Geometry paper

The Geometry and Mathematics in Japanese Origami

Yifei Huang [email protected]

Abstract The Japanese art of paper folding is obviously geometrical in nature. Some origami masters have looked at constructing geometric figures such as regular polyhedra from paper. In the other direction, some people have begun using computers to help fold more traditional origami designs. This idea works best for tree-like structures, which can be formed by laying out the tree onto a paper square so that the vertices are well separated from each other, allowing room to fold up the remaining paper away from the tree. Bern and Hayes (SODA 1996) asked, given a pattern of creases on a square piece of paper, whether one can find a way of folding the paper along those creases to form a flat origami shape; they showed this to be NP-complete. Related theoretical questions include how many different ways a given pattern of creases can be folded, whether folding a flat polygon from a square always decreases the perimeter, and whether it is always possible to fold a square piece of paper so that it forms (a small copy of) a given flat polygon.

1 Introduction

As a development of arts, origami have actually intersected in many fields with geometry and math. As for the pure origami, one of the most famous example of intersection is the Huzita-Hatori axioms, which is well defined as below: Some classical problems of geometry— namely trisecting an arbitrary angle or doubling the cube — which 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– 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. When it comes to the construction of origami, methods such as Haga's theorem have allowed paper folders to accurately fold the side of a square into thirds, fifths, sevenths, and ninths. Other theorems and methods have allowed paper folders to get other shapes from a square, such as equilateral triangles, pentagon and 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. Haga’s theorem: 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.

Citations: https://en.wikipedia.org/wiki/Mathematics_of_paper_folding