The two versions differ in their coloring-scheme, and, also, the pentagons and blue almost-triangles (actually irregular hexagons) are hidden in the second version, which you can make larger with a click:
Both images were created using Stella 4d, which you can try or buy at http://www.software3d.com/Stella.php.
I thought I would try to paint a migraine headache, and a few years ago, I did so, and painted this. I then fell asleep, and woke up, with, of course, a migraine.
The dodecahedron and the icosahedron are dual to each other, and can be combined to make this well-known compound.
In hyperspace, the analog to the dodecahedron is the hyperdodecahedron, also known as the 120-cell, as well as the hecatonicosachoron. Its dual is the 600-cell, or hexacosichoron, made of 600 tetrahedral cells. The image at the top is the compound of these two polychora, rotating in hyperspace.
These images were made using Stella 4d, available at http://www.software3d.com/Stella.php.
The cube and the octahedron are dual to each other, and can be combined to make this well-known compound (below; can be enlarged with a click).
In hyperspace, the analog to the cube is the tesseract, also known the 8-cell, the octachoron, and the hypercube. Its dual is the 16-cell, or hexadecachoron, made of 16 tetrahedral cells. The image at the top is the compound of these two polychora, rotating in hyperspace.
These images were made using Stella 4d, available at http://www.software3d.com/Stella.php.
Both of these were made using Stella 4d: Polyhedron Navigator. It’s available at http://www.software3d.com/Stella.php.
If you want to enlarge this second image, simply click on it.
I created this using Stella 4d, which is available (including a free trial download) at http://www.software3d.com/Stella.php. With adjustments in edge lengths to make the bond lengths correct, this would be the shape of a C180 fullerene molecule.
If the thirty-two faces of the truncated icosahedron are hidden, and only the sixty extra hexagons are visible, this polyhedron looks like this:
In “rainbow color mode,” it has an even more interesting appearance:
The top image here is of a zonish polyhedron based on adding zones along the five-fold symmetry axes of a rhombicosidodecahedron. All its edges are the same length, and its 62 faces include thirty elongated octagons, twelve regular pentagons, and twenty triangles. All of its edges have the same length.
The edges of this next polyhedron are also all of the same length. It was made in the same way, except that zones were added along both three- and five-fold symmetry axes of a rhombicosidodecahedron. Its 182 faces include thirty elongated dodecagons, twenty triangles, twelve regular pentagons, sixty squares, and sixty rhombi.
If only the three-fold symmetry axes are used to make a zonish polyhedron from a rhombicosidodecahedron, this next polyhedron, also with all edge lengths equal, is the result. It also has 182 faces, and they are of the same type as in the one immediately before, except that thirty elongated octagons replace the dodecagons from that polyhedron.
A rhombicosidodecahedron also has two-fold symmetry axes. If only those axes are used to make a zonish rhombicosidodecahedron, this next polyhedron is the result: a modified form of the great rhombicosidodecahedron, with unequal edge lengths.
If the two- and three-fold symmetry axes are both used, the result, once again, is a 182-faces polyhedron, but it also has unequal edge lengths, and none of its faces are regular polygons. It is shown below. There are twelve decagons, sixty rectangles, sixty hexagons of one type, twenty hexagons of another type, and thirty octagons.
Another possible combination is to use the two- and five-fold symmetry axes to create a zonish rhombicosidodecahedron. This yields a polyhedron with 122 faces, with all except the sixty squares being irregular. The other faces are twelve decagons, thirty octagons, and twenty hexagons:
Finally, there is one last combination — using the two-, three-, and five-fold symmetry axes, all at once. Here’s what it looks like:
As one should expect, this produces a zonish polyhedron with more faces than any of the earlier ones shown above: 242 in all. As in the last one shown, only the sixty squares are regular, although the sixty pink hexagons are at least equilateral. There are also sixty rectangles, twenty hexagons of a second type, thirty dodecagons, and twelve decagons.
All of these zonish rhombicosidodecahedra were created using Stella 4d, software available at http://www.software3d.com/Stella.php.
I created this using Stella 4d, which is available (including a free trial download) at http://www.software3d.com/Stella.php. It can also be viewed as a compound of the rhombic dodecahedron and another polyhedron, but I haven’t been able to identify that second polyhedron — at least, not yet. If you know what the orange polyhedron is, please leave a comment with its name.
The convex hull of this solid is the cuboctahedron. To me, it looks like a hybrid of that solid, and the Stella Octangula. I created it using Stella 4d, which is available (including a free trial download) at http://www.software3d.com/Stella.php.
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