Folding Paper and Planes - Goshen College



Folding Papers and Planes: A Critical Difference

by Rosanna Nafziger

One of my favorite occupations during middle school was folding origami. As I developed my folding precision, I could create more and more intricate works, until eventually I was folding immense geometric fusions of many origami units and some very acceptable imitations of spiders, winged horses, and three-headed dogs.

The mathematically puzzling piece is determining the limits and capabilities of origami folding. The stipulations of origami are that we begin with a square piece of paper and that a well-defined fold relies upon references—previously formed creases, points, or folds—such that the fold can be replicated fairly accurately on another piece of paper. If we consider the paper as a unit square with the bottom left corner at the origin, what points could we mark with well-defined folds? We could certainly reach the point [pic], since it would be indicated by the intersection of the creases formed by folding the paper in half vertically and horizontally. Following this type of operation, we can at least approximate any point in the square as close as can be desired.

Theorem. Let [pic] and [pic] such that [pic]and [pic]. Then there exists a series of folds of the unit square that define a point [pic] such that [pic] and [pic].

Proof. We will proceed by first proving that we can fold a vertical line within any given distance of x, and a horizontal line within any given distance of y.

First, let me define a vertical folding algorithm. At the very outset, we determine if x lies on either the left or right edge. If it does, we are done. In the first iteration, the paper is folded in half vertically, aligning the vertical edges of the paper (which are the two starting references), and then unfolded, leaving a crease we will call [pic]. If the point is on the crease, we are done. If the point is to the left of the crease, the paper is folded to align [pic]—the last crease folded—with the left edge—the reference immediately to the left of the point. If the point is to the right of the center crease, the paper is folded to align [pic] with the right edge—the reference immediately to the right of the point. Again, the paper is unfolded. If the point is on this second crease which we will call [pic], we are done. If the point is to the left of [pic], we fold the paper to align [pic]with the reference immediately to its left, which will either be [pic] or the left edge. If the point is to the right of [pic], we fold the paper to align [pic] with the reference immediately to its right, which will either be [pic] or the right edge. We continue this process of folding the nth crease between the previous crease and one of the references for the previous crease such that the point lies between the nth crease and the nearest reference for the previous crease. We continue until we either reach the point or come within a certain given distance of it. We define a horizontal folding algorithm analogously.

Now we want to prove that at the nth step of my algorithm, we have folded a vertical line that passes within [pic] of x. First, let [pic] be the closest distance from x to [pic]. We will proceed inductively. For the base case of [pic], there are three possibilities. In the case that x lies on [pic], [pic]. In the case that x is to the left of [pic], [pic] since x is neither on the left edge nor the center line and thus [pic] must be less than [pic]. In the third case, x is to the right of [pic] and [pic], so [pic] has to be less than [pic] as well. Note that since [pic] lies directly between its references, the distance between it and its references is [pic].

Next, we assume that at the nth iteration of our algorithm, when n creases have been folded, [pic] and the distance between [pic] and one of its references is [pic]. Then the [pic]th iteration will fold a crease exactly between the nth crease and one of the references used to fold the nth crease, so the distance from [pic] to either of its references will be [pic], or [pic]. Then, if the point lies on [pic], [pic] is 0 and therefore less than [pic]. If the point lies to either side of [pic], it must be between [pic] and one of its references, which implies that the horizontal distance between it and [pic] can be no more than the distance between [pic] and its references, which is [pic], so [pic]. Thus at the nth iteration of our vertical folding algorithm, we have defined a vertical line that passes within [pic] of the point [pic]. The proof that a horizontal folding algorithm produces a horizontal line within [pic] of [pic] is analogous.

To demonstrate that we can produce two lines whose intersection yields a point [pic] such that [pic] and [pic], we let k be an integer such that [pic], so [pic].[1] Then we perform both the horizontal and vertical algorithms k times for [pic] and let [pic] be the point where the horizontal [pic] ([pic]) intersects the vertical [pic] ([pic]). Since the vertical distance between [pic] and any point on [pic] is [pic], and the horizontal distance between [pic] and any point on [pic] is [pic]and [pic] lies on both [pic] and [pic], [pic] and [pic]. Therefore, since [pic]< r and [pic]< r, [pic] and [pic]. ٱ

Clearly, the algorithm I defined would not be physically possible with real-world paper, since physical paper clearly is an imperfect medium and the physical act of folding relies upon human manipulation, which is also imperfect. Not only that, but paper has thickness, so folds lack perfect precision, and thus locations on the paper lack perfect precision because they are determined by other folds or edges cut by humans.

Ideal paper, however, would also be unsatisfying. Consider the operation of folding a plane in “half” about a line in the plane, defined as follows:

Definition. Let P be a plane in three-space, l a line on P, and Q a plane containing l. A fold of P around l onto Q is a function [pic] such that the following conditions are met:

(1) All points on l remain fixed.

(2) For any point[pic] not on l and [pic]on l such that [pic]is the closest point on l to p, [pic]is also the closest point on l to [pic], and the distance from [pic]to [pic]is the distance from p to [pic].

(3) For any [pic] in Q with [pic] the point on l closest to q, if [pic] is nonnegative, then for any other point s in Q and [pic], the closest point on l to s, [pic] is also nonnegative. If [pic] is not positive, then [pic] is also not positive.

The second point provides the restriction that a fold is a reflection, and the third that we are reflecting the plane P only to the half of another plane Q that is on one side of the line. Note that this function is not onto. Observe that such an ideal fold seemingly can’t be unfolded—that is, [pic] does not exist. I will prove not only that f is not one-to-one, but that f takes pairs of points to the single points in Q. I will first need to introduce another plane, however.

Theorem. Let P and Q be planes intersecting at a line l, and let f be a fold from P to Q around l. Then for each point p in P not on l, the point [pic]on l closest to p, [pic] and p all lie on a plane perpendicular to l.

Proof. Let R be the plane containing p,[pic], and [pic]. Suppose R is not perpendicular to l. Then the line segment between [pic] and [pic] is not perpendicular to l, because it lies in R. Then [pic] cannot be the closest point on l to[pic], which is a contradiction. Thus R is perpendicular to l.ٱ

Having introduced the plane R, I can now go back to my original goal of proving that an ideal fold can’t be unfolded.

Theorem. Let P and Q be planes intersecting at a line l, and let f be a fold from P to Q around l. Then for each point p in P not on l, there exists a point [pic] such that[pic].

Proof. Let d be the shortest distance from p to l, and let [pic] be the reflection of p across l in the plane P. Then l is a perpendicular bisector of [pic], so [pic] lies on [pic], as does the point on l closest to [pic], [pic]. Since l and [pic] only intersect at one point, [pic], and [pic] contains [pic]and [pic]. Thus [pic].

Let R be the plane containing p, [pic], and [pic]. Then R contains [pic], which is the same as [pic], so [pic]. Similarly, the plane [pic]containing [pic], [pic], and [pic] contains [pic], or[pic], so[pic]. Knowing that R and [pic] are perpendicular to l and both contain p and [pic] is more than enough information to say that R and [pic] are the same plane. Then R contains f(p’), since f(p’) is in R’.

Since f(p) and f(p’) are in both R and Q, they lie on the line where R intersects Q—a line in Q perpendicular to l. We know that the distance d between p and [pic] equals the length of [pic], which is distance between [pic] and [pic], as well as between [pic] and [pic]. If [pic] and [pic] lie on the same line an equal distance away from [pic] and can’t be on opposite sides of [pic] because it lies on l, they must be the same point. Thus for each point p in P, a fold maps the reflection of [pic] to [pic].ٱ

Thus we see that folding a plane isn’t a one-to-one function, so we can’t undo our fold. In other words, a totally precise folding of completely flat paper, which seems like ideal origami, is in fact completely impractical. The paper, having no thickness, appears to absorb itself wherever it comes in contact with itself, leaving us nothing we can peel apart and unfold! Clearly there should be another model of folding.

Oftentimes in origami, the most important result of a fold is not the new position of the paper, but rather, the fact that a crease forms, marking the line of the fold. In perfect paper lacking thickness, even if we could unfold a fold, a crease wouldn’t form, because a crease results from the slight stretching and buckling of real-world paper when it bends to meet itself and then is pressed. To see that paper without thickness wouldn’t have to stretch and buckle, note that when real-world paper is folded, parts of the paper relatively far from the crease are put under no stress. It’s only when we observe the paper within half a millimeter or so of the folded edge that we can see the slight wrinkling, stretching, and buckling. On the other hand, when we observe cardboard we note that the material a centimeter away from the folded edge exhibits signs of buckling and stretching. It appears that the buckling and stress associated with forming creases is proportional to the thickness of the material being folded, from which we might infer that paper without thickness would exhibit no buckling and stretching whatsoever, and thus, not form a crease.

Once again, folding planes does not appear to model folding real-world paper very well at all. The hunt for some ideal folding model founders. If no ideal paper and no ideal precise fold exist that model real paper in any practical way, we can see that a practical model of paper folding will require an axiomatic approach that assumes a real paper can be folded accurately and still be unfolded. In fact, this assumption was inherent in the first proof I gave.

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[1] We know that since [pic] is a real number, there exists an integer N such that [pic], and that the number [pic] for any natural number n. Thus for [pic], [pic].

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