I made this using Stella 4d, software you can try for yourself right here.
I made this using Stella 4d, software you can try for yourself right here.
The cluster-polyhedron above was formed by augmenting a central isocahedron with twenty great icosahedra. The dual of this cluster is shown below.
Both these images were created using Stella 4d, which you may try for free at http://www.software3d.com/Stella.php.
After seeing my post about what I called the “double icosahedron,” which is two complete icosahedra joined at one common triangular face, my friend Tom Ruen brought my attention to a similar figure he likes. This second type of double icosahedron is made of two icosahedra which meet at an internal pentagon, rather than a triangular face. Tom jokingly referred to this figure as “a double patty pentagonal antiprism in a pentagonal pyramid bun.”
It wasn’t hard to make this figure using Stella 4d, the program I use for polyhedral manipulation and image-creation (you can try it for free here), but I didn’t make it out of icosahedra. It was easier to make this figure from gyroelongated pentagonal pyramids, or “J11s” for short. This polyhedron is one of the 92 Johnson solids.
To make the polyhedron Tom had brought to my attention, I simply augmented one J11 with another J11, joining them at their pentagonal faces. Curious about what the dual of this solid would look like, I generated it with Stella.
The dual of the double J11 appears to be a modification of a dodecahedron, which is no surprise, for the dodecahedron is the dual of the icosahedron.
I next explored the stellation-series of the double J11, and found several attractive polyhedra there. This one is the double J11’s 4th stellation.
The next polyhedron shown is the double J11’s 16th stellation.
Here is the 30th stellation:
I also liked the 43rd:
The next one shown is the double J11’s 55th stellation.
Finally, the 56th stellation is shown below. These stellations, as well as the double J11 itself, and its dual, all have five-fold dihedral symmetry.
Having “mined” the double J11’s stellation-series for interesting polyhedra, I next turned to zonohedrification of this solid. The next image shows the zonohedron based on the double J11’s faces. It has many rhombic faces in two “hemispheres,” separated by a belt of octagonal zonogons. This zonohedron, as well as the others which follow, all have ten-fold dihedral symmetry.
Zonohedrification based on vertices produced this result:
The next zonohedron shown was formed based on the edges of the double J11.
Next, I tried zonohedrification based on vertices and edges, both.
Next, vertices and faces:
The next zonohedrification-combination I tried was to add zones based on the double J11’s edges and faces.
Finally, I ended this exploration of the double J11’s “family” by adding zones to build a zonohedron based on all three of these polyhedron characteristics: vertices, edges, and faces.
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.
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.
Finally, here is the convex hull of this even-larger cluster. No one asked for it; I simply got curious.
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.
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.
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.
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.
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.
The next image is the zonohedron based on the edges of the double icosahedron.
The next zonohedron shown is based on the vertices of the double icosahedron.
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.
Zonohedrification based on vertices and faces produces the next zonohedron shown here.
The next logical step was to create a zonohedron based on the double icosahedron’s edges and faces.
Finally, here is the zonohedron based on all three characteristics: the vertices, edges, and faces of the double icosahedron.
I made this using Stella 4d: Polyhedron Navigator, which you may try for free right here.
Stella 4d was used to make this image, and you may try it for free by following this link.
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.
This is the icosahedron, followed by its first stellation.
The first stellation of the icosahedron can be stellated again, and again, and so on. The “final stellation” of the icosahedron is the one right before the stellation-series “wraps around,” back to where it started:
This final stellation of the icosahedron would serve pretty well as a “polyhedral porcupine,” but I was seeking something even better, so I turned my attention to polyhedral compounds. This is the compound of five icosahedra:
The program I use to manipulate these solids is called Stella 4d: Polyhedron Navigator (free trial download available here). My next move, using Stella, was to create the final stellation of this five-icosahedron compound . . . and, when I saw it, I knew I had found my “polyhedral porcupine.”
First, this is the 11th stellation.
Next, the 13th:
And, finally, the 15th stellation of the icosahedron:
I used Stella 4d, which you can find here, to make these.