The Double Icosahedron, and Some of Its “Relatives”

The double icosahedron is simply an icosahedron, augmented on a single face by a second icosahedron. I thought it might be interesting to explore some transformations of this solid, using Stella 4d: Polyhedron Navigator (available here), and I was not disappointed. I used Stella to produce all the images in this post.

Augmented Icosa

It is well-known that the dual of the icosahedron is another Platonic solid, the dodecahedron. Naturally, I wanted to see the double icosahedron’s dual, and here it is — a simple operation for Stella. This dual resembles a dodecahedron in its center, but gets more unusual-looking as one moves further out from its core. 

Augmented Icosa dual

I next examined stellations of the double icosahedron, but did not find any which seemed attractive enough to post, until I saw its sixteenth stellation, which features six kites as faces, in sets of three, on opposite sides of the solid.

Stellated Augmented Icosa 16th.gif

What proved most fruitful was my examination of various zonohedra based on the double icosahedron. Here’s what I found for the zonohedron based on the faces of the double icosahedron: a large number of rhombic faces, with Northern and Southern “hemispheres” separated by an “equator” of hexagonal zonogons.

Zonohedrified Augmented Icosa faces.gif

The next image is the zonohedron based on the edges of the double icosahedron.

Zonohedrified Augmented Icosa edges.gif

The next zonohedron shown is based on the vertices of the double icosahedron.

Zonohedrified Augmented Icosa vertices

All of these zonohedra have 6-fold dihedral symmetry, while the double icosahedron itself has 3-fold dihedral symmetry. The next image shows the zonohedron based on both the vertices and edges of the double icosahedron.

Zonohedrified Augmented Icosa v and e.gif

Zonohedrification based on vertices and faces produces the next zonohedron shown here.

Zonohedrified Augmented Icosa v and f.gif

The next logical step was to create a zonohedron based on the double icosahedron’s edges and faces.

Zonohedrified Augmented Icosa e and f.gif

Finally, here is the zonohedron based on all three characteristics: the vertices, edges, and faces of the double icosahedron.

Zonohedrified Augmented Icosa VEF.gif

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things, a large portion of which are geometrical. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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4 Responses to The Double Icosahedron, and Some of Its “Relatives”

  1. tyara says:

    I wonder, if you put an icosahedron on every face, what shape would turn up? Would it look/be convex? If not, what would it’s convex hull look like?

    Liked by 1 person

  2. Pingback: A Second Type of Double Icosahedron, and Related Polyhedra | RobertLovesPi.net

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