I recently ran across, via this post on Division by Zero, a way to make hyperbolic space from paper, a project that I couldn't resist. In fact, it claims to be a hyperbolic soccer ball. Digging through Reader to find the link again, I found this other recent post about shapes relating to soccer balls, so thought I'd share it as well.

Anyway, the idea is that to typically make a soccer ball, you place a ring of hexagons around a pentagon, and iterate. The pentagons introduce some positive curvature in the process (hexagons alone have 0 curvature - they tile the plane), and you end up with something fairly spherical. If you place hexagons around a central heptagon though, you get negative curvature.

The directions to actually make one yourself are at The Institute for Figuring, and are available here (pdf). Following the basic instructions, I ended up with the following:


Which my cats only took fleeting interest in:


The directions suggested that you could continue adding more rings of hexagons, with heptagons in appropriate locations, and extend the model. I was curious to see what would happen when I did, so I printed out six more of the base sheets (3 of which gave the starting model, above). It's a bit of a monster:

The Borromean rings are quite possibly my favorite link. It's 3 circles in space, no two of which are linked, but together, they can't be unlinked. Fantastic. After completing the recent Menger sponge project, I still had lots of cards leftover, and these rings came to mind pretty quickly. So, several more hours of folding later, and I've got a decent model:
Click the picture (or the upcoming text link) for the Picasa gallery, with 2 more pictures (whoo! exciting!).

I'll probably not do too many more projects like this for a while. Of course, that'd probably change quickly if I found a nice model for the trefoil knot (as I've been hoping to stumble across for a while).

My roommate, Chris, and I just completed our own level 2 Menger sponge origami project, built from 3456 business cards in something around 31 total working hours spread over 4 days (click picture for more pictures):


In all honesty, they weren't exactly business cards. A shortish chain of relations gets Chris and I to a guy with access to card stock and an industrial paper cutter, or so. For Christmas we got 4000 cards cut. They're business card size, and I think approximately the same weight. But then, I don't interact too much with business cards, so I could be mistaken.

Anyway, I should stop rambling. This is not our first large-scale origami project. A few summers ago, Chris and I, along with a then-first-year graduate student (Sean) in the department, built a modular torus:
It was a fun project, but didn't hold up under it's own weight. And since we couldn't find a place to put it, it sadly didn't last long. What's more sad about that is that we didn't even light it on fire, which would have looked simply amazing, I reckon. But, again, I digress.

So, this big sponge. It was most recently inspired by Thomas Hull's Project Origami book, but it's possible I'd heard about it before reading said book. On my sister site, I posted my calculations for finding the formula for the number of cards needed, so you can check that out, if you are interested. And over at picasa, I made an album of some of the pictures I took, which is probably more interesting. I'd also like to direct you to this picture, of somebody else's project, because they also made an 'anti-sponge', which looks awesome (like I needed another project).

In the process of building this sponge, I decided this (well, perhaps the smaller level 1 sponge) might just be the best modular origami project for somebody just starting to do modular origami. The pieces are very easy to fold (just 2 creases per unit), and pretty forgiving (if you're folds aren't quite as accurate as they could be, it should still go together pretty well). The pieces are fairly easy to put together (though, of course, it takes a little practice) - in particular because the whole thing won't fall apart if you unfold part of one piece in order to stick another piece somewhere. The final result is very nice and quite sturdy. So, if you are just starting modular origami, you might consider making this model. I recommend Hull's book, but you could probably find resources online for instructions putting these together. If you're looking for other modular origami books, my favorite is probably Tomoko Fuse's 'Unit Origami'.

I know the lighting is pretty poor in this picture, but I like it anyway:
I can't wait to take this thing outside and balance it on my chin :)

Last weekend I bought a Nikon S550 digital camera (on sale at Staples for a little while still, if you were thinking about it). I've been having a good time taking pictures (and getting them off my camera from the command line using gphoto2, totally sweet). When I went to clean up my room a bit and recycle the box, I noticed that the box itself was also pretty cool. The box had a cardboard insert separator, which, when unfolded, looked like:

What's nice about that is with the cuts it has, and the folds, it folds up through a series of rigid motions (no folding/rolling except for on the pre-scored lines) to the following separator:

The whole box itself was also folded from a single unit, which lies flat (when unfolded):

And here they are together:

I don't know. Perhaps it's not that impressive. But I really liked it. No tape on the box, just lots of nice folding. As an origami fan, it appealed to me. Plus it gave me more excuses to take pictures, and play with Picasa.

I've been wanting to make my own copy of the 72 pencil sculpture by George W. Hart since I first saw it online. Have a look at it here, and explore his other works - they're awesome. I've had two boxes of 48 pencils sitting in my closet for a few years now, waiting for the day I was ready to make this thing. Yesterday I finally decided to make it, and went to Michael's to buy some styrofoam. I spent... probably 5 hours yesterday trying to work it out, and didn't get it. I woke up this morning, and stared at it for another 4 hours or so. Finally, I started over, and within about another hour I had it. Turns out, all of my first attempts had been started wrong. Idiot.

After I got it together:
I thought I'd see what happened when I removed the rubber bands. Perhaps I removed them without enough care, but it ends up like this:


So I started over, taking pictures as I went. If you want to make your own, it really isn't that bad. I just wasn't being careful enough for my first... 10 tries or so :) What'll you need? 72 pencils (4 hexagonal cylinders, 18 pencils each) of course. I used a bit of styrofoam, a 1 inch think circular disc with diameter ~6 inches. To hold things together, I also used 4 pieces of string (each about a foot long, I'd guess) and 6 rubber bands. My cats wanted to help too:

I arranged the first 18 pencils in a cylinder poking into the styrofoam. The hexagon was bigger than it needed to be, and the pencils were wobbly, and not equally spaced. You might also notice
that I put one rubber band around the styrofoam, which I did before any pencils were in. Next, start weaving some pencils into the uprights. You may find the original picture helpful. Here's a picture of the early stage of the process, for me:
I have a couple of extra pencils (the black ones) in there for balance. They'll get removed later.

Keep stacking these pencils up, until it looks something like


After that, I tied a loop of string around each end of both horizontal cylinders. This permitted me to then put a rubber band around each end of the horizontal cylinders (and remove the strings)
Now push the rubber band off the styrofoam base, to hold the bottom of the vertical cylinder. Put a rubber band around the top of the vertical cylinder, flip the whole thing over (so the styrofoam is up in the air), and pull the styrofoam off (which I found surprisingly easy). Now you've just got the forth cylinder to get in there. I had to follow the original picture, and carefully put pencils where they should be according to the picture. After the first few, it's pretty easy to see where they should go.

So, I don't know how helpful that was. It's really not too hard, as long as you start things correctly. This was my problem the first several times. But it all worked out in the end

Of course, there's still those rubber bands. I'm not entirely sure how to get rid of them, so they'll probably stay for a while.

Update 20081201: Ok, with a little super glue one some obviously loose penciles, I was able to remove the rubber bands. Hoping it stays together for a while now.

Today I gave a (brief) talk for the Graduate Seminar in the math department here at UVA. This is an informal setting, which is only for grad students (no professors heckling the speaker). They're generally an hour, but I only went slightly over half of that time. All the same, it was a pretty comprehensible talk. At the very least, it was comprehensible material - who knows how the talk went.

The title for my talk was Origami Numbers. I walked through the first 5 of the Huzita-Hatori(-Justin-...) axioms [wikipedia], which are enough to do standard straight edge and compass constructions. The first 4 are pretty straight forward, and it's with the 5th that you start making interesting things - parabolas. Then with the 6th axiom, we are able to obtain cubic roots, showing that origami is more powerful than a straight edge and compass. A great reference, and my starting point, for all of this, is the book Project Origami by Thomas Hull (which has lots of other goodies).

One thing I found while preparing my talk was identified as Lill's method in this paper by Alperin and Lang. It's a graphical method for finding roots of polynomials, and I'd never seen it before. The paper 'Geometric Solutions of Algebraic Equations' by Riaz talks about it, as does this site, which I just found (so I should take another look at). Lill's method draws a piecewise linear path (starting, say, at O, and ending at T), where the lengths of pieces correspond to coefficients of your chosen polynomial. Then you are supposed to draw another path from O that bounces around off the lines you made and ends at T. This line shows you one of the roots for the original polynomial (see the above references for more details). It doesn't seem particularly practical if you want to find roots (I'd go for Newton's method, if all else failed), but it's fun to have a new graphical way to think about things.

The last week before the start of a semester is always a good time to mess about. While I've spent some time reading about buildings and apartments and chambers (Tits Systems and Coxeter Systems and such), I also did some origami. I've done the 5 interlocking tetrahedra before, but it wasn't very nice (I used pieces of paper that were too long). This one's a bit better, with color and all (trying to use up my color paper). Plus this gives me a chance to see how blogger does pictures. Gotta have some excuse to play with stuff, right? If you're looking to make your own, I used the plans from Thomas Hull's 'Project Origami' book, but they are also in the book I got that started my hobby: 'The Origami Handbook' by Rick Beech. In fact, Dr. Hull has the plans online.

What I'd really like to do next is an origami trefoil knot, using the Phizz units, along the lines of the torus. Now, I'm not very bright, and have never done an origami model without the plans sitting in front of me. So I asked my friend google to tell me about possible models. The only thing I've found so far is this paper (.doc, ewww), which does have plans for the trefoil. In fact, I really enjoyed the paper, and learned some things from it (the counting bits at the end). But the plans seem to lack the 3-fold symmetry I expect is possible. Perhaps I just don't know enough (any) of the geometry involved, but I figure there should be a fundamental domain that is a third of the model. Or perhaps the three things go together, but each one is 120 degrees rotated (along the meridian) to get the appropriate twisting. So, if you know of such a model, and can send me a link, it'd be much appreciated. If you know a reason such a model can't exist, please do tell me about it. Otherwise, one of these days, I may start folding some more units. That sounds productive.

Actually, now that I think about future origami plans, hot on the heals of this five tetraheda model, there's another large project I've been wanting to do. The five tetrahedra made from penultimate units. I've sat down to do it before, but you have to cut paper to specific sizes. And not something like '1.5in x 4in', but 'a 4x3 rectangle whose width is 2.58 inches'. Right. Perhaps I'm missing something (besides a functional brain).