# Proving the Kawasaki Theorem

Research

We chose this project; because we all love making origami and now we will know everything in it. Origami means the art of paper folding. It’s connected to the mathematics called geometry. When we fold origami we also create lots of surfaces. For instance, by folding a square piece of paper in half diagonally or from one tip to the opposite tip, we create two surfaces in the shape of triangles. Mathematicians’ related origami to a theorem called the Kawasaki theorem. The Kawasaki theorem states that if we add up the angle measurements of every angle around a point, the sum will be 180. It is a theorem giving a decision for an origami construction to be flat. Kawasaki theorem also states that a given crease pattern can be folded to a flat origami if all the sequences of angles , ..., are surrounding each interior vertex to the following condition

Applications of Origami:

We use origami for lots of things in life. Mathematical origami theory has been applied to produce an amazing range of practical applications. New technologies being developed include: paper product designs involving no adhesives, better ways of folding maps, unfolding space telescopes and solar sails, software systems that test the safety of airbag packing’s for car manufacturers, and self-organizing artificial intelligence systems.

Hypothesis:

Using certain folds of origami, we are going to determine if or not we can use the Kawasaki theorem to construct a pelican and a crane.

Data and Analysis:

Introduction to the Kawasaki theorem

Kawasaki’s Theorem: Given a vertex in a flat origami crease pattern, label the angles between the creases as α1, α2, ..., α2n, in order. Then we must have

or another way to say this same thing is

This is saying that if we alternately add and subtract the angles as you go around the vertex, then you’ll always get zero. If we start at one angle and then fold the creases, one at a time, around the vertex, the folds will make the angles flip back and forth, and equaling zero in the end is like coming back to where we started so that the paper won’t rip. This is basically how the proof of Kawasaki’s Theorem works. (To get the second statement of the angles, just use the fact that α1 + α2 + α3 + + α2n = 360°; add this to the previous equation and divide both sides by 2.) Solving the Kawasaki Theorem:

Basic Folds of Origami

There are lots of folds in origami these are some basic folds of origami: Valley Fold

The valley fold is formed by folding the paper toward oneself. An arrow shows where to fold the piece of paper to. Symbol: dashed line

Mountain Fold

The mountain fold is formed by folding the paper away from oneself. Symbol: alternating dashed and dotted line

Petal Fold

The petal fold lifts a point and brings it upwards so that the two edges of the point lie together. It is best to prefold both layers of paper along the shown valley and mountain folds before making the petal fold.

Rabbit Ear Fold

Prefold along the three valley folds first. Then fold the two sides down to the baseline. Fold the top point to one side to make the mountain fold.

Squash Fold

Prefold both sheets of paper along the valley and mountain fold. Then open the model, fold one layer of paper along the valley fold and flatten the model using the mountain fold. The big white arrow tells you to open the model.

Inside Reverse Fold

Prefold both sheets of paper in both directions (mountain and valley). Then open the model a little bit and bring the top point down so that the mountain fold edge becomes a valley fold edge.

Outside Reverse Fold

It is similar to the inside reverse fold except the layers of the paper have to be wrapped around outside the point.

Crimp Fold

A crimp is used as a way of incorporating two reverse folds to change the direction of a flap or point. In most cases it is easiest just to...

References: http://www.scarygami.net/basic_folds.php?lang=1&style=1

http://mathworld.wolfram.com/KawasakisTheorem.html

http://www.origami-usa.org/thefold004_hull

http://www.origami-fun.com/origami-crane.html

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