## Unit Origami as Graph Theory

The author himself has found unit **origami** not only to be a … trate the powerful link between graph theory and unit **origami**. As you read it may …

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In the search for innovative material to teach high school mathematics students more and more teachers are turning to graph theory. This subject provides an absract, powerful and very modern way to model any type of network structure, from telephone networks to the human central nervous system. One of the most engaging ways to teach graph theory is with unit origami. In this paper we will discuss examples and techniques for doing just this. 1. Introduction Unit origami has its roots in complex origami figure design. If a designer could not create the desired object with one piece of paper, several sheets might make the task easier. However, thanks mainly to the popular works of T. Fuse and K. Kasahara ([4], [5]), people nowadays think of modular origami as using several, sometimes hundreds of pieces of paper to create various geometric forms. The charm of such origami is that simple, easy to fold modules can lead to very complex, intricate structures. Also, since more than one piece of paper is involved, one can experiment with the multitude of color patterns that are possible in any one modular origami work. In exploring such geometric structures and color patterns, a good understanding of 3-dimensional polyhedral geometry is needed. In this paper, we will discuss how such a geometric understanding can be reached via a branch of mathematics called planar graph theory2 . By examining specific examples we will show how studying planar graphs can lead to the generation of more complex modular origami models and touch upon the inverse direction as well: how studying modular origami can lead to a better understanding of planar graph theory. Such a correlation would certainly be valuable to educators interested in implementing more graph theory into their curriculum. The author himself has found unit origami not only to be a great educational device, but also a great source of ideas and inspiration in his own mathematical research on planar graphs. 2. What is a planar graph? A graph is what mathematicians use to model networks. It consists of a collection of points, called vertices, which are connected by lines, called edges. Intuitively, graphs can be thought of as simply a bunch of vertices connected by edges, but for completeness a technical definition follows:

1 Author\’s Bio: Thomas Hull is a graduate student of mathematics at the University of Rhode Island. His dissertation is on graph theory, and he researches origami \”on the side\”. 2 For a more rigorous introduction to graph theory, see [2] or [3].

Also,since more than one piece of paper is involved, one can experiment with the multitudeof color patterns that are possible in any one modular origami work. In this paper, we will discuss howsuch a geometric understanding can be reached via a branch of mathematics calledplanar graph theory2. As a result, there are many known facts about planar graphs. Type I (R. e. But consider the exercise of 3-coloringthe edges of a dodecahedron, which can easily be realized using unit type I in Figure4. References[1] D. R. Graphs and Digraphs, 2nd ed. The Four-Color Problem: Assaults and Conquests,McGraw-Hill, New York, (1977).

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