Proving the Kawasaki Theorem
Introduction:
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
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
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