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.

This is the rhombic dodecahedron, the dual of the Archimedean cuboctahedron.

While the rhombic dodecahedron has 12 faces, there are many other polyhedra made entirely out of rhombi, and most of them have more than twelve faces. An example is the rhombic enneacontahedron, which has two face-types: sixty wide rhombi, and thirty narrow ones. It is one of several possible zonohedrified dodecahedra.

As the next figure shows, the wide rhombi of the rhombic enneacontahedron have exactly the same shape as the rhombic dodecahedron’s faces, so the two polyhedra can be stuck together (augmented) at those faces. These wide rhombi have diagonals with lengths in a ratio of one to the square root of two.

The next picture shows what happens if you take one central rhombic enneacontahedron, and augment all sixty of its wide faces with rhombic dodecahedra.

Since this polyhedral cluster in non-convex, it can be changed by creating its convex hull, which can the thought of as pulling a rubber sheet tightly around the entire polyhedron. Here’s the convex hull of the augmented polyhedron above.

The program I use to make these rotating images, Stella 4d (which you can try here), has a function called “try to make faces regular.” If applied to the convex hull above, this function leaves the triangles and pentagon regular, and makes the octagons regular as well. However, the rhombi become kites. The rectangles merely change, getting slightly longer, while rotating 90º, but they do remain rectangles.

After creating this last polyhedron, I started stellating it. After stellating it eight times, I obtained this polyhedron:

Once more, I applied the “try to make faces regular” function.

This polyhedron has five-valent vertices where the shorter edges of the kites meet. These are also the vertices of pentagonal pyramids which use kite-diagonals as base edges. By using faceting (the inverse operation of stellation), I next removed these pyramids, exposing their regular pentagonal faces.

In this polyhedron, all faces are regular, except for the red triangles, which are isosceles. (There are also triangles — the pink ones — which are regular.) Each of these isosceles triangles has ~63.2º base angles, and a ~53.6º vertex angle, with legs just under 11% longer than the base. This is a judgement call, for “near-miss” to the Johnson solids has not been precisely defined, but I see an ~11% edge-length difference as too great for this to be classified as a “near-miss,” even though I would love to claim discovery of another near-miss to the Johnson solids (if it even turns out I am the first one to find this polyhedron, which may not be the case). It is close to being a near-miss, though, so it belongs in the even-less-precisely defined group of polyhedra which are called, quite informally, “near near-misses.”

For the sake of comparison, here is a similar polyhedron (included in Stella 4d‘s enormous, built-in library of polyhedra) which is recognized as a near-miss to the Johnson solids. (I do not know the name of the person who discovered it, or I would include it here — I only know it wasn’t me.) It’s called the “half-truncated truncated icosahedron,” and its longest ledges are just over 7% longer than its shorter edges, with the non-regularity of faces also limited to isosceles triangles. However, this irregularity appears in all of the triangles in the polyhedron below — and in the “near near-miss above,” the irregularity only appeared in some of the triangular faces.

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):

This is the dual of the above polyhedron:

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

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.

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

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.

Next, I found the dual of this latest polyhedron.

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.

Next, I created this latest polyhedron’s dual:

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

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.

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

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.

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.

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

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.”]

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:

Has this polyhedron been found before?

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).

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.

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.