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.

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

Origami^4

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.

http://langorigami.com/science/4osme/4osme.php4

Robert Lang Interviews

I just re-found two pages online hosting interviews with Robert Lang. The first link is from an Open Source conference he spoke at, OMOSC '05:
http://www.itconversations.com/shows/detail660.html . The other interview is from an NPR show, at http://www.theconnection.org/shows/2004/11/20041117_b_main.asp .

I may as well include the link to the list of online sites with articles written by or about Lang. It's taken directly from his own site (scroll down to Articles and Media) :
http://langorigami.com/info/misclinks.php4 . There are plenty of good articles to see from there. Incidentally, I didn't realize until I looked at the first link above that there is a biographical article in Wikipedia on him. http://en.wikipedia.org/wiki/Robert_J._Lang

Lang's Crane

I realized this past weekend just how difficult a seemingly innocent looking crane can be. Robert Lang's crane in his Origami in Action has it all: 2 open unsinks, a re-sink into a different pocket, and 2 open wraps. The re-sink is particularly nasty. I wonder if this crane is the reason that Anna Kastlunger and Gerwin Sturm dedicated their "innovative folding method for the traditional crane" to Robert Lang at http://www.origamiaustria.at/articles/kranich.pdf ...

HQ PaperMaker

I just found quite an incredible site that sells handmade Thai paper at http://www.hqpapermaker.com/ . The company is based in Thailand so that while all of their prices are incredibly cheap, all of their shipping costs are incredibly expensive (it seem like they only FedEx). Bulk orders would be the way to go, after ordering their free catalog (shipping costs on that are horrendous as well). I need to explore this site a little more ... Too bad they don't have a branch in the US.

Wu's Swan

I love the fact that web archiving exists. I found again today Jake Crowley's old website on web.archive.org at http://web.archive.org/web/20040611091736/home.earthlink.net/~jakecrowley/ . I still find his work to be among the best in the world. His interpretation of Komatsu's tiger is better folded than Komatsu's! Anyway, I realized (probably for the second time) that Jake has a nicely folded Swan designed by Joseph Wu on his old site. Since I was just recently looking for swan models around Christmas, this caught my eye today. I couldn't find any trace of this model in the origamidatabase.com . I couldn't find it listed among Wu's diagrammed models on his site either. Then, searching through Google, I found on Wu's site that he made a Swan as a commissioned work. The model there looks a lot like the model shown in Jake's site, but the paper makes it hard to make a direct comparison. I assume that if it's the same model, a diagram may never become publicly available. Oh well ... Maybe I'll be able to work it out from Jake's picture.

Patricia Crawford and Neil Elias

After just buying Robert Harbin's Origami Step-By-Step and being blown away by the complexity Patricia Crawford's models displayed already in the early 70's, I decided to look into her work a little more. Of course, I'm always reminding myself that Neil Elias was at work creating very complex models around that time as well -- I never seem to remember that such complexity could be found in the world before the 80's ...

Of course, it's well known that Patricia Crawford quit origami design in the 70's. Looking through the Origami-L archives, it looks like there are several theories as to why. Most people say that her husband forced her to quit, but either way, it is then said that she picked up sculpture. What a tremendous loss to the origami world! Fortunately, Robert Harbin (and Neil Elias, more on him in a moment) had made contact with her before she quit designing and chose to popularize her work through his book. It is said that he wrote the book Step-By-Step mainly as a showcase for her work. What a great guy!

Looking in the Origami Database, it became clear that there are only two other books that feature quite a few of her works, Origami 3: The Art of Paperfolding by Robert Harbin, and Creating Origami by J. C. Nolan, both of which have 5 of her works. Of course, there are other places to find her works, such as Origami 4 by Harbin (1), Origami 2 by Harbin (1), New Adventures in Origami (1), the 1980, 1981, and 1984 BOS convention books (1 each), the Origamian vol. 12 #2,3,4 (4), Tanteidam Convention 3 (1), and a Centro Diffusione Origami publication, QQM 16, (1). Of course, all of these books are practically impossible to come by except for some of the Origami 1-4 series of Harbin, which rumor has it from BOS that they will be republished soon. Thankfully, Harbin's Origami Step by Step is currently in print as a reprint, featuring no less than 14 of her works.

Finally, the Origami Database shows two other interesting sources of her works. The first is the BOS Model Library, which has 11 of her models. Basically, as far as I can tell, the list on the Database is the list of folded works that are being housed at the BOS headquarters. So, these apparently don't represent diagrams, just finished models. Finally, the Database mentions the Complete Notebooks of Neil Elias. Assuming that these were simply out of print, I contacted the BOS Supply manager asking him of the item since they included 11 of Crawford's models including other rarities. After sending the e-mail, I realized that maybe the list simply represented all of the diagrams featured in Neil Elias' notebooks that the BOS has possession of -- sort of like the listing of the BOS Model Library. So, I e-mailed him again.

It turns out that Dave Venables from BOS, the man in charge of the Elias heritage, is soon planing on releasing a CD-ROM through the BOS Supply with as many diagrams found in the Elias notebooks such as he was able to acquire publishing rights. My several e-mails must have caused quite a splash since the next day the entire listing of the Complete Notebooks of Neil Elias was taken down from the Database. When I say entire, I mean several hundred. Currently, the list is back up, but clearly stating Not Yet Available in caps.

Somehow, Elias must have been able to make contact with Crawford as well so that he was able to diagram her works. Looking throug the list of models in his notebooks, it's quite impressive the list of models from other designers he was able to get a hold of. The list even includes several Akira Yoshizawa model diagrams! It should be noted, though, that Elias lived in the pre-Internet world, where many famous designers knew each other through snail-mail and sent each other models and diagrams in the mail. Many of these models were not widely circulated or published. So, I'm excited that Elias took the time to record diagrams of all of the models of artists he had befriended throughout the years. It's a nice record of the current status of origami at the time in the world.

I hope that the CD-ROM will be released soon so that I can soon see some of Crawford's other amazing models along with all of the other gems found in there, not the least of which are all of the incredible Neil Elias models. I wonder if Patricia Crawford is still around ... It would be nice if the origami community honored her in some way for her major contributions to the art during the early 70's. Same goes for Neil Elias.