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. 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 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.

Folding a flat model from a crease pattern has been proven by Marshall Bern and Barry Hayes to be NP complete. [1]

The loss function for folding paper in half in a single direction was given to be $L = \frac{\pi t}{6} (2^n + 4)(2^n - 1)$ , 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. It had previously been a long-held belief that paper of any size could be folded at most eight times.