Two (New?) “Near-Miss” Candidates

Yesterday, I played for the first time with GeoMag toys, which I recently purchased. I was quite surprised to have what I believe to be a near-miss to the Johnson solids appear before me, one I’ve never seen, within just a few minutes:

SANYO DIGITAL CAMERAHere’s what it looks like, when viewed from two other angles.

The faces of this three-fold dihedral polyedron are six pentagons, twelve triangles, and nine quadrilaterals. The fact that it has been proven that only 92 Johnson solids exist means that all of these faces cannot be regular. However, the irregularity is so small that I could not detect it in this model.

Next, I used Polydrons to build a net of this near-miss candidate.


What to do next was obvious: remove the “belt” of nine quadrilaterals, creating a net for a second near-miss candidate.


Having constructed this net, I then returned  GeoMags to build a 3-d model of this second, “unbelted” near-miss candidate.

I then wondered if I could make a third such solid by removal of the triangles, all of which appeared to be the lateral faces of pyramids.

Could I remove them? Yes, and I did so. Did this create a third near-miss candidate? No. The resulting polyhedron, shown immediately above, is non-convex, and therefore cannot be a near-miss. The faces with dihedral angles greater than 180° are the triangle-pairs found where the pyramids were in the previous model.

With the “belted” and “unbelted” polyhedra before this non-convex non-candidate, the next step is to share them with other polyhedra enthusiasts, get their input regarding the question of whether these are genuine near-misses, and see if these polyhedra have already been found, unknown to me, by someone else. 

[Update: please see the next two posts for more on these near-miss candidates.]

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 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.

New “Near-Miss” Candidate?

As a proposed new “near-miss” to the Johnson solids, I created this polyhedron using Stella 4d, which can be found for purchase, or trial download, here. To make it, I started with a tetrahedron, augmented each face with icosidodecahedra, created the convex hull of the resulting cluster of polyhedra, and then used Stella‘s “try to make faces regular” function, which worked well. What you see is the result.


This polyhedron has no name as of yet (suggestions are welcome), but does have tetrahedral symmetry, and fifty faces. Of those faces, the eight blue triangles are regular, although the four dark blue triangles are ~2.3% larger by edge length, and ~4.6% larger by area, when compared to the four light blue triangles. The twelve yellow triangles are isosceles, with their bases (adjacent to the pink quadrilaterals) ~1.5% longer than their legs, which are each adjacent to one of the twelve red, regular pentagons. These yellow isosceles trapezoids have vertex angles measuring 61.0154º. The six pink quadrilaterals themselves are rectangles, but just barely, with their longer sides only ~0.3% longer than their shorter sides — the shorter sides being those adjacent to the green quadrilaterals.

The twelve green quadrilaterals are trapezoids, and are the most irregular of the faces in this near-miss candidate. These trapezoids have ~90.992º base angles next to the light blue triangles, and ~89.008º angles next to the pink triangles. Their shortest side is the base shared with light blue triangles. The legs of these trapezoids are ~2.3% longer than this short base, and the long base is ~3.5% longer than the short base.

If this has been found before, I don’t know about it — but, if you do, please let me know in a comment.

UPDATE: It turns out that this polyhedron has, in fact, been found before. It’s called the “tetrahedrally expanded tetrated dodecahedron,” and is the second polyhedron shown on this page. I still don’t know who discovered it, but at least I did gather more information about it — the statistics which appear above, as well as a method for constructing it with Stella.

The Second of Dave Smith’s “Bowtie” Polyhedral Discoveries, and Related Polyhedra

Dave Smith discovered the polyhedron in the last post here, shown below, with the faces hidden, to reveal how the edges appear on the back side of the figure, as it rotates. (Other views of it may be found here.)

Smith's puzzle

So far, all of Smith’s “bowtie” polyhedral discoveries have been convex, and have had only two types of face: regular polygons, plus isosceles trapezoids with three equal edge lengths — a length which is in the golden ratio with that of the fourth side, which is the shorter base.

Smiths golden trapezoid

He also found another solid: the second of Smith’s polyhedral discoveries in the class of bowtie symmetrohedra. In it, each of the four pentagonal faces of the original discovery is augmented by a pentagonal pyramid which uses equilateral triangles as its lateral faces. Here is Smith’s original model of this figure, in which the trapezoids are invisible. (My guess is that these first models, pictures of which Dave e-mailed to me, were built with Polydrons, or perhaps Jovotoys.)


With Stella 4d (available here), the program I use to make all the rotating geometrical pictures on this blog, I was able to create a version (by modifying the one created by via collaboration between five people, as described in the last post) of this interesting icositetrahedron which shows all four trapezoidal faces, as well as the twenty triangles.

Smith's Icositetrahedron

Here is another view: trapezoids rendered invisible again, and triangles in “rainbow color” mode.

Smith's Icositetrahedron H

It is difficult to find linkages between the tetrapentagonal octahedron Smith found, and other named polyhedra (meaning  I haven’t yet figured out how), but this is not the case with this interesting icositetrahedron Smith found. With some direct, Stella-aided polyhedron-manipulation, and a bit of research, I was able to find one of the Johnson solids which is isomorphic to Smith’s icositetrahedral discovery. In this figure (J90, the disphenocingulum), the trapezoids of this icositetrahedron are replaced by squares. In the pyramids, the triangles do retain regularity, but, to do so, the pentagonal base of each pyramid is forced to become noncoplanar. This can be difficult to see, however, for the now-skewed bases of these four pyramids are hidden inside the figure.

J90 disphenocingulum

Both of these solids Smith found, so far (I am confident that more await discovery, by him or by others) are also golden polyhedra, in the sense that they have two edge lengths, and these edge lengths are in the golden ratio. The first such polyhedron I found was the golden icosahedron, but there are many more — for example, there is more than one way to distort the edge lengths of a tetrahedron to make golden tetrahedra.

To my knowledge, no ones knows how many golden polyhedra exist, for they have not been enumerated, nor has it even been proven, nor disproven, that their number is finite. At this point, we simply do not know . . . and that is a good way to define areas in mathematics in which new work remains to be done. A related definition is one for a mathematician: a creature who cannot resist a good puzzle.

The Convex Hull of a Prism-Augmented Icosidodecahedron As a (Possibly) New Near-Miss Candidate


The Convex Hull of a Prism-Augmented Icosidodecahedron As a New Near-Miss Candidate

To make this polyhedron using Stella 4d: Polyhedron Navigator (a program which is available at this website), I started with an icosidodecahedron, augmented all faces with prisms of height 1.6 times greater than their bases’ edge length, and then took the convex hull of the result. I’m proposing it as a candidate for the loosely defined group of polyhedra called near-misses to the 92 Johnson solids: convex polyhedra which are almost, but not quite, Johnson solids, due to slight irregularity in some of their faces.

In this case, the pentagons and green triangles are regular, and have the same edge length. The blue triangles, however, are isosceles, with vertex angles of ~67.6687 degrees. The yellow almost-squares are actually rectangles, with edges next to blue triangles which are ~2.536% longer than the edges next to pentagons or green triangles.

I stumbled upon this design earlier today, while simply exploring polyhedra more-or-less randomly, using Stella. Below is the prototype I found at that time, which I merely made a .gif of, but did not perform measurements on.


In this prototype, the most significant difference I can detect is in the yellow faces, which are trapezoids, rather than rectangles, since the pentagon edge-length is slightly longer than that of the green triangles.

Stella has a “try to make faces regular” function built-in to try to help improve upon polyhedra such as these, but here’s what happens when that function is used on the first polyhedron shown above:


Behold! It worked — all of the faces are perfectly regular. However, that caused another problem to appear, and you can see it most easily by looking at the blue triangle-pairs:  this polyhedron is slightly non-convex. It’s also easily described as a truncated dodecahedron, with each of the twelve decagonal faces augmented by a pentagonal rotunda.

I’ll show this to some other people who are polyhedron-experts, and will update this post with what I find after I’ve talked to them. My questions for them, as usual in such situations, are two in number:

1. Has this polyhedron been found before?

2. Is it close enough to regularity to qualify for “near-miss” status?

If it hasn’t been found before, but is judged unworthy of “near-miss” status, it will at least join the newly-described group I call “near near-misses” — polyhedra which don’t quite qualify for near-miss status, by visual inspection. Obviously, this new group’s definition is even more “fuzzy” than that of the near-misses, but there is a need for such a category, nonetheless.

[Update:  Robert Webb, who wrote Stella 4d (and is not the blogger here, despite our sharing a first name), has seen this before, so it isn’t an original discovery of mine. He doesn’t accept it as a near-miss on the grounds that it naturally “wants” to be non-convex, as seen in the last of the three images in this post, and I agree with his reasoning. I’m therefore considering this to be a “near-near-miss.”]

The Zonish Cuboctahedron: A New Near-Miss Discovery?


The Zonish Cuboctahedron:  A New Near-Miss Discovery?

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

new nearmiss before making faces regular its a face based zonish cuboctahedron

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:

    1. Has this polyhedron been found before?
    2. Is it close enough to being a Johnson solid to qualify as a near-miss?

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:

My name made the “Stella 4d” library discovery credits!


My name made the Stella library discovery credits!

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:

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:

Polyhedron Featuring Twenty Regular Enneagons, Twelve Regular Pentagons, and Sixty Isosceles Triangles


Polyhedron Featuring Twenty Regular Nonagons, Twelve Regular Pentagons, and Sixty Isosceles Triangles

If the isosceles triangles in this polyhedron were close enough to being equilateral that close inspection would be required to tell the difference, this would be a near-miss to the Johnson Solids. However, in my opinion, this doesn’t meet that test — so I’m calling this a “near-near-miss,” instead.

Software credit: visit this website if you would like to try a free trial download of Stella 4d, the program I used to create this image.

Expanded Truncated Icosahedron III


Expanded Truncated Icosahedron III

This version of an expanded truncated icosahedron was made in a similar fashion to the one two posts ago — but with the augmentation-by-prisms step altered by using prisms of somewhat greater height, with the goal being to make the rectangular faces closer to “squareness.”

As a result, this polyhedron is closer than the others to being a near-miss to the Johnson Solids — a category of polyhedra which has been discussed on this blog before. “Near-misses” are almost Johnson solids, and must therefore have faces which are regular (as the hexagon and pentagons here are) are nearly-regular (as the rectangles and triangles come close to being).

Is this a near-miss, or is it not? That’s difficult to answer, for that set of polyhedra has no precise definition, and cannot have one — it only has a “fuzzy” definition.

In my opinion, it isn’t quite a near-miss, for the triangles are too non-equilateral to qualify, to my eye — but others might disagree. In the unlikely event that mathematicians wish to start talking about “near-near-misses,” I offer this as a member of that set.

(Software credit:  see for a free trial download of the software used to make these polyhedral images.)


A Second Version of My New Near-Miss to the Johnson Solids


A Second Version of My New Near-Miss

A few days ago, I found a new near-miss to the 92 Johnson Solids. It appears on this blog, five posts ago, and looks a lot like what you see above — the differences are subtle, and will be explained below, after “near-miss” has been clarified.

A near-miss is a polyhedron which is almost a Johnson Solid. So what’s a Johnson Solid?

Well, consider all possible convex polyhedra which have only regular polygons as faces. Remove from this set the five Platonic Solids:

Next, remove the thirteen Archimedean Solids:

Now remove the infinite sets of prisms and antiprisms, the beginning of which are shown here:

What’s left? The answer to this question is known; it’s the set of Johnson Solids. It has been proven that there are exactly 92 of them:

When Norman Johnson systematically found all of these, and named them, in the late 1960s, he found a number of other polyhedra which were extremely close to being in this set. These are called the “near misses.” An example of a near-miss is the tetrated dodecahedron, which I co-discovered, and named, about a decade ago:

If you go to, you can download a free trial version of software, Stella 4d, written by a friend of mine, Robert Webb (RW), which I used to generate this last image, as well as the rotating .gif which starts this post. (The still pictures were simply found using Google image-searches.) Stella 4d has a built-in library of near-misses, including the tetrated dodecahedron . . . but it doesn’t have all of them.

Well, why not? The reason is simple: the near-misses have no precise definition. They are simply “almost,” but not quite, Johnson Solids. In the case of the tetrated dodecahedron, what keeps it from being a Johnson Solid is the edges where yellow triangles meet other yellow triangles. These edges must be ~7% longer than the other edges, so the yellow triangles, unlike the other faces, are not quite regular — merely close.

There is no way to justify an arbitrary rule for just how close a near-miss must be to “Johnsonhood” be considered an “official” near-miss, so mathematicians have made no such rule. Research to find more near-misses is ongoing, and, due to the “fuzziness” of the definition, may never stop.

My informal test for a proposed near-miss is simple:  I show it to RW, and if he thinks it’s close enough to include in the near-miss library in Stella 4d, then it passes. This new one did, but not until RW found a way to improve it, using something I don’t really understand called a “spring model.” What you see at the top of this post is the result. Unlike in the previous version, the green decagons here are regular, but at the expense of regularity in the (former) blue squares, now near-squarish trapezoids, as well as the yellow hexagons. The pink hexagons are slightly irregular in both versions, and the red pentagons are regular in both.

I’m eagerly anticipating the release of the next version of Stella 4d, for this near-miss will be in it.  If I tell my students about this new discovery, they’ll want to know how much I got paid for it, which is, of course, nothing. I don’t know how to explain to them what it feels like to participate in the discovery of something — anything — which will survive me by a very long time. There’s nothing else quite like that feeling.

Now I just need to think of a good name for this thing!

[Update:   the new version of Stella is now out, and this polyhedron is now included in it. As it turns out, I no longer need to think of a name for this polyhedron, for RW took care of that for me, naming it the “zonish truncated icosahedron” in Stella‘s built-in library of polyhedra.]