Two Views of an Icosahedron, Augmented with Great Icosahedra

Posted on 30 March 2016 by RobertLovesPi

If colored by face-type, based on face-position in the overall solid, this “cluster” polyhedron looks like this:

There is another interesting view of this polyhedral cluster I like marginally better, though, and that is to separate the faces into color-groups in which all faces of the same color are either coplanar, or parallel. It looks like this.

Both versions were created by augmenting each face of a Platonic icosahedron with a great icosahedron, one of the four Kepler-Poinsot solids. I did this using Stella 4d: Polyhedron Navigator, available here.

A Polyhedron with Exactly 200 Faces

Posted on 30 March 2016 by RobertLovesPi

200 faces 60 pentagons and 140 hexagons

Sixty of the faces of this polyhedron are pentagons (orange), and the other 140 are hexagons of three types (blue, pink, and purple). I made it using Stella 4d, a program available at http://www.software3d.com/Stella.php.

Bashing Some Democrats, for a Change

Posted on 18 March 2016 by RobertLovesPi

Image from Wikipedia: http://en.wikipedia.org/wiki/Donkey

I am sick of certain Bernie Sanders supporters who write about the “Hitlarites” who support Hillary Clinton.

I am also sick of the Hillary Clinton supporters who mock her opponent as “Barnie” Sanders, as in Barney the Clown, or perhaps Barney the purple dinosaur.

My guess is that both Hillary Clinton and Bernie Sanders, themselves, are embarrassed by these rude factions of their own supporters, and wish they would just shut up, and sit down.

They’re not helping anyone, except for Donald Trump.

My Personal, Presidential Anti-Endorsement

Image

against

I’m not a Democrat. On the other hand, the Republicans are not in the habit of giving me any other viable options.

The “Trick Johnson” (?) — A Near-Miss Johnson Solid, Surrounded by Hilariously Mistranslated Japanese

Posted on 16 March 2016 by RobertLovesPi

I did not discover this polyhedron, although I wish I had, for it has quite a clever design.

The page where I found it (poorly-translated English version, where it’s called the “Trick Johnson,” whatever that means) is at http://www.geocities.jp/ikuro_kotaro/koramu/1053_g2.htm). I generally don’t repost much work by others here, but, for the “Trick Johnson,” I’m making an exception. By appearance, it’s a near-miss to the Johnson solids, based on combining characteristics of the dodecahedron, the snub cube, and the snub dodecahedron. It has chiral four-fold dihedral symmetry.

If you understand Japanese, I’m sure there’s a lot of interesting information at that linked page. If, on the other hand, you don’t, there’s still a good reason to follow that link: making fun of Google-Chrome’s built-in translator.

“Come very! It makes it the.” Say what?

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.

Near-Miss Candidate Update #1

Posted on 13 March 2016 by RobertLovesPi

With help from friends on Facebook, I was able to figure out how to make the second of the near-miss candidates in the last post, using Stella 4d: Polyhedron Navigator, a program available here. This is quite helpful, for Stella has a “measurement mode” than lets me determine just how far off from regularity a given polyhedron is. This is what the “unbelted” polyhedron from the last post looks like, with the pentagons regular:

near near miss

In this polyhedron, although the pentagons are regular, the triangles are scalene, with angles measuring ~55.35, ~60.81, and ~63.84 degrees. Of the three edge lengths needed for this, the longest is ~9.1% longer than the shortest, and the triangles are definitely non-regular — by visual inspection alone. It is possible to “tidy up” the triangles a bit, but only at the cost of making the pentagons visibly irregular. This is enough to make the call on the “unbelted” near-miss candidate from the last post — it’s a “near near miss,” not a true “near miss.”

All polyhedra in the last post, as it turns out, are related to another near-miss, the discovery of which I had nothing to do with. It has six pentagonal faces, and four which are quadrilaterals. This near-miss may be found here: http://www.mathcurve.com/polyedres/enneaedre/enneaedre.shtml.

[Note: see the next post, also, for more about these polyhedra.]

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.]