How To Make A Pig.
Mathematics of paper folding.
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’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.
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. 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,…,a2n surrounding each vertex fulfill the condition that a1 + a3 + … + a2n-1 = a2 + a4 + … + a2n-1 = 180°; in other words, the sum of every other angle surrounding an interior vertex is always equal to 180°. (More about: Mathematics of Paper Folding)
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