On the bifurcation behavior of a folded notebook page

Folding, as a general method of transforming geometries, is found everywhere. Clothes, bi-fold doors, umbrellas, airbags, solar panels, and wings of insects and birds^1,2 are just some common examples. Folding allows a thin object to be stored compactly while remaining easily deployable, or transformed into structures with new capabilities (e.g., origami).

In mathematics, origami, a slightly constrained form of folding, has been formalized into an axiomatic system.³ Analysis and computation aid in the creation of novel origami designs.⁴ Interestingly, folding has a mathematical capability that outperforms compass-and-straightedge construction and can be used to solve classical problems such as “angle trisection” or “doubling the cube.”^5,6

Folding is also common in physical sciences. A prominent example is the protein folding problem, where one would like to establish the 3D protein structure from an amino acid sequence alone. Recently, deep learning allowed for a major breakthrough in answering this question.⁷ Interestingly, folding (together with stretching) is also a useful concept in mixing of highly viscous fluids. The diverse folding mechanisms, including mechanical, capillary, thermal, and chemical folding, can benefit a wide range of micro-scale applications.^8,9 Back at the office, a violently folded (or crumpled) piece of paper has remarkable physical insight buried in its crease network.^10,11

Admittedly, the mathematical, physical, or biological study of folds is full of surprises. This work presents one simple problem, in which folding a notebook page exhibits an interesting bifurcation behavior. Given a pocket notebook of length 1 in the x direction and l in the y direction (Fig. 1), a folded page can extend beyond the side bounds of its unfolded state, remaining visible when the notebook is closed and thus serving as a bookmark. Then, what is the optimal way to fold the page? Although preferences vary, we assume in this paper that the optimal fold maximizes the bookmark length e (Fig. 1), which is the length by which the page extends beyond its unfolded width along the x axis. In addition, the top edge of the page should not be torn off—a constraint similar to the one experienced by birds and insects when folding their wings due to the joint between the wings and body. Aside from this first constraint, the folded page should not extend past the top edge of the notebook, or it will be creased further when closing the notebook. This is the second constraint.

We will begin by considering the first constraint alone and then solve the complete problem, using only undergraduate calculus. This paper concludes by proposing several possible directions for future explorations.

We start our analysis by considering only the first constraint. Without loss of generality, we assume that the page is bound at its top edge, and the fold is in the upper-right direction (as in Fig. 1). Then the objective is to have the folded page extend rightward as much as possible, while keeping the top edge fixed.

Using the coordinate system defined in Fig. 1, the fold maps the original rectangle $Ω = [ 0 , 1 ] × [ 0 , l ]$ to a new region $Ω ′$⁠. The optimal fold is the one that maximizes the excess length, which is denoted by e in Fig. 1. One may interpret $Ω ′$ as one state of the system, and e as the negative of the state's potential energy. Then the optimal fold that maximizes e is the one that minimizes the system's energy.

Folding the page amounts to choose a straight crease in Ω and reflect the region below it—which we will call the “folded region”—with respect to this line. For a rectangular page, there are only two possible cases: either the crease crosses the left and bottom edges and the folded region is a triangle (Fig. 2, case 1), or it crosses the left and right edges in which case the folded region is a trapezoid (Fig. 2, cases 2a and 2b). Regardless of the crease location, e will be determined by the location of a corner point of Ω (otherwise, there is always a neighboring point that, once reflected, is further to the right). For this reason, we only need to consider the corners of the folded region.

In both cases, we will first find the x coordinate of the rightmost point of the folded region, which is denoted by x_e. x_e may correspond to different page corners, depending on how we fold the page (refer to Fig. 2) and is related to e by $e = x e − 1$⁠.

Case 1:

When $l ≤ 1$⁠, meaning the page is “short and wide” (including the square page shown in Fig. 1), substituting a = b = l in x_e gives $x e ∗ ( l , l ) = l$ (⁠ $∗$ denotes optimal values). As a result, $e ∗ = l − 1 ≤ 0$⁠. This means that, for a short page, it is impossible to fold a bookmark using case 1 creases.

Case 2:

The crease line crosses the left edge at a distance a from the lower-left corner and the right edge at a distance b from the lower-right corner (Fig. 2, cases 2a and 2b). In this case, the folded region is trapezoid OCBA, with $a ≥ b$ for the fold to be in the upper-right direction. We define θ as the angle between the crease line and the x-axis. Clearly, $tan θ = a − b$⁠. The rightmost point in $Ω ′$ depends on the value of θ: It is $C ′$ when $θ ≤ 45 °$ (Fig. 2, case 2a) and $O ′$ when $θ > 45 °$ (Fig. 2, case 2b).

All cases considered, the optimal fold is Eq. (3) of case 2a and resembles the illustration Fig. 2, case 2a.

We continue our analysis by including the second constraint: The folded page should stay within one side of the bind, that is to say, “below” the bind—so that it does not get accidentally creased when one closes the notebook. We start from case 2a, which gave us the optimal fold considering only the first constraint, and improve it to satisfy the second constraint. One will see that the additional constraint leads to bifurcation behaviors of the optimal solution.

The accessible region of the (a, b) phase space according to Eq. (7) is shown in Fig. 4 (shaded area) for $l ≥ 1$⁠. Each point in the accessible region represents one way to fold the page without violating any of our constraints. At low a, the region is bounded by the condition $b ≤ a$⁠. At higher values, it is determined by $2 a ≤ l ( 1 + ( a − b ) 2 )$ and is, therefore, more complicated. Still, one can show that $∂ e / ∂ a ≥ 0$⁠, so that $e ∗$ occurs on the curve boundary. One might also notice that e = 0 along the other two (straight) boundaries of the accessible region.

The equation of the curved boundary is $b = a − 2 a l − l 2 / l$⁠, which is defined over the interval $a ∈ [ l / 2 , l ]$ (the other solution $b = a + 2 a l − l 2 / l$ violates $b ≤ a$ and is discarded). Points on this boundary correspond to a special kind of folding: the ones for which the page's lower-left corner overlaps with its top edge. This physical picture explains the lower limit $l / 2$ for a. The remaining constraint, $b ≥ a − 1$⁠, is automatically satisfied on this boundary.

Because the solutions are such that $a i n ( l )$ is within the corresponding range in a (⁠ $a ∈ [ l / 2 , l ]$⁠), we call them internal solutions. The crease lines intersect with the page's left edge but do not pass through either of its two end points.

The boundary and internal solutions become equal at the critical value l_cr = 1.20711, which can only be found numerically by solving $e ̂ i n ( l ) = e ̂ b d ( l )$ or $e ( a i n ( l ) , l ) = l − 1$⁠. The solution steps for mathematica are provided in the  Appendix as an example.

To further understand the disappearance of the internal solution, Fig. 5 plots its trajectory as l is increased (dotted curve). In addition, Fig. 5 plots the trajectory of the local minima (dashed curve). It is clear that, as l increases, the two solutions approach each other. They coincide when $l = l c r 2$ and, as a pair, spontaneously disappear (Fig. 6 will also show this).

In the top part of Fig. 6, three pages with increasing aspect ratios are each folded in many different ways, and the trajectories of the lower-right page corner (⁠ $C ′$ in Fig. 2) are recorded by thick black curves. The internal and boundary solutions are marked in the leftmost example. As l increases, the optimal solution clearly changes from the internal to the boundary solution.

Overall, the solution to $e ∗$ as a function of the page aspect ratio l has three distinct behaviors, which is surprising for such a simple problem. A graphical summary is found at the bottom of Fig. 6, which plots the internal solution $e ̂ i n ( l )$ and the boundary solution $e ̂ b d ( l )$⁠; the maximum of the two gives $e ∗$⁠. The two curves intersect at the critical point $l c r ≈ 1.20711$⁠, which signals an abrupt change of the optimal fold state. Figure 6 also plots the excess length given by the local minima (gray curve). This curve is the dashed curve in Fig. 5 but plotted as a function of l instead of a(l). It intersects with the internal solution at $l = l c r 2$⁠, forming a cusp where the two solutions annihilate. The jump of the optimal solution from the internal solution branch to the boundary solution branch as well as the spontaneous disappearance of extrema in pair are all well-known manifestations of bifurcations. Interestingly, the behavior of $e ∗$ can be loosely likened to the (negative of) free energy in first-order phase transitions.

In this paper, we investigated an everyday physics question, “how to optimally fold a rectangular notebook page into a bookmark?” so that the folded page remains most visible but stays within one side of the notebook bind. The solution to this problem is unexpectedly rich given that the problem is governed by a single control parameter, the page's aspect ratio. In particular, this system exhibits a bifurcating behavior found in dynamical systems.

While related to origami, the problem has a fixed edge and makes a single fold. To distinguish the two, we may name ours as the “1-fold bookmark problem.” A few extensions can be readily proposed for future studies.

1. Use, for example, circular or even concave pages. For a circular page, one may fix its topmost point, as there is no straight edge. Regardless of its shape, the page can be contained within a bounding rectangle. Or more intuitively, it can be glued on top of a larger rectangular “bounding page.” The maximum excess length of the bounding page is known from the current work and is always greater than that of the bounded page. Thus, the current work provides useful upper limits even for folding pages of arbitrary shapes.

2. Extend to higher dimensions. This is challenging to represent. While the page in our problem is two-dimensional, during folding, it temporarily relies on the third dimension. Similarly, folding a three-dimensional shape relies on the fourth dimension.

3. Allow more than one fold to tackle the n-fold bookmark problem. Because each fold yields a more complex shape, the geometric complexity increases rapidly. However, a quick look at the “2-fold” problem of a square page shows that a simple 2-fold solution easily beats our optimal 1-fold solution: first fold the square along its diagonal to have an isosceles right triangle and then fold the triangle to align its longer edge with its top edge. This gives $e = 2 − 1 ≈ 0.41421$⁠, while our optimal solution from Eq. (8) is only

   $e [ 1 3 ( 3 33 + 17 3 − 2 3 33 + 17 3 − 1 ) , 1 ] ≈ 0.13488 .$

Although the simple 2-fold solution outperforms the best 1-fold solution, it is unclear whether it is the optimal way to fold a square page twice into a bookmark, and whether its performance generalizes to pages of different aspect ratios.

The author would like to thank the reviewers for their comments and Dr. Rodolfo R. Rosales for the discussions.

The author has no conflicts to disclose.

As an example, we list the steps of finding l_cr using mathematica:

1. Find the roots by Solve[a^∧4̂==(2a − l)l, a], which gives four roots.

2. To pick the right root, one can substitute some test values of l. Suppose that root is named aSol.

3. Substitute aSol to have the internal solution eIn =−2+l/a+Sqrt[(2a−l)l]/.{a->aSol}.

4. Find l_cr numerically by providing a guess of its value FindRoot[eIn == l − 1, {l, 1.2}].