A (Possible) New Near-Miss to the Johnson Solids

near-miss candidate

In the polyhedron above, the octagons, hexagons, and triangles are regular. The only irregularities are found in the near-squares, which are actually isosceles trapezoids with three edges of equal length: the ones shared with the octagons and hexagons. The trapezoid-edges adjacent to the triangles, however, are ~15.89% longer than its other three edges. As a result, two of the interior angles of the trapezoids measure ~85.44º (the ones nearest the triangles), while the other two (adjacent to the shorter of the two trapezoid bases) measure ~94.56º. In a rotating model, it can be difficult to see the irregularities in these trapezoids. Were someone to build an actual physical model, however, the fact that they are not squares would be far more obvious.

In case someone would like to build such a model, here is a net you can use.

near-miss candidate

As you can see on this Wikipedia page, near-misses are not precisely defined — nor can they be, without such a definition (including something such as “no edge may be more than 10% longer than any other) being unjustifiably arbitrary. Instead, new near-miss candidates are discussed among members of the small community of polyhedral enthusiasts with an interest in near-misses, and are either admitted to the set of recognized near-misses, or not, based on consensus of opinion. This isn’t an entirely satisfactory system, but it’s the best we have, and may even be the best system possible.

The shortest definition for “near-miss Johnson solid” is simply “a polyhedron which is almost a Johnson solid.” Recently, a new (and even more informal) term has been created: the “near near-miss,” for polyhedra which are almost near-misses, but with deviations from regularity which are too large, by consensus of opinion, to be called near-misses. This polyhedron may well end up labeled a “near near-miss,” rather than a genuine near-miss.

Several questions remain at this point, and once I have found the answers, I will update this post to include them.

  • Is this close enough to being a Johnson solid to be called a near-miss, or merely a “near near-miss?”
  • Has this polyhedron already been found before? It looks quite familiar to me, and so it is entirely possible I have seen it before, and have simply forgotten when and where I saw it. On the other hand, this “I’ve seen it before” feeling may be caused by this polyhedron’s similarity to the great rhombcuboctahedron (also known as the truncated cuboctahedron, and a few other names), one of the Archimedean solids.
  • Does this polyhedron already have a name?
  • If unnamed at this time, what name would be suitable for it?

All the images in this post were created using Stella 4d, and I also used this software to obtain the numerical data given above. A free trial download of this program is available, and you can find it at http://www.software3d.com/Stella.php. Also, since it was mentioned above, I’ll close this post with a rotating image of the great rhombcuboctahedron. Perhaps a suitable name for the near-miss candidate above would be the “expanded great rhombcuboctahedron,” although it is entirely possible that a better name will be found.

Trunc Cubocta

Update #1: I now remember where I’ve seen this before: right here on my own blog! You can find that post here. I could delete this, as a duplicate post, but am choosing not to. One reason: the paths I took to create these two identical polyhedra were entirely different. Another reason is that this post includes information not included the first time around.

Update #2: This was already discussed among my circle of polyhedral enthusiasts. As I now recall, the irregularity in the quadrilaterals was agreed to be too large to call this a true “near-miss,” so, clearly, it’s a “near near-miss” instead.

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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