# Folding

## Contents

### Origami Meets Mathematics

Origami is the Japanese art of paper folding. Cranes, frogs and fishes usually are what people think of when origami is mentioned. Now the Japanese art of folding brings algorithms, equations and axioms to mind. Little has been done to analyze origami mathematically, but we can expect this to be a new field in mathematics. It relates to other disciplines such as engineering, art, and biology.

#### Single-Cut and Carpenter's Problem

The single-cut problem goes as far back as ancient China. In fact folding and cutting played a large role in designing the American Flag. Betsy Ross was asked to create a flag with thirteen six-pointed stars. However, she suggested and proved that it would be easier if the stars had five points by folding a piece of paper and using one single cut. Also, the single-cut problem has been the source of many magic tricks performed by Harry Houdini. He would produce cut-outs of any animal.

The theorem states that any origami shape can be made with folding and a single cut. This includes polygons, birds, or something as complex as a hyperbolic parabaloid. This is very surprising that every polygonal shape can be produced, in Dr. Erik Demaine, the leading theoretician of origami mathematics, solved the single-cut problem using two methods.

The first method called the straight skeleton shrinks the region between cuts so that the edges are parallel and move in a perpendicular direction until the region intersects itself. This is done at a fixed rate.

The second method is the circle-packing solution. Circle packing is the arrangement of circles in a given region such that the circles do not overlap but are mutually tangent.

Carpenter's Problem

#### Huzita Axioms

The Huzita Axioms are a set of rules related to the mathematical principles of paper folding. They were discovered by Humiaki Huzita. The operations describe what folds are possible in origami assuming the paper is a plane and all folds are linear.

The Huzita Axioms (HAs) originally had six operations. These six operations define a single fold bringing together line and points, considering all possible alignments between points, lines, fold lines, and folded images. The first operation allows a fold to connect two points. The second operation states that a point can be folded onto another point. The third folds a line onto another line. The fourth states a fold can pass through a point that's perpendicular to a line. The fifth states a point can be placed onto a line while passing through another point. Lastly, the sixth states that two points can be placed onto two different lines simultaneously.

A seventh operation was discovered by Jacques Justin thus making it now the Huzita Justin Axioms (HJAs). The seventh overlooked operation allows a perpendicular fold with a point and two line, and is somewhat similar to operation 6. This seventh operations though raises the question: are there other undiscovered single-fold axioms?

These seven axiomatic models of geometric construction of paper folding shows what origamists have been able to do so far. However, the limits of origami constructing is an open problem. Paper folding can be quite complex. There are many mathematical theories and operations that can be performed to draw lines connecting point, circles, bisecting angles, and perpendicular lines without the need of a straight edge or compass. With the seven operations many extravagant origami figures have been made.

Here is a video where someone performs and explains the Huzita Axioms.

#### Why It's Interesting

- Origami mathematics links to the major scientific question of our time: how do proteins fold up? Proteins are made of amino acids that fold up in complex shapes. In order to carry out their function efficiently, proteins need to be folded into their correct shape. By studying folding, scientists can understand proteins better and help design specialized proteins as drugs.