A History of Folding in Mathematics

Most mathematicians, I think, would be surprised that there is a history of folding in mathematics. The author begins by asking how and why a subject with roots in geometry that dates at least to the sixteenth century became so marginalized. He notes that duplication of the cube, and thereby construction of a segment of length the cube root of two, while impossible using only straightedge and compass, is comparatively easy using folding. (See here for a construction using a method due to Margherita Piazzolla Beloch.) Much more recently, work by Eric Demaine and his coauthors have introduced or re-introduced geometrical folding algorithms to a wider audience. But what happened? How did the mathematics of folding never reach the mathematical mainstream?

While the author devotes most of the book to the historical background, the first part grapples primarily with the question of marginalization. The author notes that “other cultural material techniques, such as weaving, braiding and knotting … began to be considered mathematically long before the end of the nineteenth century”, so he finds it ridiculous that folding would not have been considered “mathematical”. He presents a description of a philosophical framework and logic in which this marginalization occurred.

According to the author folding was marginalized for several reasons: a notation for folding wasn’t available; folding was considered just a useful tool and not a concept for investigation; folding appeared only within the framework of another well-established concept (such as when it was used to exemplify concepts of symmetry); and items written about folding were unpublished or appeared only at the social margins of the mathematical community far from the centers of mathematical research. The author even draws on ideas of the French philosopher Derrida to argue that folding “crosses every distinction between inside and outside”, ending up neither on the inside with geometry or outside and thus found no place of its own.

The author’s history focuses on Europe and the work beginning in the sixteenth century with Albrecht Dürer. Dürer presented a new way of representing and producing three-dimensional solids by unfolding them into polyhedral nets. The author argues that Dürer gave folding legitimacy by thinking about it mathematically. Following Dürer mathematicians in the mid eighteenth into the nineteenth century began to use folding as an inference method in proofs — to show congruence between figures and symmetry between and within figures.

The author argues that Beltrami’s work on non-Euclidean geometry in the nineteenth century owed a good deal to the folded paper and cardboard models he made. He suggests that these physical models preceded the development of Beltrami’s mathematical theory and that the action of folding was essential to his theoretical work.

A breakthrough for folding was the 1893 publication of Geometric Exercises in Paper Folding by Tandalam Sundra Row; it presented folding as a method on which geometry could be based. Then work by Beloch in 1934 showed how paper folding could be used to solve cubic equations.

It was only comparatively recently that certain aspects of folding have been formalized. Huzita and Justin presented an axiomatic basis for folding-based geometry at a conference in 1989. Demaine and O’Rourke’s Geometric Folding Algorithms: Linkages, Origami, Polyhedra, published in 2007, brought the ideas of folding to a much larger audience.

This text seems to be aimed largely at mathematical historians. For the general mathematical reader it presents a story of folding and its marginalization stretched out over several loosely connected periods of time. It is a difficult story to read because it has many individual threads that don’t naturally fit together. The author is, after all, largely trying to explain why something didn’t happen.

The author does not consider the contributions to folding coming from China and Japan because, as he says, he does not know Japanese or Chinese well enough to explore original sources. Yet the tradition of origami may well have had a mathematical component. One sees possible evidence of that in the twentieth century in the engineering and design work of the astrophysicist Koryo Miura (see this for example), used to create a large foldable solar array that was easily unfoldable in space. This is the same Miura whose Miura-ori create those charming maps that are unfoldable and foldable with a single motion.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.