Near-Miss Candidate Update #2

Posted on 14 March 2016 by RobertLovesPi

With some work, I was able to figure out how to make my second near-miss candidate from two posts ago, using Stella 4d (available here), but the results show it is a “near near miss,” not a near miss. Like the first one, the triangles are visibly irregular — and so are the green rectangles; there are also four edge lengths, the longest of which is ~11% longer than the shortest. This is not close enough to qualify as a near-miss.

Not long after I made the image above, a friend I shall simply call T. (until and unless I have his permission to publish his full name) e-mailed me his own versions he made, also using Stella. Here’s what they look like. Each can be enlarged with a click.

These are improved in the sense that the triangles (and squares, in the second one) are regular, but this was done at the expense of the pentagons. At the top and bottom of the figures, the edges where pentagons meet other pentagons are ~6.8% shorter than the other edges of each figure.

These last two are more likely to qualify for actual “near-miss” status — that has yet to be decided — but I need to make it clear than I did not discover them alone, but as part of a team. In my versions, after all, the flaws are more severe. Also, we do not yet know whether or not a different individual or team found these same polyhedra earlier, as often happens.

Two (New?) “Near-Miss” Candidates

Posted on 13 March 2016 by RobertLovesPi

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:

Here’s what it looks like, when viewed from two other angles.

SANYO DIGITAL CAMERA

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.

SANYO DIGITAL CAMERA

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.

SANYO DIGITAL CAMERA

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.]

Expanded Truncated Icosahedron III

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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

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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.

Speculation About a Zonish Polyhedron

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Speculation About a Zonish Polyhedron

If the longer hexagon/octagon sides were all shortened, could this become nearly regular, thus qualifying it to join the “near-misses?” As it is, those long edges are too long to call it by that term, but perhaps this can be fixed.

(Note: If you’d like to make stuff like this yourself, you should check out http://www.software3d.com/stella.php.)

Speculation Related to the Johnson Solids

Posted on 20 October 2012 by RobertLovesPi

Consider all possible convex polyhedra which have 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:

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, which I used to generate this last image. This program has a built-in library of near-misses . . . 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.

I’ve played a small part in such research, myself. I’ve also been asked how much I’ve been paid for doing this work, but that question misses the point. I’ve collected no money from this, and nobody gets involved in such research in order to get rich. Those of us who do such things are motivated by the desire to have fun through indulgence of mathematical curiosity. Our reward is the pure enjoyment of trying to figure things out, and, on really good days, actually doing so.

I’m having a good day. I’m looking at the Johnson Solids in a different way, purely for fun. I have found something that may be a blind alley, but, if my fellow geometricians show me that it is, that won’t erase the fun I have already had.

Here’s what I have found today. It is not a near-miss in the same way as the tetrated dodecahedron, but is related to the Johnson Solids in a different way. Other than a “heptadecahedron” (for its seventeen faces) it has no name, as of yet:

How is this different from traditional near-misses? Please examine the net (third image). In this heptadecahedron, all of these triangles, pentagons, and the one decagon are perfectly regular, unlike the situation with traditional near-misses. However, some faces, as you can see in the 3-d model, are made of multiple, coplanar equilateral triangles, joined together. In the blue faces, two such triangles form a rhombus; in the yellow faces, three such triangles form an isosceles trapezoid. Since they are coplanar and adjacent, they are one face each, not two, nor three. The dashed lines are not folded in the 3-d model, but merely show where the equilateral triangles are.

Traditional near-misses involve relaxation of the rules for Johnson Solids to permit polyhedra with not-quite-regular faces to join a new “club.”

Well, this heptadecahedron is in a different “club.” To join it, a polyhedron must fit the criteria for “Johnsonhood,” except that some faces may be formed by amalgamation of multiple, coplanar regular polygons.

My current subject of speculation is this: would this new club have an infinite or a finite number of members? If finite, it will, I think, be a larger number than 92. If finite, it will also be a more interesting topic to study.

I don’t know, yet, what answer this new problem has. I do know I am having fun, though. Also known: no one will pay me for this. No one needs to, either.