Origami and mathematics

Summary: People explore many interesting mathematical questions using the art and principles of paper folding.

Origami is the famous Japanese art of paper folding. Historically, it has been used for a variety of purposes, including document certification and as a way to represent religious symbols. In traditional origami, a single piece of paper is folded to construct one of a variety of objects. The art has grown to include compound forms that involve connecting several individual origami pieces together, with modular origami specifying geometrically equal pieces.

Origami art, mathematics, and science have many explicit interconnections, and in the 1990s and 2000s there have been several conferences specifically devoted to these links. In the twenty-first century, computational origami is an emerging discipline that applies mathematical theory and computational algorithms to formulate and solve complex folding problems, many of which have applications in engineering, industrial design, and a variety of sciences. Such solutions are often called “origami technology.” For example, engineers and mathematicians explored origami lenses for use in space telescopes, and precision folding technology is already being used to optimize manufacturing processes.

Origami forms are inherently mathematical. Their geometry can be identified as reflections with respect to the folding line. The possible operations for points and lines in origami, using a single fold, are described by seven axioms generally known as the “Huzita–Hatori axioms,” named for mathematicians Humiaki Huzita and Koshiro Hatori. However, mathematician Jacques Justin may have been the first to enumerate these seven axioms. The axioms allow mathematicians to answer interesting questions, such as the classic problems of trisecting an angle and doubling the cube, which are impossible using only ruler and compass constructions. More generally, it is possible to solve any equation up to degree three with origami geometry. Further, although origami forms are usually produced using finite sheets of paper, origami folding can theoretically be extended to the infinite plane.

Use of Origami in Modern Mathematics

In the late twentieth century, mathematicians got interested in the foundations of this art. For this community of scientists, the creation of models in origami is not a matter of inspiration; it is spurred by the search for understanding of the concepts and limitations of Euclidean geometry, properties of geometric figures, symmetry, angles, lines, and mathematical communication, among others.

There are several major topics in the practice and study of origami, including the following:

• Its geometry and relationship between this and other geometries, in particular, Euclidean geometry
• The straightening of the bend—whether a model can be unfolded (which has been studied by Marshall Bern and Barry Hayes)
• Rigid origami—the possibility of constructing models if the paper were replaced by metal (which has already been used for solar panels of satellites in space)

Mathematics teaching techniques increasingly use origami. Moreover, paper folding is used to develop manual dexterity, as well as to teach aesthetics appreciation and topics such as proportions, foundations of geometry, and measurements. Origami is also a handy resource for other areas, like mathematical communication, problem solving, and investigation of three-dimensional objects and spatial relationships.

Huzita–Hatori Axioms

• 1. Given two points P₁ and P₂, we can fold a line connecting them.
• 2. Given two points P₁ and P₂, we can fold P₁ onto P₂.
• 3. Given two lines l1 and l2, we can fold line l1 onto l2.
• 4. Given a point P and a line l, we can make a fold perpendicular to l passing through the point P.
• 5. Given two points P₁ and P₂ and a line l, if the distance between P₁ and P₂ is equal to or larger than the distance between P₂ and l, we can make a fold that places P₁ onto l and passes through the point P₂.
• 6. Given two points P₁ and P₂ and two lines l₁ and l₂, if the lines aren’t parallel and if the distance between the lines isn’t larger than the distance between the points, we can make a fold that places P₁ onto line l₁ and places P₂ onto line l₂.
• 7. Given a point P and two lines l₁ and l₂, if the lines aren’t parallel, we can make a fold perpendicular to l₂ that places P onto line l₁.

Robert Lang proved that this list of axioms covers all possible cases for a single folding. If one of them is removed from the list, it is no longer complete.

Bibliography

Demaine, Erik, and Joseph O’Rourke. Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge, England: Cambridge University Press, 2007.

Hull, Thomas C. Project Origami: Activities for Exploring Mathematics. Wellesly, MA: A K Peters, 2006.

Lang, Robert J. “Origami and Geometric Constructions.” http://www.langorigami.com.

———. Origami Design Secrets: Mathematical Methods for an Ancient Art. Natick, MA: A K Peters, 2003.