### 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

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

## 2 Comments:

I think a better name is "origami constructions".

Robert Lang's excellent article on his webpage is one source. Also look for Tom Hull's new book, "Project Origami", which should have some relevant info. A recent Tanteidan magazine contains a much simplified proof (by Tom Hull) that the 5 axioms are complete.

I'm discovering exactly what you said -- a better name instead of axioms is either "origami constructions", as you suggest, or even "origami moves". I like constructions.

Thanks for the references, in particular the Tanteidan reference. I haven't yet started putting money into subscribing to their magazines. I just looked, and it's $40 for a year's subscription, 6 issues. That seems very reasonable to me. Thanks for the inspiration.

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