A Polyhedral Journey, Beginning with Face-Based Zonohedrification of an Icosahedron

To begin this, I took an icosahedron, and made a zonish polyhedron with it, with the new faces based on the zones of the existing faces. Here’s the result.

1 face-based zonish icosahedron

Next, I started stellating the polyhedron above. At the sixth stellation, I found this. It’s a true zonohedron, and the first polyhedron shown here is merely “zonish,” because one has triangles, and the other does not. (One of the requirements for a polyhedron to be a zonohedron is that all its faces must have an even number of sides.)

2 6th stellation face-based zonish icosahedronAfter that, I kept stellating, finding this as the 18th stellation of the first polyhedron shown here.

3 18th stellation face-based zonish icosahedron

With this polyhedron, I then made its convex hull.

4 Convex hull of 18th stellation of face-based zonish icosahedronAt this point, the irregular hexagons were bothering me, so I used Stella 4d‘s “try to make faces regular” option. (Stella 4d is polyhedron-manipulation software you can try for free, or purchase, right here.)

5 spring model of convex hull of 18th stellation of face-based icosahedron

The next step I chose was to augment all the yellow trapezoids with prisms, each with a height 1.6 times the trapezoids average edge length.

6 Augmented sping model of convex hull of 18th stellation of FBZI

The next step was, again, to make the convex hull.

7 Convex hull of augmented convex hull

At this point, I tried “try to make faces regular” again, and was pleased with the result. The green rectangles became so thin, however, that I had to stop displaying the edges and vertices, in order for then to be seen.

8 spring model of last oneNext, I augmented both the blue faces (decagons) and the yellow faces (dodecagons) with antiprisms, again using a height 1.6 times that of the augmented faces’ average edge-lengths.

9 Augmented Poly 9th in series

Next, I made the convex hull again — a step I often take immediately after augmenting a polyhedron.

10 Convex hull

This one surprised me, as it is more complicated than I expected. To clean things up a bit, I augmented only the trapezoids (dark pink) with prisms, and dodecagons (green) with antiprisms, again using the factor 1.6 for the augmentation-height.

11 augmented Convex hull

The next step I chose was to take the convex hull, once more. I had not yet noticed that the greater height of the trapezoidal prisms would cause the dodecagonal antiprisms to be “lost” by this step, though.

12 convex hull

Next, “try to make faces regular” was used again.

13 spring model

This last result had me feeling my polyhedral journey was going in circles, so I tried augmentation again, but in a different way. I augmented this polyhedron, using prisms, on only the red trapezoids (height factor, 1.6 again) and the blue rectangles (new height factor, 2.3 times average edge length).

14 augmented spring model

After that, it was time to make another convex hull — and that showed me that I had, indeed, taken a new path.

15 Convex hullI found the most interesting faces of this polyhedron to be the long, isosceles trapezoids, so I augmented them with prisms, ignoring the other faces, using the new height-factor of 2.3 times average edge length this time.

16 augmented Convex hull

Of course, I wanted to see the convex hull of this. Who wouldn’t?

17 Convex hull

I then started to stellate this figure, choosing the 14th stellation as a good place to stop, and making the edges and vertices visible once more.

18 the 14th stellation of the previous Convex hull

A Zonish Icosahedron, and Some of Its “Relatives”

To begin this, I used Stella 4d (available here) to create a zonish polyhedron from the icosahedron, by adding zones along the x-, y-, and z-axes. The result has less symmetry than the original, but it is symmetry of a type I find particularly interesting.

zonohedrified icosahedron xyz

After making that figure, I began stellating it, and found a number of interesting polyhedra in this polyhedron’s stellation-series. This is the second such stellation:

zonohedrified icosahedron xyz 2nd stellation

This is the 18th stellation:

zonohedrified icosahedron xyz 18th stellation

The next one, the 20th stellation, is simply a distorted version of the Platonic dodecahedron.

zonohedrified icosahedron xyz 20th stellation

This one is the 22nd stellation:

zonohedrified icosahedron xyz 22nd stellation

This is the 30th stellation:

zonohedrified icosahedron xyz 30th stellation

The next really interesting stellation I found was the 69th:

zonohedrified icosahedron xyz 69th stellation

At this point, I returned to the original polyhedron at the top of this post, and examined its dual. It has 24 faces, all of which are quadrilaterals.

zonohedrified icosahedron xyz dual

This is the third stellation of this dual — and another distorted Platonic dodecahedron.

zonohedrified icosahedron xyz dual 3rd stellation

This is the dual’s 7th stellation:

zonohedrified icosahedron xyz dual 7th stellation

And this one is the dual’s 18th stellation:

zonohedrified icosahedron xyz dual 18th stellation

At this point, I took the convex hull of this 18th stellation of the original polyhedron’s dual, and here’s what appeared:

Convex hull of 18th stellation of dual of zonish icosahedron xyz

Here is this convex hull’s dual:

dual of Convex hull of 18th stellation of dual of zonish icosahedron xyz

Stella 4d, the program I use to make these (available here), has a built-in “try to make faces regular” function. When possible, it works quite well, but making the faces of a polyhedron regular, or even close to regular, is not always possible. I tried it on the polyhedron immediately above, and obtained this interesting result:

spring model of Dual of convex hull of stellation of zonish xyz icosahedron

While interesting, this also struck me as a dead end, so I returned to the red-and-yellow convex hull which is the third image above, from right here, and started stellating it. At the 19th stellation of this convex hull, I found this:

19th stellation of Convex hull of 18th stellation of dual of zonish icosahedron xyz

I also found an interesting polyhedron as the 19th stellation of the dual which is three images above:

19th stellation of dual of Convex hull of 18th stellation of dual of zonish icosahedron xyz

Faced-Based Zonish Versions of the Icosahedron and the Icosidodecahedron


Faced-Based Zonish Versions of the Icosahedron and the Icosidodecahedron

I’ve had some success lately finding near-misses to the Johnson solids by making face-based zonish versions of various polyhedra. These were found during that search, and are certainly not near-misses, but I still find them interesting, primarily due to their symmetry. Like the others, they were found using Stella 4d, which you can try or buy at http://www.software3d.com/Stella.php.

The top image was formed by making this modification to the icosahedron, and the one below (which you can enlarge with a click) was created by doing the same thing to an icosidodecahedron.


The Seven Zonish Rhombicosidodecahedra Based On Symmetry Axes


Zonish Versions of the Rhombicosidodecahedron

The top image here is of a zonish polyhedron based on adding zones along the five-fold symmetry axes of a rhombicosidodecahedron. All its edges are the same length, and its 62 faces include thirty elongated octagons, twelve regular pentagons, and twenty triangles. All of its edges have the same length.

The edges of this next polyhedron are also all of the same length. It was made in the same way, except that zones were added along both three- and five-fold symmetry axes of a rhombicosidodecahedron. Its 182 faces include thirty elongated dodecagons, twenty triangles, twelve regular pentagons, sixty squares, and sixty rhombi.

182 faces incl 30 elongated dodecagons and 12 pentagons and 60 squares and sixty rhombi and twenty triangles

If only the three-fold symmetry axes are used to make a zonish polyhedron from a rhombicosidodecahedron, this next polyhedron, also with all edge lengths equal, is the result. It also has 182 faces, and they are of the same type as in the one immediately before, except that thirty elongated octagons replace the dodecagons from that polyhedron.

zonish rid

A rhombicosidodecahedron also has two-fold symmetry axes. If only those axes are used to make a zonish rhombicosidodecahedron, this next polyhedron is the result:  a modified form of the great rhombicosidodecahedron, with unequal edge lengths.

zonish rid

If the two- and three-fold symmetry axes are both used, the result, once again, is a 182-faces polyhedron, but it also has unequal edge lengths, and none of its faces are regular polygons. It is shown below. There are twelve decagons, sixty rectangles, sixty hexagons of one type, twenty hexagons of another type,  and thirty octagons.

zonish rid

Another possible combination is to use the two- and five-fold symmetry axes to create a zonish rhombicosidodecahedron. This yields a polyhedron with 122 faces, with all except the sixty squares being irregular. The other faces are twelve decagons, thirty octagons, and twenty hexagons:

zonish rid

Finally, there is one last combination — using the two-, three-, and five-fold symmetry axes, all at once. Here’s what it looks like:

zonish rid

As one should expect, this produces a zonish polyhedron with more faces than any of the earlier ones shown above: 242 in all. As in the last one shown, only the sixty squares are regular, although the sixty pink hexagons are at least equilateral. There are also sixty rectangles, twenty hexagons of a second type, thirty dodecagons, and twelve decagons.

All of these zonish rhombicosidodecahedra were created using Stella 4d, software available at http://www.software3d.com/Stella.php.

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.

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.]

Zonish Icosidodecahedron


Zonish Icosidodecahedron

To make this, I started with an icosidodecahedron, then added zonogons to the existing faces, in the ten zones along the three-fold symmetry axes.

Software used: Stella 4d. You may find it at www.software3d.com/stella.php.

Rotating Zonish Polyhedron


Rotating Zonish Polyhedron

Creating zonish polyhedra (related to zonohedra) and creating rotating polyhedral .gifs are two features newly added to Stella, the program used to make this, which may be tried for free at http://www.software3d.com/stella.php