The reason I am not calling this a compound of three octahedra is that the faces of the dipyramids aren’t quite equilateral. They are, however, isosceles.
This was created with Stella 4d, which you can buy, or try for free, right here.
The world ended on this day in 2012 — December 21 — when the Mayan calendar began a new cycle. We now secretly live in a computer simulation run by highly advanced ancient Mayan aliens. They have authorized me to wish you a happy second anniversary of the end of your previous existence.
[Image credit: within this simulation, you can find this picture at http://wall.alphacoders.com/by_sub_category.php?id=206132.]
The polyhedron above originally had thirty yellow square faces, but I rendered them invisible so that the interior structure of this polyhedron could be seen.
When stellating such a partially-invisible figure, the new faces “inherited” from the “parent polyhedron” are either visible or invisible, depending on which type of face they are derived from. This makes for a very unusual look for some stellations, such as this, the rhombicosidodecahedron’s 50th:
I created these images using a program called Stella 4d: Polyhedron Navigator. You may try it for yourself at http://www.software3d.com/Stella.php.
If a zonish dodecahedron is created with zones based on the dodecahedron’s vertices, here is the result.
If the same thing is done with edges, this is the result — an edge-distorted version of the great rhombicosidodecahedron.
Another option is faces-only. Although I haven’t checked the bond-lengths, this one does have the general shape of the most-symmetrical 80-carbon-atom fullerene molecule. Also, this shape is sometimes called the “pseudo-truncated-icosahedron.”
The next zonish dodecahedron has had zones added based on the dodecahedron’s faces and edges, both.
Here’s the one for vertices and edges.
Here’s the one for faces and vertices.
Finally, the last of this set of seven has had zones added based on all three: faces, vertices, and edges.
All seven of these were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.
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.
This tiling-pattern could be continued indefinitely, while still maintaining its five-fold radial symmetry, giving it the overall appearance of a pentagon.
The golden triangles, in yellow, are acute isosceles triangles with a leg:base ratio which is the golden ratio. Golden gnomons, shown in orange, are related, for they are obtuse isosceles triangles where the golden ratio shows up as the base:leg ratio, which is the reciprocal of the manifestation of the golden ratio which appears in the yellow triangles.
This tessellation can be viewed in at least two ways: it can be seen as being composed of overlapping octagons which are equilateral, but not equiangular — or it can be viewed as a periodically-repeating pattern of golden gnomons, as well as golden triangles of two different sizes. Both golden triangles and golden gnomons are isosceles triangles with sides in the golden ratio, but golden triangles are acute, while golden gnomons are obtuse.
If a zonish icosahedron is created with zones based on the icosahedron’s vertices, here is the result.
If the same thing is done with edges, this is the result.
Another option is faces-only.
The next zonish icosahedron has had zones added based on the icosahedron’s faces and edges, both.
Here’s the one for vertices and edges.
Here’s the one for faces and vertices.
Finally, the last of this set of seven has had zones added based on all three: faces, vertices, and edges.
All seven of these were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.
An enantiomorphic-pair compound requires a chiral polyhedron, for it is a compound of a polyhedron and its mirror image. Among the Archimedeans, only the snub cube and snub dodecahedron are chiral. For this reason, only threir duals are chiral, among the Archimedean duals, also known as the Catalan solids.
That’s a compound of two mirror-image snub cube duals (pentagonal icositetrahedra) above; the similar compound for the snub dodecahedron duals (pentagonal hexacontahedra) is below.
Both these compounds were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.
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