Kawasaki and Origami Megan Morgan Inquiry IV Presentation April 27, 2010

Motivation I love origami It always amazes me how something as simple as a square can turn into something new and exciting. A few pieces I have never been able to make To understand the mathematics behind the folds In order to perfect my technique.

Background Origami has been used throughout the ages for religious use, decorations, gifts, teaching, entertainment, and much more. Egyptians would fold fabric into shapes for decoration and function. The art of paper folding is 2000 years old (China invented paper). Japanese made origami famous. It is a widely believed that origami even has healing powers. Kasahara 3 Coerr 1 “[The crane is] supposed to live for a thousand years. If a sick person folds one thousand paper cranes, the gods will grant her wish and make her healthy again”

Kawasaki’s Theorem A single-vertex crease pattern defined by angles ϴ 1 + ϴ 2 + …+ ϴ n = 360 o is flat foldable if and only if n is even and the sum of the odd angles ϴ 2i+1 is equal to the sum of the even angles ϴ 2i, or equivalently, either sum is equal to 180 o O’Rourke

Design 5 different shapes per origami base Note: origami is the art of paper folding. True origami does not allow for paper cutting. In addition, this project will become exceedingly difficult if the shape requires more than one piece of paper to make. The top and bottom of the sheet should be a different color Square Clear protractor Fold all of the shapes, according to the directions. For each shape do the following: Count each fold as you unfold the object Count all the vertices Find the most used vertex Number all the angles intersecting it (1,2,3,4…) Measure the degree of the odd numbered angles Calculate the total of said angles Perform χ 2 test on the total of the odd angles to prove Kawasaki’s Theorem.

Results Upon performing the χ 2 test, the probability was found to be 1. (The value of χ 2 is 10.) A vertex composed of six angles was rarely symmetrical, which caused the combined value of the angles to be either less or more than the 180 stated in the theorem.

Analysis There is no correlation between the number of angles a given shape has, the number of vertices, and or the number of folds.

Number of Measureable Angles Several shapes used the same number of angles. There are typically 4, 6, or 8 angles around a vertex

The Problem with Reference Folds A common problem that kept occurring in measuring the angles is that there were an odd number of them. Do not make reference folds Mark the reference folds so that you can calculate them out at the end If you do not, then the theorem (and laws of geometry) are invalid and cannot be used.

180 = 180 The average measure of the angles: 181 o

Conclusions One must first exclude the reference points and then the math works perfectly. My χ 2 test showed that the data collected is 100% acceptable and not at all by random chance. Odd angles do equal 180 o which is half of 360 o which is how many degrees are in a circle.

References Coerr Eleanor. Sadako and the Thousand Paper Cranes. New York, 1977, Puffin Modern Classics. Pg 1. Kasahara Kunihiko. The Art and Wonder of Origami.Japan, 2004, Gijutsu Hyoron-sha Publishing. O’Rourke Joseph. Geometric Folding Algorithms: Linkages, Origami, Polyhedra, 2007, Library of Congress. Pg 169,