mathematics of paper folding

The tradition & historical record

The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper folds to solve mathematical equations up to the third order. Computational origami is a recent branch of computer science that is concerned with studying algorithms that solve paper-folding problems. The field of computational origami has also grown significantly since its inception in the 1990s with Robert Lang's TreeMaker algorithm to assist in the precise folding of bases. Computational origami results either address origami design or origami foldability. In origami design problems, the goal is to design an object that can be folded out of paper given a specific target configuration. In origami foldability problems, the goal is to fold something using the creases of an initial configuration. Results in origami design problems have been more accessible than in origami foldability problems. ¹

In 1893, Indian civil servant T. Sundara Row published Geometric Exercises in Paper Folding which used paper folding to demonstrate proofs of geometrical constructions. This work was inspired by the use of origami in the kindergarten system. Row demonstrated an approximate trisection of angles and implied that the construction of a cube root was impossible. In 1922, Harry Houdini published "Houdini's Paper Magic," which described origami techniques that drew informally from mathematical approaches that were later formalized. ¹

In 1936 Margharita P. Beloch showed that use of the 'Beloch fold', later used in the sixth of the Huzita–Hatori axioms, allowed the general cubic equation to be solved using origami. In 1949, R C Yeates' book "Geometric Methods" described three allowed constructions corresponding to the first, second, and fifth of the Huzita–Hatori axioms. The Yoshizawa–Randlett system of instruction by diagram was introduced in 1961. ¹

In 1980 a construction was reported which enabled an angle to be trisected. Trisections are impossible under Euclidean rules. Also in 1980, Kōryō Miura and Masamori Sakamaki demonstrated a novel map-folding technique whereby the folds are made in a prescribed parallelogram pattern, which allows the map to be expandable without any right-angle folds in the conventional manner. Their pattern allows the fold lines to be interdependent, and hence the map can be unpacked in one motion by pulling on its opposite ends, and likewise folded by pushing the two ends together. No unduly complicated series of movements are required, and folded Miura-ori can be packed into a very compact shape. ¹

The construction of origami models is sometimes shown as crease patterns. The major question about such crease patterns is whether a given crease pattern can be folded to a flat model, and if so, how to fold it; this is an NP-complete problem. Related problems when the creases are orthogonal are called map folding problems. There are three mathematical rules for producing flat-foldable origami crease patterns: ¹

Maekawa's theorem: at any vertex the number of valley and mountain folds always differ by two. It follows from this that every vertex has an even number of creases, and therefore also the regions between the creases can be colored with two colors. Kawasaki's theorem or Kawasaki-Justin theorem: at any vertex, the sum of all the odd angles (see image) adds up to 180 degrees, as do the even. A sheet can never penetrate a fold. Paper exhibits zero Gaussian curvature at all points on its surface, and only folds naturally along lines of zero curvature. Curved surfaces that can't be flattened can be produced using a non-folded crease in the paper, as is easily done with wet paper or a fingernail. Assigning a crease pattern mountain and valley folds in order to produce a flat model has been proven by Marshall Bern and Barry Hayes to be NP-complete. Further references and technical results are discussed in Part II of Geometric Folding Algorithms. ¹

Some classical construction problems of geometry — namely trisecting an arbitrary angle or doubling the cube — are proven to be unsolvable using compass and straightedge, but can be solved using only a few paper folds. Paper fold strips can be constructed to solve equations up to degree 4. The Huzita–Justin axioms or Huzita–Hatori axioms are an important contribution to this field of study. These describe what can be constructed using a sequence of creases with at most two point or line alignments at once. Complete methods for solving all equations up to degree 4 by applying methods satisfying these axioms are discussed in detail in Geometric Origami. ¹

As a result of origami study through the application of geometric principles, methods such as Haga's theorem have allowed paperfolders to accurately fold the side of a square into thirds, fifths, sevenths, and ninths. Other theorems and methods have allowed paperfolders to get other shapes from a square, such as equilateral triangles, pentagons, hexagons, and special rectangles such as the golden rectangle and the silver rectangle. Methods for folding most regular polygons up to and including the regular 19-gon have been developed. A regular n-gon can be constructed by paper folding if and only if n is a product of distinct Pierpont primes, powers of two, and powers of three. ¹

The side of a square can be divided at an arbitrary rational fraction in a variety of ways. Haga's theorems say that a particular set of constructions can be used for such divisions. Surprisingly few folds are necessary to generate large odd fractions. For instance 1⁄5 can be generated with three folds; first halve a side, then use Haga's theorem twice to produce first 2⁄3 and then 1⁄5. The accompanying diagram shows Haga's first theorem: ¹

B Q = 2 A P 1 + A P . {\displaystyle BQ={\frac {2AP}{1+AP}}.} ¹

The function changing the length AP to QC is self inverse. Let x be AP then a number of other lengths are also rational functions of x. For example: ¹

The text above is adapted from the Wikipedia article “Mathematics of paper folding”, available under the Creative Commons Attribution-ShareAlike 4.0 licence. ¹

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