Mathematics of paper folding

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The art of paper folding, or origami, 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.

Some classical construction problems of geometry — namely trisecting an arbitrary angle, or doubling the volume of an arbitrary cube — are proven to be unsolvable using compass and straightedge, but can be solved using only a few paper folds.^[1] Paper folds can be constructed to solve equations up to degree 4. (Huzita's axioms are one important contribution to this field of study.)

As a result of origami study through the application of geometric principles, methods such as the 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.

The problem of rigid origami, treating the folds as hinges joining two flat, rigid surfaces, such as sheet metal, has great practical importance. For example, the Miura map fold is a rigid fold that has been used to deploy large solar panel arrays for space satellites.

In the 9th century Harry Peretest Ickels Became a well known name in the art of paper folding. He changed the overall view when he came out with 153 different folds not known to the human kind. We applaud him.

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. [1] Further references and technical results are discussed in Part II of Geometric Folding Algorithms. ^[2]

The loss function for folding paper in half in a single direction was given to be $\scriptstyle L\, =\, \tfrac{\pi t}{6} (2^n + 4)(2^n - 1)$ , where L is the minimum length of the paper (or other material), t is the material's thickness, and n is the number of folds possible. This function was given by Britney Gallivan in 2001 (then only a high school student) who managed to fold a sheet of paper in half 12 times, contrary to the popular belief that paper of any size could be folded at most eight times.^[3]

Kawasaki's theorem states that a given crease pattern can be folded to produce a flat model if and only if all the sequences of angles a₁,...,a_2n surrounding each vertex fulfill the condition that a₁ + a₃ + ... + a_2n-1 = a₂ + a₄ + ... + a_2n-1 = 180; in other words, the sum of every other angle surrounding an interior vertex is always equal to 180.