10 | June | 2015 | RobertLovesPi.net

There’s a binary dwarf-planet-candidate / large satellite pair, way out in the outer solar system, called Orcus and Vanth. Much like the “double dwarf planet” Pluto/Charon, and the other satellites in that system, Orcus and Vanth orbit the sun in a 3:2 resonance with Neptune, and this orbit crosses that of Neptune, as well. The Orcus/Vanth binary system is sometimes referred to as the “anti-Pluto,” because, unlike most “plutinos” (as such distant objects, in orbital resonance with Neptune, are called), Orcus and Vanth have a strange — and, so far, unexplained — relationship with the Pluto/Charon system. When Pluto and Charon are closest to the sun (perihelion), Orcus and Vanth are at their furthest from the sun (aphelion), and vice-versa. So far as I have been able to determine, this is not true for any other known plutinos. For more on the real Orcus and Vanth, please check this Wikipedia page.

Those are the scientific facts, as we know them . . . and now, it’s time for some silliness. On Facebook, recently, I mentioned that “Orcus” and “Vanth” really would make good names for comic book characters, but that I couldn’t decide what they should look like, nor what powers they should have. A discussion with some of my friends followed, and, together, we decided that Orcus should be a tough fighter-type, while “Vanth” sounded like a name for some sort of spell-caster. It didn’t take long before I decided I should visit one of the numerous create-your-own-comic-book-character websites, and go ahead and make quasi-anthropomorphized images of Orcus and Vanth — the characters, not the outer solar-system objects.

I used a website called Hero Machine for this diversion, which you can find here. First, I created an image for a character named Orcus.

Unfortunately, I didn’t discover (until it was too late) that this website allows the user to change the background . . . and I didn’t want to re-make Orcus, so I went ahead and created an image of his companion, Vanth, instead.

I don’t have the time, nor the artistic talent, to write and illustrate actual comic book stories featuring this pair of characters . . . but perhaps someone will read this, and decide they want to take on such a project. That’s fine with me . . . but I want credit (in writing, each issue) for creating them, and, if the endeavor makes any money, I want at least 20% of the profits, and that’s if I have nothing more to do with creating Orcus and Vanth stories, beyond what is posted here. If I do have additional involvement, of course, we’ll need to carefully negotiate the terms of a contractual agreement. I consider 20% fair for simply creating images of this pair of characters, but actually co-creating stories would be something else altogether.

By the way, although Orcus certainly looks scarier, Vanth is actually the more formidable of the pair. She just pretends to play the “side-kick” role, in order to preserve the element of surprise, for situations when, during their adventures, Orcus finds himself in over his head, and Vanth then needs to really cut loose with the full extent of her abilities.

I stumbled upon this interesting hybrid of two well-known polyhedra, while simply playing around with Stella 4d, the software I use to make these rotating polyhedral images (you can try a free trial download of it here).

The faces of the above polyhedron are twelve modified regular pentagons, each with a triangular piece removed which contained one of the pentagon’s edges. Therefore, it would also be correct to refer to these modified pentagons as non-convex hexagons. These modified pentagons interpenetrate, so all that can be seen are triangular “facelets” — the parts of the faces which are not hidden inside the polyhedron. Each of these facelets is a golden gnomon (an obtuse, isosceles triangle with a base:leg ratio which is the golden ratio), and these golden gnomons come in two sizes. The larger ones were “inherited” from Jessen’s icosahedron, and there are twelve of them. The smaller golden gnomons, on the other hand, were “inherited” from the great dodecahedron, and are twenty-four in number, in eight sets of three. Like Jessen’s icosahedron itself, but unlike the great dodecahedron, this hybrid has pyritohedral symmetry.

For more information about Jessen’s icosahedron, please visit this site at Wolfram Mathworld. Also, here is an image of Jessen’s icosahedron, one of the two “parents” of the hybrid above.

While Jessen’s icosahedron is a relatively new discovery (Børge Jessen revealed it to the world in 1967), the hybrid’s other “parent,” the great dodecahedron, has been known for much longer; Louis Poinsot discovered it in 1809, according to this source. Here’s an image of the great dodecahedron.

As you can see, the smaller golden gnomons found in the hybrid above were “inherited” from the great dodecahedron, while the larger ones came from the six indented face-pairs found in Jessen’s icosahedron.

A well-known property of Jessen’s icosahedron is that it is “shaky,” unlike most polyhedra, which are rigid. A physical model of Jessen’s icosahedron, made from paper and tape, can, in fact, be collapsed to form an octahedron. While I suspect that a physical, paper-and-tape model of this newly-discovered hybrid polyhedron would share these properties (“shakiness,” and at least some degree of collapsibility), I have not (yet) tested this conjecture.