This compound has three parts: two tetrahedra, plus one smaller cube. I made it using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.
I made this using Stella 4d, software you can try out for free at this website: http://www.software3d.com/Stella.php.
The 18th stellation of the rhombicosidodecahedron, shown above, is also an interesting compound. The yellow component of this compound is the rhombic triacontahedron, and the blue-and-red component is a “stretched” form of the truncated icosahedron.
This was made using Stella 4d, which you can try for free right here.
To make this polyhedron using Stella 4d (available here), I began with the dodecahedron, dropped the symmetry of the model from icosahedral to tetrahedral, and then stellated it thirteen times.
This stellated polyhedron has pyritohedral symmetry, but this is easier to see in its convex hull:
The eight blue triangles in this convex hull are equilateral, while the twelve yellow ones are golden isosceles triangles.
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.
To make this polyhedron, I first changed the symmetry-type of a dodecahedron from icosahedral to tetrahedral, then stellated it twice. This was done using Stella 4d, a program you may try for free at http://www.software3d.com/Stella.php.
If you stellate the snub dodecahedron 43 times, this is the result. The yellow faces are kites, not rhombi.
Like the snub dodecahedron itself, this polyhedron is chiral. Here is the mirror-image of the polyhedron shown above.
Any chiral polyhedron may be combined with its own mirror-image to create a compound.
This is the dual of the snub dodecahedron’s 43rd stellation.
This dual is also chiral. Here is its reflection.
Finally, here is the compound of both duals.
I used Stella 4d: Polyhedron Navigator to create these images. You may try this program for yourself at http://www.software3d.com/Stella.php.
I made this using Stella 4d: Polyhedron Navigator, which you may try for free right here.
This is the truncated dodecahedron. It is one of the Archimedean solids.
This polyhedron has a long stellation-series, from which I selected several on aesthetic grounds. The figure immediately below is the truncated dodecahedron’s 16th stellation.
Here is the 21st stellation.
It’s easy to stellate polyhedra rapidly, and make many other changes to them, with Stella 4d: Polyhedron Navigator. You can try it for free at http://www.software3d.com/Stella.php.
The stellation shown immediately above is the 25th, and the one shown immediately below is the 27th.
Here is the next stellation: the 28th. Unlike the ones shown above, it is chiral.
This is the truncated dodecahedron’s 31st stellation.
This one is the 38th stellation.
This one is the 44th.
The last one shown here is called the truncated dodecahedron’s final stellation because, if it is stellated once more, it returns to the original truncated dodecahedron.
I created these with Stella 4d, which you may try for free at this website. To make a given polyhedral stellation appear larger, simply click on it.
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