Paper folding, more commonly known as origami, is the art of creating objects from a piece of paper, usually without using scissors and glue. While there are differing accounts on the origins of origami, it's generally acknowledged that the art of paper folding reached its heights in Japan. Practiced since the early 1600s, origami is also popular in many countries around the world. There are many popular origami creations such as paper butterflies, cranes, buildings, fans, umbrellas, fish, and others.

Origami tekkei, or technical origami, is especially useful in some branches of mathematics such as geometry, topology, and fractions. The crease patterns of models to form a certain structure allow mathematicians to analyze and solve the intricacies of such problems. There are many mathematicians who have spent a lot of time studying the significance of paper folding in mathematics. Humiaki Huzita and Toshikazu Kawasaki are among those who have developed principles to define origami geometrically and some theories have even been named after them.

Comprehensive sites that offer information on the uses of origami in mathematics include:

• Origami Index: Modular Origami, Tessellating Origami, and other Mathematical Origami
• Huzita-Hatori Axioms: Set of 7 Mathematical Principles of Paper Folding
• Wolfram MathWorld: Kawasaki's Theorem

Paper folding helps solve classic geometric problems such as cube duplication (creating a second cube that's twice the volume of the first) and angle trisection (angle division in three equal parts). With origami, various shapes such as an equilateral triangle, pentagon, hexagon, and even a decagon can be created. The geometric theorem on paper folding is that any straight and flat design can be made through a single cut after correct folding is made on a single piece of paper. Origami constructions can actually solve equations up to the fourth degree.

Although some people often mistakenly think that topology is just another term for geometry, it is actually an entirely different concept. With geometry, you cannot stretch or bend things, but this is exactly how it works with topology. Topology, otherwise called "rubber-sheet geometry", works around the principle that stretching or reshaping a certain object will not change the object at all. Using origami, topologists believe that any planar or flat-surfaced object can only be colored in two tones, such as in crease patterns. For instance, if you unfold your origami, you will see that the paper has many crease patterns. In order to make sure that no two borders are the same, you only need two colors to make this possible.

The use of origami on fractions has been less utilized in schools nowadays. However, mathematicians know that visualization of a math problem is important to be able to "crack" it. Through paper folding, fractional problems become more concrete because you can actually "see" the fraction. This is especially applicable in introducing fractions to grade-schoolers. By folding a piece of paper in infinite number of ways, they would be able to understand fractional concepts better.

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