Proving the Kawasaki Theorem

Intro:

Exploration

We chose this project; because we all like making origami and now we will know everything in it. Origami means the art of conventional paper folding. It can connected to the math called geometry. When we flip origami we all also create lots of areas. For instance, simply by folding a square piece of paper in half diagonally or in one tip for the opposite hint, we make two surfaces in the form of triangles. Mathematicians' related origami to a theorem called the Kawasaki theorem. The Kawasaki theorem says that whenever we add up the angle measurements of every position around a level, the sum will be 180. It is a theorem giving a decision for a great origami development to be smooth. Kawasaki theorem also claims that a given crease routine can be folded to a smooth origami if perhaps all the sequences of sides,..., are encircling each interior vertex to the following condition

Applications of Origami:

We use origami for lots of issues in life. Mathematical origami theory has been used on produce an amazing range of sensible applications. Fresh technologies staying developed include: paper merchandise designs concerning no adhesives, better means of folding roadmaps, unfolding space telescopes and solar sails, software systems that evaluation the safety of airbag packing's for car manufacturers, and self-organizing unnatural intelligence devices.

Hypothesis:

Applying certain retracts of origami, we are going to see whether or not we can utilize the Kawasaki theorem to construct a pelican and a crane.

Info and Evaluation:

Introduction to the Kawasaki theorem

Kawasaki's Theorem: Given a vertex within a flat origami crease style, label the angles involving the creases while О±1, О±2,..., О±2n, in order. Then we should have

or another method to say a similar thing can be

This really is saying that whenever we alternately add and subtract the aspects as you go surrounding the vertex, then you'll always obtain zero. If we start at one particular angle and then fold the creases, one at a time, around the vertex, the folds up will make the angles flip back and forth, and equaling zero in the end is like coming back to in which we began so that the paper won't grab. This is essentially how the proof of Kawasaki's Theorem works. (To get the second statement in the angles, simply use the fact that О±1 & О±2 & О±3 + + О±2n = 360В°; add this to the past equation and divide both sides by installment payments on your ) Solving the Kawasaki Theorem:

Standard Folds of Origami

There are several folds in origami these are generally some basic retracts of origami: Valley Collapse

The pit fold is created by flip-style the newspaper toward yourself. An arrow shows best places to fold the piece of paper to. Symbol: dashed line

Mountain Flip

The hill fold is formed by folding the newspaper away from yourself. Symbol: switching dashed and dotted line

Petal Fold

The petal fold lifting a point and brings this upwards so the two sides of the level lie with each other. It is best to prefold both tiers of conventional paper along the proven valley and mountain folds up before making the petal fold.

Rabbit Ear Flip

Prefold over the three pit folds 1st. Then collapse the two attributes down to the baseline. Flip the top point out one aspect to make the mountain fold.

Squash Flip

Prefold both sheets of paper along the valley and mountain collapse. Then wide open the unit, fold one layer of paper along the valley flip and trim the unit using the pile fold. The top white arrow tells you to spread out the version.

Inside Reverse Fold

Prefold both equally sheets of paper in both guidelines (mountain and valley). In that case open the model somewhat and bring the top level down so that the mountain collapse edge turns into a valley fold edge.

Outside Change Fold

It truly is similar to the inside reverse collapse except the layers with the paper have to be wrapped about 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 perhaps point. Generally it is easiest just to...

Referrals: 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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