Facetings are created by joining vertices to other vertices, but not choosing the vertices in the usual manner, which results in new positions for edges and faces. Faceting is also the reciprocal-function for polyhedral stellation. This is one of many possible facetings of the snub dodecahedron, and I created it using Stella 4d, which you can find here.
This faceting was created using Stella 4d, available here.
To make a faceted polyhedron, vertices of the original polyhedron are connected in new ways, to create a different set of faces and edges. To make this particular faceting of the truncated icosahedron, I first connected all vertices in the configuration shown below, to make irregular decagonal faces in the interior of the solid.
In this next pic, the decagonal faces formed above are shown in red, and a new set of pentagonal faces is being created.
For a polyhedron to be considered mathematically valid, faces must meet in pairs at each edge. To accomplish that, I had to create another set of pentagonal faces, this one smaller than the last, as shown below.
Here’s the completed polyhedron, with each face-type having its own color.
This next image is of the same polyhedron, but with a different coloring-scheme. In this second version, each face has its own color — except for faces which are parallel, with those faces given the fame color.
I created these images using Stella 4d: Polyhedron Navigator, a program which is available here.
Check out http://www.software3d.com/stella.php to try the software used to make this image.
The snub dodecahedron is chiral, meaning it appears in left- and right-handed forms. This faceted version, where the same set of vertices is connected in different ways (compared to the original), possesses the same property.
Chiral polyhedra can always be tranformed into interesting polyhedral compounds by combining them with their own mirror-images. If this is done with the polyhedron above, you get this result, presented with a different coloring-scheme.
Both of these images were created using Stella 4d: Polyhedron Navigator, and you may try it at www.software3d.com/Stella.php.
I created this using Stella 4d: Polyhedron Navigator, available at www.software3d.com/Stella.php. Faceting involves connecting different sets of vertices (relative to the original polyhedron) to form new edges and faces. The new edges and faces, both, typically intersect each other, although often not as many times as in this particular example of a faceted polyhedron.
I made this with Stella 4d, a program you can find at http://www.software3d.com/Stella.php.
If you look at the second image from the post two entries ago, and wonder what it would look like without the pink faces, wonder no longer: it’s what you see above.
Next, the red polygons are hidden, and this is what is left (you may click these smaller images if you wish to enlarge them).
The green faces are hidden next.
The next step is to remove the pink faces visible in the interior.
Next, removal of the blue faces leaves only the yellow ones left.
The last step: change the color scheme, so as to more easily be able to tell one face from another.
All of this polyhedron-manipulation, I did with Stella 4d, software I consider an indispensable research-tool. It is available at http://www.software3d.com/Stella.php.
The last post had two images, and this is the dual of the second one. I was therefore surprised when I ran into this while playing around with Stella 4d, a program which allows easy polyhedron manipulation. (See http://www.software3d.com/stella.php for free trial download.)
Why did it surprise me?
Well, isn’t a polyhedron. for one thing. It is a collection of irregular and concentric polygons which intersect, but they don’t meet at edges. This doesn’t normally happen, so it requires explanation. I figured it out pretty quickly.
I’ve been using the loosest possibly definition for “faceting,” not insisting that faces meet at each edge in pairs, and even making some faces invisible in order to see the interior structure of the “polyhedra.” Since this breaks the faceting-rules, it isn’t surprising that the dual would fail to be a true polyhedron.
That’s my guess, anyway.
In this variation of the snub cube, twenty of the triangular faces have been excavated with short triangular pyramids. Since the snub cube is chiral, it’s possible to make a compound out of it and its mirror-image:
A polyhedron which is somewhat similar to the first one shown here can be produced by faceting a snub cube:
Stella 4d was used to create these images. You can find this program at http://www.software3d.com/Stella.php.
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