Four Different Clusters of Multiple Rhombicosidodecahedra

Octa 5 augmented with 40 RIDs

To make the cluster above, I began with the compound of five octahedra, which has 5(8) = 40 faces, all of them equilateral triangles. Next, I augmented each of those triangular faces with a single rhombicosidodecahedron — forty in all.

Next, I started anew with the compound of five dodecahedra, which has 5(12) = 60 pentagonal faces, all of them regular. Each of these sixty pentagons was then augmented by a single rhombicosidodecahedron.

Dodecahedra 5 augmented by 60 RIDs

For the next cluster, I started with the most well-known compound of ten tetrahedra. There are actually two such compounds; I used the one which is the compound of the chiral five-tetrahedron compound, combined with its mirror image. Since 10(4) = 40, this cluster, like the first one in this post, contains forty rhombicosidodecahedra. Unlike the other models shown here, this one has “holes,” which you can see as it rotates, but the reason for this is a mystery to me. The same is true for the first cluster shown in this post.

Tetrahedra 10 augemnted with 40 RIDs

There also exist two compounds of eight tetrahedra each, and I used one of them for this next cluster, using the same procedure, so this cluster is composed of 8(4) = 32 rhombicosidodecahedra.

Tetrahedra 8 augmented with 32 RIDs

All four of these clusters were created with Stella 4d, a program you may try for free here.

The Compound of the Truncated Octahedron and the Rhombic Dodecahedron

RD trunct oct compound

I created this compound using Stella 4d, software you may try at this website.

Orcus and Vanth

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 diversionwhich you can find here. First, I created an image for a character named Orcus.

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.

vanth

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.

A Hybrid Polyhedron: The “Offspring” of Jessen’s Icosahedron and the Great Dodecahedron

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).

Jessens icosa meets the great dodeca

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.

Jessens Icosa

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.

Great Dodeca

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.

On the Verge

Mysterious

I made this with Stella 4d, which you can download here.

The Great Rhombicosidodecahedron, Built from Rhombic Triacontahedra, and Its Dual

The great rhombicosidodecahedron is also known as the truncated icosidodecahedron (and, confusingly, several other names). Regardless of what it’s called, these pictures demonstrate that this Archimedean solid can be constructed using rhombic triacontahedra as building-blocks.

First, here’s one in the same color I used for the decagonal ring of rhombic triacontahedra in the last post:

GRID of Rhombic Triaconta

The next one is identical, except I used “rainbow color mode” for it.

GRID of Rhombic Triaconta RB

Also, just in case you’re curious, here’s the dual of this polyhedron-made-of-polyhedra — this time, colored by face-type.

dual of GRID of Rhombic Triaconta

These virtual models were all built using Stella 4d, software you may buy, or try for free, right here.

Decagonal Ring of Rhombic Triacontahedra

ring of ten Rhombic Triaconta

Ten rhombic triacontahedra fit perfectly into a decagonal ring. It’s not a “near-miss” — the fit is exact.

I made this with Stella 4d, software you can try for free, or purchase, at http://www.software3d.com/Stella.php.

A Rhombic Dodecahedron, Built from Icosahedra and Octahedra

RD made of octas and icosas

I assembled this using Stella 4d, software available here.

A Table of Known Masses for Numerous Objects in the Solar System, in Kilograms, Solar Masses, Jovian Masses, Terran Masses, and Lunar Masses

solar system object masses

The source of the information in the first two columns is this Wikipedia page. I calculated the numbers in the other columns, so any errors there are my own.

There are many other objects of known mass in the solar system, but I tried not to skip any, as I worked from larger-mass objects down toward those of smaller mass. Skipping some was necessary, though, for there are many objects (the likely dwarf planet Sedna is but one example) for which the mass is simply unknown. The next one I encountered after the asteroid Pallas did not have a name, but merely an alphanumerical designation, so I decided to stop there.

An Attempt to Blend Five Snub Cubes with One Snub Dodecahedron

snubby

Viewers will be the judges of how successful this attempt to blend these polyhedra actually is. I made it using Stella 4d, software you can try right here.