Both of these were made using Stella 4d, software you can try at this website.
These two compounds, above and below, are duals. Also, in each of them, one polyhedron with icosidodecahedral symmetry is combined with a second polyhedron with cuboctahedral symmetry to form a compound with pyritohedral symmetry: the symmetry of a standard volleyball.
A program called Stella 4d was used to make these compounds, and create these images. It may be purchased, or tried for free, at this website.
I made these using Stella 4d, available here.
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.
In the next image, “color by face type,” based on symmetry, was used.
The next image shows the dual of this polyhedral cluster, with face color chosen on the basis of number of sides.
Here is another version of the dual, this one in “rainbow color” mode.
Finally, this image of the dual is colored based on face type.
These six images were made with Stella 4d, which may be found here.
Pyritohedral symmetry, seen by example both above and below, is most often described at the symmetry of a volleyball:
[Image of volleyball found here.]
To make the rotating polyhedral compound at the top, from an octahedron and an icosahedron, I simply combined these two polyhedra, using Stella 4d, which may be purchased (or tried for free) here.
In the process, I demonstrated that it is possible to combine a figure with octahedral (sometimes called cuboctahedral) symmetry, with a figure with icosahedral (sometimes called icosidodecahedral) symmetry, to produce a figure with pyritohedral symmetry.
Now I can continue with the rest of my day. No matter what happens, I’ll at least know I accomplished something.
Polyhedra are one of the areas (there are at least a few others) where the fields of mathematics and art intersect. Stella 4d, the program I used to make this image, is a great tool for the exploration of this region of intersection. This software may be tried for free right here.
Note: icosidodecahedral symmetry, a term coined (as far as I know) by George Hart, means exactly the same thing as icosahedral symmetry. I simply use the term I like better. Also, a few of these, but not many, are chiral.
The images directly above and below show the shape of the most symmetrical 240-carbon-atom fullerene.
The image above is of the compound of five tetrahedra. This compound is chiral, and the next image is the compound of the compound above, and its mirror-image.
In the next two, I was experimenting with placing really big spheres at the vertices of polyhedra. The first one is the great dodecahedron, rendered in this unusual style, with the faces rendered invisible.
I made these using Stella 4d: Polyhedron Navigator. You may try this program for free at http://www.software3d.com/Stella.php.
All of these were made with Stella 4d, a program you can find at http://www.software3d.com/Stella.php.
I’ve received a request to slow down the rotational speed of the polyhedral models I make and post here, and am going to try to do exactly that. First, though, I need to empty my collection of already-made image files which haven’t yet been posted, so that I can start again, with models which rotate more slowly, after deleting all the “speedy” ones. From my backlog of polyhedral images to post, then, here are most of the ones with icosidodecahedral symmetry.
The next one shown has 362 faces — the closest I have come, so far, to a polyhedron with a number of faces which matches the number of days in a year.
The next one is a variant of the rhombic enneacontahedron, with that polyhedron’s wide rhombic faces replaced by kites, and its narrow rhombi replaced by pairs of isosceles triangles.
In the remaining polyhedral images in this post, some faces have been rendered invisible. I do this, on occasion, either so that the front and back of the polyhedra can be seen at the same time, or simply for aesthetic reasons.
All of these images were created using Stella 4d: Polyhedron Navigator. If you’d like to try this program for yourself, the website to visit for a free trial download is www.software3d.com/Stella.php.
I made all of these using Stella 4d: Polyhedron Navigator. You may try this software for yourself at www.software3d.com/Stella.php.