Two More Abstract Models

After building my "Esses" model (see previous post) I set to work on two more "abstract" models (models where the faces interweave without enclosing an area). The first is a model I designed myself that is based on the first stellation of the icosahedron. For those of you who don't know what that means, a stellation of a model is when the faces of a model are extended until they intersect. And if that still makes no sense, what stellation does in practice is make convex (round) polyhedra into stars (from the greek "stella", or "star").

Anyhow, the first stellation of the icosahedron is a pretty dull looking model, which is why I chose it as the subject for an interesting looking abstract model. The model has 20 pinwheel-shaped parts, which is why I chose bright colors to build it with. The way in which they interweave leaves 12 large pentagonal holes (shown below) which I think look pretty neat.

The second model in this post is a paper model I made of a sculpture by George Hart. It's a fairly complex model, so I hope the pictures can at least give you a sense of the thing. It has two parts to it that interweave with each other without actually being glued togother. The core of the model (in light green) is basically a rhombic triacontahedron with sections cut out of the faces. The outer red structure is a stellation of this core, also with sections cut out of the faces.

They each are made of 30 similar-looking parts, which were very difficult to cut out of cardstock! Since the red and green parts of the model interweave but are not held together by glue or tape, they can move around slightly relative to each other. This is an *extremely* delicate model! As you can see by this next image (photos courtesy of Sam Scheidler, by the way), this model also has neat pentagonal holes.

So, there you have it! These are my current most recently completed models (except for a quasitruncated hexahedron, but that was just for fun), but I have lots more on the way. Particularly a great dodecicosidodecahedron in 7 shades of pink and purple. But maybe I'll save that for Valentine's day.

Esses

A month or so ago I was fooling around on Great Stella, faceting stellations of the rhombic triacontahedron, when I came across this cool shape. It is made up of 30 "S" shaped pieces (or backwards S's, depending on whether you make it left- or right-handed), and they interweave in very interesting ways. The really neat thing about this model is that even though the pieces interweave among eachother quite a bit, they don't actually touch except at the twelve points of the model. Also, each point is not directly connected to any of the points directly next to or across from it; only to points exactly two points away. This was my first "open-faced" model; in other words, it is made up of strips of paper that don't actually have any thickness.

Unfortunately, a few days after building this model I found two other designs similar to it. The first is by Robert Webb, who created the Great Stella program that I like so much. His model is topologically the same, though he made the peices a lot thicker. And the second similar design is the sculpture "Compass Points" by George Hart (the "inside" structure of this model is the same basic shape as mine). Well, it was still fun to build something that I at least thought I had discovered.

Inspired by this model, I have already built two more "open-faced" models. One is a direct copy of a model by George Hart, and the other is my own design (and this time it really is my own). As soon as I can get Sam to photograph them I'll post on them.

Super-Hemi-Twister-thingy

After around 14 hours of cutting, scoring, folding, and gluing, my model of a stellation of the small (or great) dodecahemicosahedron is finally finished! Since most of you reading this blog probably have no idea what that means, you can call it a Super-Hemi-Twister or something like that. It has (for all of you out there who like ot know these kinds of things) 32 faces, which, since they intersect a lot, are divided into 180 visible facelets (small sections of faces). Since some adjacent facelets are the same color, they were combined into the same actual piece that I cut cut of paper (aka, net). So... the actual number of parts I cut out was 120, making this my fourth model with exactly 120 parts! However, because of the intricacy of this model, it was by far the most difficult. Since you can't tell from the picture, I should mention that it has 7 colors: white for the icosahedral faces and red, blue, yellow, orange, purple, and green for the dodecahedral faces.

One of the neat things about this model are the 12 holes that go right to the center of the model. You can see one of them in this close-up.
Since this model was so intricate, getting the last part in was extremely difficult. Eventually I had to use a hot glue gun for the last few joints. But the nice thing about a polyhedron model is that you can always turn it the other way. So no one who reads this blog will ever know. Except that I just gave it away... ah, well.

Before I forget, photo credits for this post (and I guess all of them so far) are due to my brother Sam.

Speaking of photos, I just put an actual photo of my rhombic enneacontahedron up, so check that post to see it.

Compounds Galore!

When I first began this blog, I had a notion that I would have a post for every one of my twenty-some polyhedron models. Of course, I now realize that I build them a lot more often than I do blog posts, so I am going to have to do them in groups. If I still had a set of the Platonic Solids (the most basic polyhedra), they would be a logical first post, but ironically I don't. I have made 3 sets of these already, but each time they seem to slowly get destroyed. A logical next post would be my set of Archimedean solids, but I still haven't completed them either. So this post is on my three compounds of Tetrahedra. I used to have a compound of two tetrahedra as well, but it
had an unfortunate encounter with the cat. Anyhow, here are my compounds of 5, 6, and 10 tetrahedra. The compound of five (in primary colors) is especially cool because it is chiral, meaning that it had both a left handed and a right handed for, like a pair of gloves (albeit one with 20 fingers). If you combine both forms together you get a compound of ten tetrahedra (shown at top). I don't mean to brag, but I like the colors that I used in that model. I used to have a compound of five tetrahedra that matched, but my sister Mary got into it and that was that. And the model sandwiched in between the compounds of five and ten tetrahedra (because I don't know how to rearrange the images on this post) is a model of six tetrahedra in pastel. I like pastels for polyhedra, though they can get boring after a while.

Well, it's dinner time!

Great Stella

After finally persuading my Dad to let me get it, I now have my own copy of the program Great Stella! Great Stella is one of the only available computer programs for viewing and printing out nets for polyhedra. But it can also stellate, facet, truncate, augment, and much more. Great Stella is the second in a series of polyhedron-making software by Robert Webb. The first, Small Stella, comes with a library of about 300 polyhedra and can print out nets for all of them, but cannot modify them at all other than changing their size. Great Stella has the capacity to, as I mentioned earlier, play around with the models quite a bit. It also comes with a built in library of several hundred polyhedra, including many not included in Small Stella. And finally, there is Stella 4D, which is a lot like Great Stella, but also can handle 4-dimensional models. Wait, 4-dimensional? Sounds like something out of a science fiction novel, right? Technically, though, there is no way of proving that the fourth dimension doesn't exist (I think), and besides, four dimensional objects (projected into our familiar three) look really cool. And who knows? Maybe we are just poor shortsighted three dimensional beings wandering around on the three dimensional surface of a four dimensional hypersphere. But I digress.
So far I have made one model with nets from Great Stella, the compound of four cubes. My model is in red, yellow, orange, and black, with an edge length of 3 inches (so a diameter of about 5.2 inches). I will get a picture of this model up soon. The model matches my compound of 3 cubes in edge length, and I am currently building a compound of 6 tetrahedra with an edge lenth of three inches as well.

The Rhombic Enneacontahedron

I recently finished a model of the Rhombic Enneacontahedron, the also known as the "zonohedrified dodecahedron". To own the truth, I have very little idea what it means to zonohedrify a polyhedron, but I do know that the dodecahedral "pattern" is very clear in this model. It has 90 faces: 60 wide rhombus faces and 30 slim. I built the model in 6 colors, with all the slim rhombi in orange and the rest in blue, red, purple, green, and yellow, with each color appearing once around each five-faced vertex. I think that if you stellated these faces you would come up with a compound of 5 dodecahedra, but I'm not entirely sure that it would be the compound of five dodecahedra.

Of course, the day after I finished this model, my little sister Ada decided that it would make a nice ball. She threw it into the air, where, due to the natural force of gravity, it did not stay, and instead returned to earth at once. Being a very spherical model, it caved in quite easily, but at least it landed on its bad side, and I was able to repair the damage. I will be more careful with this model from now on.

I have an entire collection of polyhedra (about 20) just waiting to be posted on, so keep checking this blog for updates (if anyone reads my blog, which I doubt).

And by the way, I will get a picture of the actual model up here soon.

UPDATE: I now have a picture of the actual model up. Photo credits go to Sam.

First Post

This is my first post. Hopefully it will not also be my last.
About me: I am 15 years old, Homeschooled, Byzantine Catholic, the oldest of 8 children, and I don't watch television.
I enjoy making polyhedron models out of paper, drinking Earl Grey tea, and flying model airplanes (or at least wanting to).
If I was a kitchen utensil, I would be a garlic press.
Well, that pretty much sums it up!

-Nate