its eight hexagonal faces with trianglular cupolae, this is the result.

AUGMENTED TRUNCTAED OCTAHEDRON

Seeing this, I did a quick check of its dual, and found it quite interesting:

DUAL OF AUGMENTED TRUNCATED OCTAHEDRON

After seeing this dual, I next created its convex hull.

Convex hull x

After seeing this convex hull, I next creating its dual:  one of several 48-faced polyhedra I have found with two different sets of twenty-four kites as faces, one set in six panels of four kites each, and the other set consisting of eight sets of three kites each. I think of these recurring 48-kite-faced polyhedra as polyhedral expressions of a simple fact of arithmetic: (6)(4) = (8)(3) = 24.

48 KITES AGAiN

I use Stella 4d (available at http://www.software3d.com/Stella.php) to perform these polyhedral transformations. The last one I created in this particular “polyhedral journey” is shown below — but, unfortunately, I cannot recall exactly what I did, to which of the above polyhedra, to create it.

Convex hull OF AUGMENTED CUBOCTAHEDRON

A Cluster of Nine Octahedra, and Related Polyhedra

If one starts with a central octahedron, then augments each of its eight triangular faces with identical octahedra, this is the result.

9 Octahedra

It is then possible to augment each visible triangle of this cluster with yet more octahedra, which produces this result, in which some octahedra overlap each other.

Meta-9 Octahedra

After making this, I wanted to see its convex hull:  the smallest, tightest-fitting convex polyhedron which can contain a given non-convex polyhedron. (I use Stella 4d: Polyhedron Navigator to perform these manipulations of polyhedra, and this program makes this a fast and easy process. If you’d like to try this software, even as a free trial download, the website to visit is http://www.software3d.com/Stella.php.) Here’s what this convex hull, which bears a resemblance to the rhombcuboctahedron, looks like.

Convex hull of meta-9-octahedron

Looking for previously-unseen, and interesting, polyhedra, I then starting stellating this convex hull. I did find something interesting — to me, anyway — after only two stellations.

2nd stellation of Convex hull of meta-9-octahedron

That concluded my latest polyhedral investigation, but I certainly don’t intend it to be my last.