A Polyhedral Journey, Beginning with a Near-Miss Johnson Solid Featuring Enneagons

When Norman Johnson first found, and named, all the Johnson solids in the latter 1960s, he came across a number of “near-misses” — polyhedra which are almost Johnson solids. If you aren’t familiar with the Johnson solids, you can find a definition of them here. The “near-miss” which is most well-known features regular enneagons (nine-sided polygons):

ennneagonal-faced near-miss

This is the dual of the above polyhedron:

ennneagonal-faced near-miss dual

As with all polyhedra and their duals, a compound can be made of these two polyhedra, and here it is:

ennneagonal-faced near-miss base=dual compound

Finding this polyhedron interesting, I proceeded to use Stella 4d (polyhedron-manipulation software, available at http://www.software3d.com/Stella.php) to make its convex hull.

Convex hull of near-miss base-dual compound

Here, then, is the dual of this convex hull:

dual of Convex hull of near-miss base-dual compound

Stella 4d has a “try to make faces regular” function, and I next used it on the polyhedron immediately above. If this function cannot work, though — because making the faces regular is mathematically impossible — one sometimes gets completely unexpected, and interesting, results. Such was the case here.

attempt no make latest polyhedron have regular faces

Next, I found the dual of this latest polyhedron.

attempt no make latest polyhedron have regular faces's dual

The above polyhedron’s “wrinkled” appearance completely surprised me. The next thing I did to change it, once more, was to create this wrinkled polyhedron’s convex hull. A convex hull of a non-convex polyhedron is simply the smallest convex polyhedron which can contain the non-convex polyhedron, and this process often has interesting results.

Convex hull of wrinkled dual

Next, I created this latest polyhedron’s dual:

dual of Convex hull of wrinkled dual

I then attempted “try to make faces regular” again, and, once more, had unexpected and interesting results:

dual of latest polyhedron

The next step was to take the convex hull of this latest polyhedron. In the result, below, all of the faces are kites — two sets of twenty-four each.

convex hull of last polyhedron with two sets of two dozen kites each

I next stellated this kite-faced polyhedron 33 times, looking for an interesting result, and found this:

33rd stellation of latest polyhedron

This looked like a compound to me, so I told Stella 4d to color it as a compound, if possible, and, sure enough, it worked.

33rd stellation of latest polyhedron colored as a compound

The components of this compound looked like triakis tetrahedra to me. The triakis tetrahedron, shown below, is the dual of the truncated tetrahedron. However, I checked the angle measurement of a face, and the components of the above compound-dual are only close, but not quite, to being the same as the true triakis tetrahedron, which is shown below.

Triakistetra -- ANGLES AREN'T QUITE A MATCH for last polyhedron

This seemed like a logical place to end my latest journey through the world of polyhedra, so I did.

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 http://www.software3d.com/Stella.php):

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:  https://en.wikipedia.org/wiki/Near-miss_Johnson_solid

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: http://www.software3d.com/Stella.php.

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.

Symmetrohedron Featuring Eighteen Regular Octagons, Eight Equiangular Hexagons, and Twenty-four Isosceles Trapezoids


Symmetrohedron Featuring Eighteen Regular Octagons, Eight Equiangular Hexagons, and Twenty-four Isosceles Trapezoids

The regular octagons are of the same size, but of two different types, when one considers the pattern of other faces surrounding them. This is why six of them are yellow, and twelve are red.

If the hexagons and isosceles trapezoids were closer to regularity, this would qualify as a near-miss to the Johnson solids, but it falls short on this test. Is is, instead, a “near-near-miss” — and not the first such polyhedron to appear on this blog, either.

(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)

An Expansion of the Truncated Icosahedron — and, Perhaps, a New Near-Miss


An Expansion of the Truncated Icosahedron -- and, Perhaps, a New Near-Miss

To create this using Stella 4d (see http://www.software3d.com/stella.php), I started with a truncated icosahedron, augmented each of its faces with a prism that was 1.5 times as tall as the base edge length, and took the convex hull of the result. It may qualify as a near-miss to the Johnson Solids — for that to be the case, all faces would have to be close to regular, but “close to” has no precise definition. I’ll have to consult with the experts on this one!

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 http://www.software3d.com/stella.php for a free trial download of the software used to make these polyhedral images.)


Polyhedron Featuring Decagons and Pentagons


Polyhedron Featuring Decagons and Pentagons

There are twelve regular decagons in this polyhedron, and sixty irregular pentagons. If the pentagons were closer to regularity, this would qualify as a near-miss to the ninety-two Johnson Solids. It is not known how many of these “near-misses” exist — primarily because this group of polyhedra lacks a precise definition.

This polyhedron was discovered with the aid of Stella 4d, software you can try for yourself at http://www.software3d.com/stella.php.

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 http://www.software3d.com/Stella.php, 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.]