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.
Each of the smaller pictures below may be enlarged by clicking on them.
All of these images were produced using Stella 4d, which you may try or buy at http://www.software3d.com/Stella.php.
This one is a variant of the icosidodecahedron.
This one is based on the rhombcuboctahedron.
This one is made of squares, rhombi, and irregular pentagons.
This one is composed entirely of pentagons and hexagons, none of which are regular.
This one has faces which include squares, rhombi, and isosceles triangles.
In this one, the hexagons and squares are regular. Only the isosceles triangles are irregular.
This is the dual of the last one shown here. Its faces are all either kites or rhombi.
I hoped to make this one into a near miss to the Johnson solids, but the octagons of both types, especially, are too far from regularity to get that to work. The only faces which are regular are the green triangles.
This one is a variant of the icosahedron.
I found this one interesting.
And this one is its dual:
Finally, here’s one made of kites and regular hexagons.
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.
The Stella Octangula is another name for the compound of two tetrahedra. In this variant, each triangular face is replaced by a panel of three irregular pentagons. I used Stella 4d to make it, and you can find that program at http://www.software3d.com/Stella.php.
I’ve done quite a bit on this blog involving modifiying polyhedra via augmentation, in which polyhedra are attached to faces of another polyhedra. This was made using the opposite process, excavation, where parts of a given polyhedron are removed. First, a pentagonal pyramid was excavated from each pentagonal face of the icosidodecahedron. Next, octahedra were excavated from each triangular face. Finally, the bottom face of each of these octahedra was hidden, making it possible to see all the way through this solid. I constructed this with Stella 4d, software you can try or buy at www.software3d.com/Stella.php.
Here’s the same polyhedron again, but with a different color-scheme.
If one starts with a cuboctahedron, and then creates a zonish polyhedron from it, adding zones (based on the faces) to the faces which already exist, here is the result, below, produced by Stella 4d: Polyhedron Navigator (software you may buy or try at http://www.software3d.com/Stella.php):
The hexagons here, in this second image, are visibly irregular. The four interior hexagon-angles next to the octagons each measure more than 125 degrees, and the other two interior angles of the hexagons each measure less than 110 degrees — too irregular for this to qualify as a near-miss to the Johnson solids. However, Stella includes a “try to make faces regular” function, and applying it to the second polyhedron shown here produces the polyhedron shown in a larger image, at the top of this post.
It is this larger image, at the top, which I am proposing as a new near-miss to the 92 Johnson solids. In it, the twelve hexagons are regular, as are the eight triangles and six octagons. The only irregular faces to be found in it are the near-squares, which are actually isosceles trapezoids with two angles (the ones next to the octagons) measuring ~94.5575 degrees, and two others (next to the triangles) measuring 85.4425 degrees. Three of the edges of these trapezoids have the same length, and this length matches the lengths of the edges of both the hexagons and octagons. The one side of each trapezoid which has a different length is the one it shares with a triangle. These triangle-edges are ~15.9% longer than all the other edges in this proposed near-miss.
My next step is to share this find with others, and ask for their help with these two questions:
Once I learn the answers to these questions, I will update this post to reflect whatever new information is found. If this does qualify as a near-miss, it will be my third such find. The other two are the tetrated dodecahedron (co-discovered, independently, by myself and Alex Doskey) and the zonish truncated icosahedron (a discovery with which I was assisted by Robert Webb, the creator of Stella 4d).
More information about these near-misses, one of my geometrical obsessions, may be found here: https://en.wikipedia.org/wiki/Near-miss_Johnson_solid
These polygons are known to virtually all speakers of English as the triangle and the quadrilateral, but that doesn’t mean I have to like that fact, and, the truth is, I don’t. Why? There are a couple of reasons, all involving lack of consistency with the established names of other polygons.
Consider the names of the next few polygons, as the number of sides increases: the pentagon, hexagon, heptagon, and octagon. The “-gon” suffix refers to the corners, or angles, of these figures, and is derived from Greek, The end of the word “triangle” also refers to the same thing — but not in Greek. For the sake of consistency, triangles should, instead, be called “trigons.”
In the case of the quadrilateral, the problem is twofold. The suffix “-lateral” refers to sides, not angles. For the sake of consistency, “-gon” should be used instead. The prefix “quadri-” does mean four, of course, but is derived from Latin, not Greek. We use the Greek prefix “tetra-” to refer to four when naming polyhedra (“tetrahedron”), so why not use it for polygons with four sides, also? The best name available for four-sided polygons requires a change in both the prefix and suffix of the word, resulting in the name “tetragon” for the figure on the right.
When I listed the names of higher polygons above, I deliberately stopped with the octagon. Here’s the next polygon, with nine sides and angles:
I’m guilty of inconsistency with the name of nine-sided polygons, myself. All over this blog, you can find references to “nonagons,” and the prefix “nona-” is derived from Latin. Those who already know better have, for years, been calling nine-sided polygons “enneagons,” using the Greek prefix for nine, rather than the Latin prefix, for reasons of consistency. I’m not going to go to the trouble to go back and edit every previous post on this blog to change “nonagon” to “enneagon,” at least right now, but, in future posts, I will join those who use “enneagon.”
Here’s one more, with eleven sides:
I don’t remember ever blogging about polygons with eleven sides, but I have told geometry students, in the past, that they are called “undecagons.” I won’t make that mistake again, for the derivation of that word, as is the case with “nonagon,” uses both Latin and Greek. A better name for the same figure, already in use, is “hendecagon,” and I’m joining the ranks of those who use that term, derived purely from Greek, effective immediately.
With “hendecagon” and “enneagon,” I don’t think use of these better names will cause confusion, given that they are already used with considerable frequency. Unfortunately, that’s not the case with the little-used, relatively-unknown words “trigon” and “tetragon,” so I’ll still be using those more-familiar names I don’t like, just to avoid being asked “What’s a trigon?” or “What’s a tetragon?” repeatedly, for three- and four-sided polygons. Sometimes, I must concede, it is necessary to choose the lesser of two irritations. With “triangle” and “quadrilateral,” this is one of those times.
This is the dual of the one polyhedral cluster found here which has more than one color-scheme shown: https://robertlovespi.wordpress.com/2014/05/29/the-great-rhombcuboctahedron-as-a-building-block/
It’s the dual of a rhombcuboctahedron made of great rhombcuboctahedra, and was created using software called Stella 4d: Polyhedron Navigator. This software may be purchased at http://www.software3d.com/Stella.php — and there is a free trial version available to download there, as well.
Stella’s creator just came out with a new version of Stella 4d, and a discovery of mine made the built-in library that comes with that software. This is my blog, so I get to brag about that, right? My legal name appears in the small print on the right side, at the end of the first long paragraph. I added the red ellipses to make it easier to find.
You can see the earlier posts related to my discovery of this zonish truncated icosahedron here:
https://robertlovespi.wordpress.com/2013/05/10/a-new-near-miss-to-the-92-johnson-solids/
https://robertlovespi.wordpress.com/2013/05/13/a-second-version-of-my-new-near-miss/
If you’d like to try (as a free trial) or buy this software (I recommend Stella 4d over the other available options), here’s the link for that: http://www.software3d.com/Stella.php.
I’ve been asked by a reader of this blog to post nets for this polyhedral compound. Printing nets with Stella 4d is easy, and I’m happy to post them here, in response to that request. Warning, though: there are many nets needed for this compound.
Each of these smaller images may be enlarged with a single click.
Here’s the first net type needed (above). You’ll need thirty copies of this net. The gray parts show, and the white parts are tabs to help put it together. Below is the second type needed, of which you need sixty copies.
There’s also a third type of net, and these last two types may need to be rescaled before you print them, to fit the net of the first type, also. You’ll need sixty copies of this third net (below) as well, It’s the mirror-image of the net of the second type.
Finally, here’s a non-rotating image of the completed polyhedron, to help with the construction:
I recommend using card stock or posterboard, and trying to get as much tape as possible on the inside of the model, making an uncolored version — and then painting it with five different colors of your choice, after the model is assembled. Happy building!
[Software credit: I used Stella 4d: Polyhedron Navigator to create all these images. It’s available at http://www.software3d.com/Stella.php. Downloading and trying a trial version is free, but you have to buy the fully-functioning version to print nets, or to make these rotating .gif files I post all over this blog.]
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