Generalisation of a Paper-Folding Axiom and its Exploration using Interactive Geometry Software

by Colin McAllister

22 July 2010

Abstract The Huzita-Justin paper-folding Axiom 5, adapted for a circle, is explored using interactive geometry software. The axiom is generalised to other shapes, and applied to a triangle with rounded corners. An interesting configuration of folds is discovered when the triangle is equilateral. The properties of this configuration are explained by drawing a circle, of which the folds are diameters. An adjustable simulation of a folded paper triangle is used to demonstrate this explanation. A folding hypothesis is postulated for arbitrary shapes.

Acknowledgement

I wish to thank Maria Droujkova, Linda Fahlberg-Stojanovska and my former school teacher Kenneth Blair for sharing their ideas and for their enthusiasm in exploring and teaching mathematics.

1. Introduction

The Huzita-Justin or Huzita-Hatori axioms [1] state the mathematical principles of paper folding or Origami. In the discussion Circle Origami Axioms [2] on the Math 2.0 Interest Group [3], the axioms are adapted for folds on a plane sheet of paper with a circular boundary. Huzita-Hatori Axiom 5 states: “Given two points p1 and p2 and a line l1 we can make a fold that places p1 onto l1 and passes through the point p2.” Huzita-Hatori Axiom 5 is adapted as Axiom 5-C for a circle. It applies for two points p1 and p2 in a circle, and the possibility of folds through p2 that place p1 onto the boundary of the circle. Axiom 5-C states "If the distance between p1 and p2 is greater than the distance between p2 and the circle, there are two such folds, if the distances are equal, one such fold. If the distance between p1 and p2 is smaller than the distance between p2 and the circle, the fold is impossible."

I propose generalising Axiom 5-C to shapes more general than circles. What family of shapes does the axiom define? Does it restrict the radius of curvature of arc sections of the boundary? Would an angular shape with sharp corners be excluded? Would a starfish (sea star) shape, with inward curving corners, be included? We can explore such questions by using interactive geometry software to create, perform and verify geometry experiments. Programs like CaRMetal [5] and Geogebra [6] are flexible tools for teaching mathematics, and they are easy for students to learn [7]. I used the CaRMetal cross-platform application for this study. The software is accessed graphically via geometry diagrams that serve as both input and output of the computer programs. The key diagrams are included in this paper, and explain the evolution of this investigation. The geometry files are available for download from the geometry file sharing site http://i2geo.net [4].

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2. Interactive Geometry SoftwareGeometry software facilitates more abstract imagination of constructions than folding a sheet of paper. Folding a sheet of paper against itself makes a straight crease. This is equivalent to reflection of part of the sheet in a straight line on the Euclidean plane. Thus paper folding, which occurs in 3-dimensional space, becomes accessible to geometry software that models a 2-dimensional plane. Geometry software represents each point internally as a pair of floating point numbers, under an orthogonal coordinate system. The components are represented internally as objects, and drawn on the screen as a diagram. When free points are moved by the mouse, dependent objects are instantaneously recalculated and redrawn. This interaction facilitates experimental investigation. It provides familiarity with the problems, and confidence in our discoveries and solutions. Such a solution may be useful and educational, but it does not constitute mathematical proof. It may be the foundation of a mathematical proof, given further analysis, calculation and deduction.

3. Geometric Model of a Folded Circle

Figure 1. Geometric Model of Origami Circle Axiom 5-C

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Paper folding is modelled by configuring the initial points and constraints and defining our problem to be discovery of the folds and image points that satisfy those constrains. We use the tool panel or tool bar of the geometry software to define our problem by building a construction of dependent objects. An object point and its image on the boundary of the shape define a line segment that can be envisaged as a ray of light. The line of reflection is the perpendicular bisector of the ray, and corresponds to the fold that would be made on a paper circle. Figure 1 shows a model of Axiom 5-C, constructed using the CaRMetal geometry software. Points p1 and p2 are free to move. The dashed circle is drawn around p2 and through p1. Intersection of the dashed circle with the boundary circle defines the image points p1' and p1''. The line segments ray' and ray'', represent the translation of p1 to the boundary. The origami creases, fold' and fold'', are derived as perpendiculars of ray' and ray''. Folding a paper circle along fold' would place p1 on the boundary at p1'. Alternatively, folding it along fold'' would place p1 on the boundary at p1''. If the dashed circle intersects the boundary of the shape, two folds can be drawn. If the distance from p1 to p2 equals the distance from p2 to the nearest point on the boundary, one fold exists, because the two circles meet at a single tangent point. If the dashed circle does not intersect the boundary, there is no fold that solves the problem for the given positions of p1 and p2. Thus the geometry software allows us to explore and solve the problem of identifying the folds of the paper circle.

4. Generalising Axiom 5-C to Shapes other than CirclesThe property of Axiom 5-C that most concerns us is that there are no more than two folds. That is not a natural limit of the axiom for shapes other than circles, so we must devise a way to impose it.

The generalised shape that I choose to study is a triangle with rounded corners. For simplicity, I examine a triangle with the same radius of curvature on each corner. You could make this shape by taping three paper plates and three sheets of photocopier paper together. To construct it with compass and straight edge, draw three identical circles and connect them with three straight segments that meet the circles at a tangent. The boundary of the shape is composed of one arc section of each circle, and the line segments that connect them.

I propose that Axiom 5-C holds true if the radius of curvature of all arc sections of the boundary is greater than or equal to the distance between p1 and p2. Each corner has the same radius, so we can enforce this condition by drawing a line segment from p2, of fixed length equal to that radius. Let p1 move along that line segment, inclusive of its end point. This construction is shown in figure 2.

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5. Modelling a Triangle with Rounded Corners

Figure 2. Origami Circle Axiom on a Triangle with Rounded Corners

I used the CaRMetal interactive geometry software to create this construction. (CaR stands for Compass and Ruler). (p1 is represented as a black point and p2 as a red point on the diagram). The calculated folds or lines of reflection are shown as red lines. The ray lines that connect p1 to its image points on the boundary are shown as green line segments. By experimenting with this construction, we build up confidence that Axiom 5-C holds true. Moving p1 (black) and p2 (red) with the mouse, we can observe the translation and change in size of the dashed circle, and the movement of its intersection points on the boundary of the rounded triangle. A limitation of this model is that the boundary is not a continuous curve on which the intersection points with the dashed circle can move freely. We can easily work around this by using the software's tool bar to assign intersection points, to either the straight segments or the arc segments of the boundary, as the dashed circle is moved around. After experimenting with a variety of positions of p1 and p2 in the triangle, maintaining the restriction on the distance between them, I came to the reasonable conclusion that Axiom 5-C holds true. Would a triangle with sharp corners be excluded by the axiom? Yes, because the radius of curvature of a sharp corner is zero. Would a starfish (sea star) shape, with inward curving corners, be included? I suggest that you carry out your own geometry experiments to answer that question.

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Explore Axiom 5-C by moving points p1, p2 and p3 using the mouse. p1 is the object point that must fold onto the boundary. p2 is the point through which the fold must pass. The line segment p2-p3 has fixed length equal to the radius of curvature of the corners of the triangle. p1 is free to move on this line segment, thus setting a maximum distance between p2 and p1. As the points are moved, the (red) folds are recalculated, permitting exploration of the geometry.

6. Multiple Folds of a Rounded Triangle

Figure 3. Folds of a Triangle with Rounded Corners

Axiom 5-C stated there were up to two folds of the circle that satisfied the requirements. We were able to enforce the same limit of two folds on a rounded triangle, by restricting the distance between p1 and p2 to the radius of curvature of the corners. On removing this restriction, up to six folds are possible. By dragging p1 (black) and p2 (red) around with the mouse, we can see that there are positions that result in two, four or six different (red) fold lines that project p1 onto the boundary. With six fold lines, there are six (blue) image points, and these may fall on the straight or circular sections of the boundary. Positions with an odd number of folds are difficult to find by moving p1 with the mouse, as they occur only when the circle is precisely tangential to the boundary.

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7. Multiple Folds of a Rounded Equilateral TriangleNow, there is an interesting discovery to be made. Using the mouse, move the three equal circles so that they make contact and form an equilateral triangle with rounded corners, as shown in figure 4. Move point p2 (red) to the centre of the triangle. Explore the changing geometry by moving point p1 (black) around the triangle. We discover a position of p1 for which the six image points coincide with joints of the arc sections and straight segments of the boundary. The diagram shows that the points do not coincide exactly, because p1 and p2 are manually positioned at approximate positions, using the mouse. Notice the dashed circle in the construction, which is centred on p2 and goes through p1. It is the intersection of this circle with the boundary that defines the image points. The geometry data files are available on i2geo.net [4] for both CaRMetal and Geogebra software.

Figure 4. Folds of an Equilateral Triangle with Rounded Corners

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8. Analysis of Folds of an Equilateral TriangleThe boundary of a rounded equilateral triangle (figure 4) is composed of arc and line segments that meet at six junction points. The discovery, of six image points coincident with six junction points needs to be explained. My first conclusion was that this coincidence was due to the three inscribed circles being in contact with each other. I disproved that by moving the circles apart, to make a larger equilateral triangle, and observing that the coincidence remained. The important property of the rounded equilateral triangle is that the six junction points, between the straight and arc sections, all lie on a circle. It is easy to deduce that the junction points lie on a circle. By three-fold symmetry about the centre of the triangle and symmetry of reflection in the lines of altitude, it is clear that all six joints are equidistant from the centre. This intersecting circle can vary in size, from a small circle that inscribes the triangle, to a large circle that circumscribes the triangle. The object point p1 can be anywhere on that circle, as we can see by moving it left or right of the altitude line.

Figure 5. Circle Intersecting an Equilateral Triangle

I made another geometric model of an equilateral triangle in which the intersecting circle has adjustable radius. The adjustment is made by moving the point labelled “Control” with the mouse. Intersection of the circle with the triangle defines the six image points. The object point p1 is placed on the circle, and may be moved using the mouse. The (green) rays from the object point to the six image points are drawn. The six (red) folds are drawn as perpendicular bisectors of the rays. The diagram shows that the rays are chords of the circle, and the folds are diameters. The perpendicular bisector of a chord of a circle is always a diameter, and diameters pass through the centre of the circle. Therefore the six folds all pass through the centre of the circle. That explains the coincidence of image points that was observed in figure 4. The effect does not depend on the radius of curvature of the corners. It also applies to an equilateral triangle that does not have rounded corners.

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9. Simulated Folding of a Paper TriangleI created a more realistic model of a folded paper triangle using the CaRMetal geometry software. The (red) lines of reflection represent folds of paper equilateral triangle. Folding of paper is simulated by fading out the portion of the triangle above the fold line, and drawing its reflection below the fold line. A line segment of the base of the triangle is hidden, to give the appearance of two layers of paper. The simulation is dynamic. Moving the “Control” point with the mouse changes the radius of the intersecting circle, causing the fold line to rotate about the centre of the triangle. Figures 6 and 7 illustrate this simulation for two different radii of the intersecting circle.

Figure 6. Simulated Fold of an Equilateral Triangle (1 of 2)

In figure 6 a blue circle intersects the black equilateral triangle. The control point is moved to the right to give the circle a larger radius. The six possible positions where p1 may be folded onto the boundary of the triangle are marked at the intersection of the circle and the triangle. There are six corresponding folds through p2, shown as red lines. Only one of these fold lines is selected for simulation. Point p1 is folded onto the boundary of the triangle at point p1'. The green line between p1 and p1' is perpendicular to the fold. The parallel green line shows how the apex of the triangle has been folded to its new position.

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Figure 7. Simulated Fold of an Equilateral Triangle (2 of 2)

In figure 7 the control point was moved to the left to make the circle smaller. The circle still intersects the triangle at six points, but their positions have moved. By comparing figure 6 and figure 7, you can see how variable control of the radius of the circle causes p1' on the folded part of the triangle to move, while remaining on the boundary of the triangle. The reflected apex of the triangle has also moved. This simulation demonstrates that the pattern of green ray lines and red fold lines is consistent for a circle of varying radius. The green ray lines are chords of the circle and the perpendicular red fold lines are diameters of the circle. That is why the folds always pass through the centre of the circle, even as we adjust its radius and change the angles of the fold lines.

10. Generalisation of Circle-Folding Axiom 5-C for other ShapesWe began this study with Circle-Folding Axiom 5-C, an adaptation of Huzita-Hatori Axiom 5. Using the example of an equilateral triangle, I make a more general adaptation, which I call Hypothesis 5-S, where S stands for an arbitrary shape. I call it a hypothesis rather than an axiom, because it is a suggestion for experiment, not a declaration of fact. It applies for two points p1 and p2 in a shape, and the possibility of folds through p2 that place p1 onto the boundary of the shape.

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We may state, as Hypothesis 5-S. “If a circle is drawn centred on p2 and through p1, then the number of folds through p2, which place p1 on the boundary, is equal to the number of points at which the circle intersects or touches the boundary.” In the case of an equilateral triangle, where the centre (p2) of the circle is the centre of the triangle, there are six folds, grouped in pairs, with 3-fold symmetry. It was this symmetry of the reflected points p1' that caught our attention while exploring folds of a triangle with rounded corners. Our analysis and simulation reveals that the symmetry can be explained by a circle centred on p2 and through p1. Our simulation gives confidence that the generalised Hypothesis 5-S applies to an equilateral triangle. Its application to other shapes is open to further study. The geometry data file for the paper folding simulation is available on i2geo.net [4] for both CaRMetal and Geogebra software.

11. SummaryThis study focussed on exploration of interactive geometry diagrams. I generalised circle origami Axiom 5-C beyond a circle to other shapes. A valid fold projects an object point p1 onto the boundary of the shape, and goes through point p2. I identified the minimum radius of curvature of an arc segment of the boundary as the constraint that limits the number of possible folds. The minimum radius of curvature must be greater than or equal to the distance between the object point p1, and the point p2 that lies on the fold. I experimented with a particular shape, a triangle with rounded corners, each corner having the same radius. Exploration using interactive geometry software gives confidence that the axiom holds true, and there are at most two folds under the given constraints. On removing the constraint of the axiom, there are up to six folds of the triangle that project p1 onto the boundary. By repositioning the circles to create an equilateral triangle, a surprising configuration is discovered, in which the six image points coincide with junction points of arcs and line segments on the boundary. The explanation is that the junction points lie on a circle, and the six fold lines are diameters of this circle. I simulated this explanation, using the geometry software to give the illusion of folded paper. Building on these observations, I propose a hypothesis that uses an intersecting circle to predict the number of valid folds of an arbitrary paper shape. The hypothesis is not noteworthy, but the methods of exploration and simulation may interest some.

LicenceThis work is licenced under the Creative Commons Attribution 2.0 UK: England & Wales License. To view a copy of this licence, visit http://creativecommons.org/licenses/by/2.0/uk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California 94105, USA.

References and Resources

[1] Huzita-Hatori axioms, at http://www.langorigami.com/science/hha/hha.php4[2] "Circle Origami Axioms" Bradford Hansen-Smith, Alexander Bogomolny, Maria Droujkova, Katherine Droujkov in Math 2.0 Interest Group, 24 June 2010 at http://mathfuture.wikispaces.com/Circle+origami+axioms [3] Math 2.0 Interest Group, at http://mathfuture.wikispaces.com/[4] The geometry file sharing site http://i2geo.net, this document is Paperfoldinggeometry.pdf[5] CaRMetal, Compass and Ruler interactive geometry software, at http://carmetal.sourceforge.net/[6] Geogebra, interactive geometry and algebra software, at http://www.geogebra.org/

[7] Compass and Straightedge with Geogebra, Linda Fahlberg-Stojanovska, http://geogebrawiki.pbworks.com/ciit10

[8] Author's mathematics blog: http://cmcallister.vox.com

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