Origami Axioms and Mathematical Constructions: A History and Exploration Derek Chen Maggie L. Walker Governor’s School ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 2 Abstract The development and application of a mathematically rigorous theory of origami has been a very recent phenomenon, considering the historical origins of origami and paperfolding in general. Many civilizations discovered paperfolding as a recreational activity soon after the invention of paper thousands of years ago, while in contrast, the upsurge in popularity of the discipline, as well as the rise in mathematical interest surrounding origami, have all taken place largely since the mid-20th century. Among the most extensively studied mathematical interpretations of origami are the seven Huzita-Hatori or Huzita-Justin Axioms, which describe the full set of single folds possible on a plane of paper. The operations described by these ‘axioms’ are similar to those allowed with a compass and straightedge – they provide a framework under which certain numbers are constructible. One particular Huzita-Hatori axiom, the sixth, is equivalent to finding a common tangent of two distinct parabolas in the plane, which may have up to three solutions. As a result, single-fold origami constructions are more powerful than compass and straightedge constructions, allowing for a larger set of “origami-constructible” numbers than just the Euclidean or Thalian numbers. Thus, many problems that stymied the Ancient Greeks, such as doubling the volume of a given cube, solving a general cubic equation, and trisecting an arbitrary angle, are possible with origami. Even more powerful operations are possible if multiple simultaneous folds are allowed, paving the way to solving polynomials of any degree and constructing regular polygons of increasing size. However, even the single-fold axioms present an intriguing take on paperfolding that offers concrete solutions to famous mathematical problems. ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION Table of Contents I. Introduction 4 II. Definitions 5 III. The Huzita-Hatori Axioms 6 IV. Geometric and Algebraic Consequences 8 3 IV.i The Construction of √2 9 IV.ii The Trisection of an Angle 12 IV.iii The Real Solutions of a Cubic 13 V. Two-Fold Axioms and More 16 Bibliography 18 3 ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 4 I. Introduction The origins of paperfolding (“origami”) trace their roots to the invention of paper, with various traditions beginning contemporaneously in many regions of Eastern Asia (Hatori, n.d.). Origami, the Japanese offshoot of this phenomenon, began following the arrival of paper in the archipelago, introduced by Buddhist monks during the 6th century C.E. Paperfolding appeared in Europe during or prior to the 15th century, though it may have been discovered much earlier by the Moors. In the early 20th century, origami was popularized as a tool for child development and education, and colorful kami was used to teach origami in the Bauhaus design school in Germany. Origami underwent a renaissance in the middle of the 20th century thanks to the contributions of Akira Yoshizawa, an origami designer and author. In 1954, Yoshizawa introduced a notational system to indicate origami folds which evolved into the internationally recognized Yoshizawa-Randlett system still used today (Fox, 2005). Further publications by other authors popularized the tradition immensely, and the foundation of origami societies like OrigamiUSA and an increasing online presence have resulted in a substantive rise in awareness of the craft worldwide. Along with this revolution came a realization of the mathematical contexts of paperfolding. For much of its existence, origami has been based on linear folds – only recently have innovations in wet-folding and techniques employing curved folds begun to depart from the rigidly defined constructions of ‘pure’ origami. Like the compass and straightedge, linear origami presents a method of constructing numbers, geometries, and algebraic analogues (i.e. finding real roots of polynomials). However, the seven Huzita-Hatori axioms that describe paperfolding are more powerful than the compass and straightedge (Hull, 2011). Namely, ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 5 paperfolding under the Huzita-Hatori axioms allows for the trisection of angles, doubling the cube, and solving general cubic equations (Alperin & Lang, 2006). II. Definitions Point A point (𝑥, 𝑦) is an ordered pair of Cartesian coordinates in two dimensions. Line A line defined by the ordered pair (𝑋, 𝑌) is the set of points that satisfy 𝑋𝑥 + 𝑌𝑦 + 1 = 0. This definition of a line gives a unique representation of every describable line, except for lines that pass through the origin. This problem is resolved by translating any given system so no such lines exist. Folded Point The folded image 𝐹𝐿𝐹 (𝑃) of a point 𝑃 = (𝑥, 𝑦) across a line 𝐿𝐹 is the reflection of point 𝑃 across 𝐿𝐹 . The coordinates of 𝐹𝐿𝐹 (𝑃) can be expressed as ( 𝑥(𝑌 2 −𝑋 2 )−2𝑋(1+𝑦𝑌) 𝑦(𝑋 2 −𝑌 2 )−2𝑌(1+𝑥𝑋) , 𝑋 2 +𝑌 2 𝑋 2 +𝑌 2 ). Folded Line The folded image 𝐹𝐿𝐹 (𝐿) of a line 𝐿 = (𝑋, 𝑌) across fold line 𝐿𝐹 = (𝑋𝐹 , 𝑌𝐹 ) is the reflection of 𝐿 𝑥(𝑋𝐹 2 −𝑌𝐹 2 )+2𝑋𝐹 𝑌𝑌𝐹 across 𝐿𝐹 . The coordinates of 𝐹𝐿𝐹 (𝐿) are given by (𝑋 𝐹 2 −2𝑋𝑋 𝐹 −2𝑌𝑌𝐹 +𝑌𝐹 2 𝑦(𝑋𝐹 2 −𝑌𝐹 2 )−2𝑋𝑋𝐹 𝑌𝐹 ,𝑋 𝐹 2 −2𝑋𝑋 −2𝑌𝑌 +𝑌 2 𝐹 𝐹 𝐹 ). Note that folding is the inverse operation of itself – that is, the folded image of the folded image of a point 𝑃 or line 𝐿 under an arbitrary fold line 𝐿𝐹 is itself (𝐹𝐿𝐹 (𝐹𝐿𝐹 (𝑃) = 𝑃 and 𝐹𝐿𝐹 (𝐹𝐿𝐹 (𝐿) = 𝐿) (Alperin & Lang, 2006). Point Alignment ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 6 In general, an alignment, or incidence, between two objects 𝐴 and 𝐵 is notated as 𝐴 ↔ 𝐵. Two points, 𝑃1 = (𝑥1 , 𝑦1 ) and 𝑃2 = (𝑥2 , 𝑦2 ) are said to be aligned, i.e., 𝑃1 ↔ 𝑃2 , iff 𝑥1 = 𝑥2 and 𝑦1 = 𝑦2 . Line Alignment Two lines 𝐿1 = (𝑋1 , 𝑌1 ) and 𝐿2 = (𝑋2 , 𝑌2 ) are aligned (𝐿1 ↔ 𝐿2 ) iff 𝑋1 = 𝑋2 and 𝑌1 = 𝑌2 . Point-Line Alignment A point P=(x,y) and a line L=(X,Y) are aligned (P↔L) iff xX+yY+1=0. That is, P and L are aligned if and only if P is a point whose coordinates satisfy the equation defining line L (P is on line L). III. The Huzita-Hatori Axioms The seven elementary single-fold operations in origami were discovered first by Justin, crediting Peter Messer in the publication (Alperin & Lang, 2006). In 1989, Huzita again introduced six of the seven known axioms, defining possible folds given an initial set of points (in context, intersections of creases or where a crease meets the edge of the paper) and lines (creases, paper edges, etc.). These six axioms have been shown to encompass all Euclidean constructions possible with a straightedge and compass – Martin further showed that Huzita’s sixth axiom alone allowed all constructions possible by the first six operations (Alperin R. C., 2000). On the other hand, Auckly and Cleveland showed that the set of constructible numbers from the first 4 axioms is smaller than that of compass and straightedge construction (Alperin & Lang, 2006). In 2001, Hatori rediscovered the existing seventh axiom (first introduced by Justin two decades prior), completing the set of single-fold operations. It does well to note that the first ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 7 five “axioms” are all constructible using solely the sixth, but following nomenclatural precedent the set of seven operations are all named the Huzita-Hatori or Huzita-Justin Axioms, as follows: 1. Given two points P1 and P2, there exists a unique fold that connects them. Axiom 1 is the operation creating a line that passes through two points. This operation has exactly one solution. 2. Given two points P1 and P2, there exists a unique fold that aligns P1 with P2. The second axiom describes the operation that folds the perpendicular bisector of P1P2. Again, there is always exactly one solution to this axiom. 3. Given two lines L1 and L2, there is a fold that aligns L1 with L2. Axiom 3 is equivalent to finding an angle bisector of the angle between two lines. Note that when L1 and L2 are parallel, this operation has only one solution, and it has two solutions given skew lines. 4. Given a point P and a line L not aligned with P, there exists a unique fold that passes through P and is perpendicular to L. Axiom 4 describes the construction of a perpendicular passing through a point. There exists one solution for this operation. 5. Given two points P1 and P2 and a line L, there exists a fold that aligns P1 with L and that passes through P2. The fifth axiom may have 0, 1, or 2 distinct solutions. The operation described finds the point(s) of intersection of a circle with center P2 passing through P1 and the line L. 6. Given two points P1 and P2, and two lines L1 and L2, there exists a fold that aligns P1 with L1 and aligns P2 with L2. ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 8 The sixth operation finds a line simultaneously tangent to two parabolas, with foci at P1 and P2, and directrices defined by L1 and L2. An expanded discussion on this interpretation of HHA 6 may be found in Section IV.iii. Since two parabolas in the plane have at most three distinct lines of tangency, this axiom has 0, 1, 2, or 3 solutions, allowing for the solving of cubic equations. 7. Given a point P and two lines L1 and L2, there exists a fold that aligns P with L1 and is perpendicular to L2. The final axiom has no concise geometric equivalent other than the operation described. It may have 0 solutions (i.e. if L1 and L2 are parallel and P is not already aligned with L1) or 1 solution. It has been shown that, though this is not equivalent to any of the previous six operations, HHA 7 does not expand the set of origami-constructible numbers (Alperin & Lang, 2006). IV. Geometric and Algebraic Consequences As discussed, the seven Huzita-Hatori Axioms (HHAs) permit the construction of all numbers and figures possible using a compass and straightedge (Alperin R. C., 2000). However, it has been shown that several constructions not possible with the compass and straightedge are possible with origami – for example, constructing the cube root of 2 and trisecting an arbitrary angle (Lee, 2017). There exist limitations to the HHAs, among them the quintisection of an angle and the construction of the endecagon, as well as the solving of quintic equations (Alperin & Lang, 2006). All of these constructions are possible under origami, though they require operations outside the HHAs, specifically the simultaneous creation of two folds. ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 9 It has been proven that the seven HHAs cover all possible single-fold operations in the plane through exhaustive enumeration of all possible alignments of points and lines (Alperin R. C., 2000). For two-fold operations, there exist 489 distinct operations, which lie firmly outside the scope of this paper. Further, the definition of three-fold operations allows for the solving of quintic operations – as a general rule, an nth-degree general equation may be solved with operations of n-2 folds (Alperin & Lang, 2006). Noting the limitations of the HHAs, they nonetheless represent a more powerful set of operations than those defined for the straightedge and compass. A discussion of several applications of the HHAs follows: the doubling of the cube, the trisection of an angle, and the solving of cubic equations. 3 IV.i The Construction of √2 The construction of the cube root of 2 is an ancient problem – that of doubling the volume of a cube, which was tackled by the Ancient Greeks with compass and straightedge operations, which are not powerful enough to complete the construction (Hull, 2011). However, this problem is possible under one-fold planar origami, as described here. To begin, consider the Beloch Square. Given two distinct points P1 and P2, and two lines, L1 and L2, the Beloch Square is a square formed by points WXYZ such that adjacent corners X and Y are aligned with L1 and L2, respectively, and the two parallel lines defined by WX and YZ are aligned with P1 and P2, respectively. Such a square can be constructed with origami. First, two lines L1’ and L2’ are constructed, such that L1’ is parallel to L1 and L1 is equidistant to P1 and L1’, and L2’ is parallel to L2 and L2 is equidistant to P2 and L2’. Then, the sixth Huzita-Hatori operation is performed, aligning P1 with L1’ and P2 with L2’. The fold line created, LF, intersects ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 10 L1 and L2 at points X and Y, respectively. Further, since L1 is halfway between P1 and L1’, X is collinear with P1 and its folded image, and thus similarly, Y is collinear with P2 and its image. Further, LF is necessarily perpendicular to the line segments connecting P1 and P2 with their folded images, and as a result, we know P1X and P2Y are parallel and both perpendicular to XY. Thus, we let XY be one side of the Beloch Square, with adjacent sides to XY aligned with P1X and P2Y. Fig. 1: The Beloch Square and its Construction ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 11 The construction of Beloch’s Square can be applied to the construction of the cube root of 2. In this case, we define L1 to be the y-axis and L2 to be the x-axis. Further, we let P1 = (-1,0) and P2 = (0,-2). Consequently, L1’ is the line x = 1, and L2’ is the line y = 2. Using HHA 6, a fold can be made that aligns P1 with L1’ and P2 with L2’, which will intersect the x- and y-axes at two points, X and Y, respectively. Calling the origin O, notice that the right triangle OP1Y is similar to the two triangles OXY and OP2X, since XY is perpendicular to P1FLF(P1) and P2FLF(P2). Thus, |OY| |OX| we know that |OP | = |OY| = 1 |OX| |OP2 | |OX| . We know |OP1| = 1 and |OP2| = 2, and therefore substituting 2 3 yields |OY| = |OY| = |OX|. Solving this for |OY| yields |OY|3 = 2, and thus |OY| = √2 and Y = 3 3 (0, √2). Note that with an arbitrary P = (0, −k), Y will have coordinates (0, √k), allowing the construction of the cube root of any real (Hull, 2011). 3 Fig. 2: Using Beloch’s Square to Construct √2 ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 12 IV.ii The Trisection of an Angle Based on the work of Hull, Alperin, and Lang, Hisashi Abe developed a method of trisecting any given angle θ using single-fold operations in 1980 (Lee, 2017). As with the doubling of the cube, this was a problem attempted by the Greeks, but left unsolved, as its generalization requires the construction of solutions to a cubic equation, which cannot be done with a compass and straightedge. Given an arbitrary angle θ represented by two lines L1 and L2 intersecting at point P in the plane, position L2 horizontally and fold a perpendicular to it passing through P. This new line, L3, can be thought of as our y-axis, and L2 as our x-axis. Let Q be an arbitrary point on L3. Fold a perpendicular, LP, to L3 through Q, and then make the fold that aligns LP with L2 – since LP and L2 are parallel, this will construct a line L4 that is also parallel to the two lines and halfway between them. Next, HHA 6 is used to align Q onto L1 and P onto L4. The fold, LF, that accomplishes this is unique, and the line connecting P and its folded image from this operation θ will form an angle of 3 with L2. To prove this, we will consider P’ and Q’ to be the folded images of P and Q across LF. Let R be the intersection of PQ’ and P’Q, S be the intersection of L3 and L4, and T be the intersection of PP’ and LF. We know that R is aligned with LF since P’ and Q’ are the reflections of P and Q across LF, and we also know that |PS| = |QS|. Further, since P’ lies on L4, we know P’QP is an isosceles triangle. Also, |PT| = |P’T| since P’ is the reflection of P across LF, and since R lies on a perpendicular equidistant from P and P’, RPP’ is also an isosceles triangle. letting m∠RPP′ = 𝛼, 𝑚∠𝑃𝑃′𝑆 = 𝛽, & 𝑚∠𝑅𝑃′𝑆 = 𝛾, we then know that 𝛽 = 𝛾 and 𝛼 = 𝛽 + 𝛾. Since L2 and L4 are parallel, the angle formed by PP’ and L2 also has measure 𝛽. Therefore, 𝜃 = 𝛼 + 𝛽 = 1 (𝛽 + 𝛾) + 𝛽 = 3𝛽, and so 𝛽 = 3 𝜃. ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 13 Fig. 3: Trisection of an Arbitrary Angle While the process described assumes an acute angle θ, the method can be extended to obtuse angles by dividing said angle into a right angle and an acute angle, for example by folding a perpendicular to one of the initial rays defining θ (Lee, 2017). Following the trisection process on the two new angles will construct the trisection of θ between the two respective segments, P1P1’ of the acute angle, and P2P2’ of the right angle. IV.iii The Real Solutions of a Cubic The sixth Huzita-Hatori Axiom may also be called the Beloch Fold, as it was independently discovered in 1936 by Margharita P. Beloch, who was the first to realize and ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 14 prove that origami constructions can find the real solutions of general cubic equations (Hull, 2011). First, consider an arbitrary fold line LF that aligns a given point P and line L, given P and L are not already aligned. This fold aligns some point P’ on L with P. A perpendicular of L aligned with P’ will necessarily intersect with LF at some point X. Note that, since P’ is the folded image of P across line LF, the point X on LF must be equidistant from P and P’. Since the line segment between P’ and X is perpendicular to L, consequently X is equidistant from P and L, and thus is a point on the parabola defined by focus P and directrix L. Thus, any arbitrary fold that aligns P and L will be a tangent line of this parabola (Lucero, 2017). Since HHA 6 aligns two given points, P1 and P2, with two given lines, L1 and L2, it can be thought of as finding a single line, L’, that is tangent to the two parabolas defined by foci P1 and P2, and directrices L1 and L2, respectively (Lucero, 2017). As mentioned, two parabolas in the plane can have at most three unique common tangents, suggesting that this operation is equivalent to solving a general cubic equation. Beloch’s development of a method to construct real roots relies on an 1867 paper by Eduard Lill, an Austrian engineer (Hull, 2011). This paper contains a geometric procedure for finding real roots of any given polynomial. An abridged description of Lill’s method follows. Given some polynomial 𝑓(𝑥) = 𝑎0 + 𝑎1 𝑥1 + 𝑎2 𝑥 2 +. . . +𝑎𝑛 𝑥 𝑛 , we can geometrically create a path representing the coefficients of 𝑓(𝑥), and in so doing find the real solutions of the polynomial. To construct this path, begin at the origin and face in the positive x-direction, and move a distance of an. Then, turn 90° counterclockwise and move a distance of an-1. This process repeats until the final segment, of distance a0, is traveled, at which point the path ends at some point T. For any negative coefficients, movement is opposed to the direction faced. Then, ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 15 beginning at the origin, a line at angle θ from the positive x-axis is extended until it intersects with the extension of the segment corresponding to an-1, at which point it will turn 90° towards the segment corresponding to an-2. This process repeats until the new path intersects the line extending from the a0 segment. If this new path also ends at the point T, then x = -tanθ is a root of f(x) (Hull, 2011). The proof of this, which relies on the recognition that the second path is composed of hypotenuses of similar right triangles, whose legs lie exclusively along lines defined by the first path, has been omitted for brevity, but is left as an exercise to the reader. Fig. 4: Lill’s Method for Cubic Polynomials (Note that a1 < 0) Lill’s method can be applied to polynomials of any degree, but Beloch recognized the power of HHA 6 to execute Lill’s method for cubic equations, and therefore to solve the general cubic. First, Lill’s method is followed, constructing a path of four segments corresponding to a3, a2, a1, and a0 of the given cubic equation. Considering the origin as P1, and the ending point T as P2, and further considering the lines extending from the a2 segment and a1 segment as L1 and L2, respectively, then the sides of the Beloch Square (see IV.i) constructed from these givens will lie ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 16 along the lines forming the second path, thus constructing the desired angle θ with the a3 segment extending from the origin (Hull, 2011). Folding a perpendicular to the a3 segment of unit distance from the origin then results in a construction of |x|, where x is a solution of the given cubic polynomial. Since a maximum of three unique Beloch Squares can be constructed from a given set of two points and two lines, all real roots of any cubic may be found with this method. V. Two-Fold Axioms and More It does well to establish that the Huzita-Hatori Axioms do not encompass all possible operations in origami – rather, they describe only those performed with a single fold. Allowing two, three, or more simultaneous folds to be made drastically increases the number of possible operations – as noted before, the exhaustive list of two-fold operations contains 489 distinct combinations (Alperin & Lang, 2006). The full list of these is omitted, though it is notable that just two such operations, in combination with the seven HHAs, allow for the solving of some quintic polynomials and the quintisection of an angle. One limitation of two-fold axioms is that it is unknown whether or not they allow for the solution of a general quintic, only a select subset of them. However, it can be shown that any quintic polynomial can be solved by expanding to three-fold axioms. The method devised by Alperin and Lang to find such solutions again relies on Lill’s Method, and it has been proven that the application of Lill’s Method using N-Fold axioms allows for the solution of general polynomials of degree N + 2. In considering two- and three-fold axioms, it is clear that the theory yields broad consequences in angle division and polynomial solutions, but even two-fold axioms become increasingly impractical as a physical method for solving quartic equations, as methods to ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 17 perform such operations simply do not exist in most cases (Alperin R. C., 2000). As a consequence, the theory developed for single-fold operations has far more meaningful implications for practical mathematics – that is, for defining and exploring the physical limits of origami in the real world. ORIGAMI AXIOMS AND MATHEMATICAL CONSTRUCTIONS: A HISTORY AND EXPLORATION 18 Bibliography Alperin, R. C. (2000). A Mathematical Theory of Origami Constructions and Numbers. New York Journal of Mathematics, 119-133. Alperin, R. C., & Lang, R. J. (2006, December 6). One-, Two-, and Multi-Fold Origami Axioms. 1-25. Fox, M. (2005, April 2). Akira Yoshizawa, 94, Modern Origami Master, Dies. Retrieved from New York Times: https://www.nytimes.com/2005/04/02/arts/design/akira-yoshizawa-94modern-origami-master-dies.html Hatori, K. (n.d.). History of Origami. Retrieved from K's Origami: https://origami.ousaan.com/library/historye.html Hull, T. C. (2011, April). Solving Cubics With Creases: The Work of Beloch and Lill. The Mathematical Association of America Monthly(118), 307-315. Lee, H. Y. (2017). Origami-Constructible Numbers. Georgia, United States of America. Lucero, J. C. (2017). On the Elementary Single-Fold Operations of Origami: Reflections and Incidence Constraints on the Plane. Forum Geometricorum, 17, 207-221.