Totally Real Origami and Impossible Paper Folding

Origami is the ancient Japanese art of paper folding. It is possible to fold many intriguing geometrical shapes with paper [M]. In this article, the question we will answer is which shapes are possible to construct and which shapes are impossible to construct using origami. One of the most interesting things we discovered is that it is impossible to construct a cube with twice the volume of a given cube using origami, just as it is impossible to do using a compass and straight edge. As an unexpected surprise, our algebraic characterization of origami is related to David Hilbert's 17th problem. Hilbert's problem is to show that any rational function which is always non-negative is a sum of squares of rational functions [B]. This problem was solved by Artin in 1926 [Ar]. We would like to thank John Tate for noticing the relationship between our present work and Hilbert's 17th problem. This research is the result of a project in the Junior Fellows Program at The University of Texas. The Junior Fellows Program is a program in which a junior undergraduate strives to do original research under the guidance of a faculty mentor. The referee mentioned two references which the reader may find interesting. "Geometric Exercises in Paper Folding" addresses practical problems of paper folding [R]. Arnong many other things, Sundara Row gives constructions for the 5-gon, the 17-gon, and duplicating a cube. His constructions, however, use more general folding techniques than the ones we consider here. Felix Klein cites Row's work in his lectures on selected questions in elementary geometry [K]. In order to understand the rules of origami construction, we will first consider a sheet of everyday notebook paper. Our work with notebook paper will serve as an intuitive model for our definition of origami constructions in the Euclidean plane. There are four natural methods of folding a piece of paper. The methods will serve as the basis of the definition of an origami pair. We construct the line L1, by folding a crease between two different corners of the paper. Another line may be constructed by matching two corners. For example, if corners or and Sy are matched, the crease formed, L2, will be the perpendicular bisector of the segment orSy. Another natural construction is matching one line to another line. For instance, ,8^y, the paper's edge, and L2 are lines. If we lay ,8^y upon L2 and form the crease, then we obtain L3 which is the angle bisector of the two lines. If we start with two parallel lines in this third construction, then we will just get a parallel line half way in between. The fourth and final construction which seems natural is consecutive folding. This is similar to rolling up the sheet of paper only one does not roll it up, he folds it up. More explicitly, start with a piece of paper with two creases on it as in Figure 2. Fold along line Ll and do not unfold the piece of paper. Notice that line L2 lies