Two Views of an Icosahedron, Augmented with Great Icosahedra

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

Augmented Icosa using grt icosas

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.

Augmented Icosa using grt icosas parallel faces colored together

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

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

OLYMPUS DIGITAL CAMERA
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.

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

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

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.

BELTED POLYHEDRON 11 PERCENT EDGE DEVIATION 4 EDGE LENGTHS

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

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.]