Thursday, February 09, 2006

Albertino's guépard (cheetah)

I just finished making Lionel Albertino's guépard from his book Safari Origami. While I don't have a very extensive library of origami books, I still find this model unique. It has to do with the cheetah's back.

Decades ago, many origami animals were open-backed. After a while, it became fashionable to make closed-back models. Satoshi Kamiya, in his book The Works of Satoshi Kamiya, mentions that many closed-back models suffer from very thin layers on their back. So, he started pleating his models along their backs in order to give them the proper thickness. This does mean that his closed-back models have a seam running along the center of the back.

Albertino's cheetah certainly is a closed-back model, and it has a seam running along the back's center. There are actually many layers that come together right at this seam, though not through pleating. These many layers look somewhat ugly directly exposed, almost defeating the purpose of a closed-back. So, Albertino devised a quite clever way to make the model look better.

The cheetah's tail and head come from two opposite corners of the paper, while the four legs come from the edge of the paper. This means that the two remaining corners go unused, and normally would be tucked inside the model. Albertino's ingenious idea is to liberate one of the corner flaps, and at the end of the model use that flap to cover over the exposed layers coming together at the seam. The flap goes around and over the back and gets tucked between the layers on the side. The idea works quite well. It doesn't cover the layers in the neck area, but it does a good job on the back, going all of the way to the tail.

This is a new idea to me, even if it's not new in the origami world, and I don't see it repeated in any other of his designs. Overall, it's a nice model. And, since the head gets formed from a frog base, there is considerable flexibility for making other animals of the cat family. In my model, I modified the head to look like a tiger.

Saturday, February 04, 2006

Another Interview with Robert Lang

In keeping in line with my previous post on interviews with Robert Lang, here's another one. Today on the Origami List, I was made aware of this interview video with Robert Lang on Google Video. The interview centers around his book Origami Design Secrets, but it also mentions his origami career.

The interviewer asks him a question I often wonder about when I see models by Lang, Kamiya, Hojyo, and others -- "how long did it take you to fold the moose? " I'm really glad that he's a full-time origami artist. There would be a lot less Lang models in the world otherwise.

Origami Numbers

Origami numbers sparked my interest last week, and I decided to look them up again. The area of origami numbers is a little bigger than first meets the eye, and somewhat disorganized. As I spent more and more time researching this area, I found that it will take some sorting through things until I am able to write a nice review of this area of research. So, in the coming weeks I hope to bring my findings to this blog. I'll start with some basics.

Take a sheet of paper. Call the bottom left corner the origin, (0,0) and the bottom right corner (1,0). Then, make some creases on the paper and find all of the points of intersection of creases. From the points' x and y coordinates, we can make the number p = (x,y) = x + iy, where we're assuming that the sheet of paper represents the complex plane. The bottom edge of the paper represents the real x-axis and the left edge of the paper represents the imaginary y-axis. What kinds of points are constructible this way? Well, it all depends on the types of folds you use.

It is interesting to think about how many different ways there are to fold a straight crease between points and lines on a piece of paper. In 1989, Humiaki Huzita and Benedetto Scimemi showed at the First International Meeting of Origami Science and Technology (1IMOST) 6 different methods, the Huzita axioms . Then, more than ten years later, in 2001, Koshiro Hatori found a seventh axiom . This long-lost axiom raised questions as to how many ways there are to fold a straight crease between points and lines. Robert Lang researched this, and proved that the 7 axioms are the only origami axioms. Finally, it is possible to ask how many different methods there are for folding multiple creases at once between points and lines. This last issue has not been fully resolved, nor discussed in formal literature, as far as I can tell.

There have been very few papers published concerning origami numbers. The first 6 axioms were published only in the conference proceedings of the 1IMOST. Subsequently, Robert Geretschlager published a paper, “Euclidean Constructions and the Geometry of Origami” in Mathematics Magazine, vol. 68, no. 5, December, 1995, pp. 357–371. David Auckly and John Cleveland wrote a paper on origami numbers as well, but their paper used a very limited set of origami axioms. Then, Roger Alperin published 3 papers on origami numbers including a paper published in Origami^3, the conference proceedings of the 3rd International Meeting of Origami Science, Math, and Education. All of his papers can be found here. As far as I can tell, searching through the American Mathematical Society’s MathSciNet , these have been the only papers published in journals on origami numbers. Other relevant sources are Lang’s nicely written paper on the topic and Hatori’s website . The world awaits further publication, especially concerning multiple creases at once – hopefully soon at the upcoming 4th OSME.

List of relevant links:
Lang's site, with commentary and paper
Roger Alperin's site with all his papers
Koshiro Hatori's site with discussion of his findings
MathSciNet, a nice resources for finding published papers on origami
The arXiv, a great resource for finding pre-prints of papers in many different academic areas

Friday, February 03, 2006


I found out from going to Lang's website that the next conference on Origami, Science, Mathematics, and Education will be coming up this September. I did not see this advertised anywhere else, though it is listed in OrigamiUSA's calendar of events. I suppose it's a close knit community and all those likely to be speakers were well aware of this date for a while now. Someday I'd love to find a connection between origami and string theory. It seems like every other branch of math is being used in theoretical physics, even knot theory! Someday perhaps.