Geometry and design of popup structures

Abstract

Origami and Kirigami, the famous Japanese art forms of paper folding and cutting, have inspired the design of novel materials & structures utilizing their geometry. In this article, we explore the geometry of the lesser known popup art, which uses the facilities of both origami and kirigami via appropriately positioned folds and cuts. The simplest popup-unit resembles a four-bar mechanism, whose cut-fold pattern can be arranged on a sheet of paper to produce different shapes upon deployment. Each unit has three parameters associated with the length and height of the cut, as well as the width of the fold. We define the mean and Gaussian curvature of the popup structure via the discrete surface connecting the fold vertices and develop a geometric description of the structure. Using these definitions, we arrive at a design pipeline that identifies the cut-fold pattern required to create popup structure of prescribed shape which we test in experiments. By introducing splay to the rectangular unit-cell, a single cut-fold pattern is shown to take multiple shapes along the trajectory of deployment, making possible transitions from negative to positive curvature surfaces in a single structure. We demonstrate application directions for these structures in drag-reduction, packaging, and architectural facades.

Figures (8)

• Figure 1: Geometry of popup structures. (A) A single popup unit is made from a flat sheet by two straight cuts and three parallel folds (cut-fold pattern shown at top left), parameterized by lengths $l_x, l_z$, and width $l_y$. Increasing actuation angle $\Psi$ (from 0 to $\pi/2$) drives deployment through intermediate states to the final 3D configuration. The fold vertex $\mathbf{r}(\Psi)$ is located at the center of the width. The image on the right in red is a popup unit from experiments. (B) Assembly of five popup units used to define curvature of a popup structure with a central unit parameterized by ($r$, $\phi$); $r, \phi$ are the distance and the angle made by the vector joining the fold vertex of central unit-cell with the origin, while $\lambda$ is scaling factor that changes the lengths of of opposite pair of unit-cells. For $\lambda > 1$, the diagonal units are larger than the rest and vice-versa for $\lambda < 1$. (C) Gaussian curvature $K(r,\lambda)$ of the five-unit-cell assembly is computed by triangulating the fold vertices of the units and using the Gauss-Bonnet theorem (described in detail in SI Sec. \ref{['Gaussian-and-mean']}). The $K=0$ solid line divides positive ($K>0$, convex) and negative ($K<0$, saddle/hyperbolic) regions, enabling different curvature regimes by tuning $(r, \lambda)$. (D) Mean curvature $H(r,\lambda)$ of the popup assembly computed using the discrete Laplace-Beltrami operator (described in main text). Locus of $\lambda, r$ corresponding to minimal surface, $H=0$ divides locally convex and concave regions of the deployed structure. $K(r,\lambda)$ and $H(r,\lambda)$ describe the geometry of popup structures with fixed widths.
• Figure 2: Pipeline for popup structure design. (A) A popup structure with uniform width is designed to transform to a target 3D surface by first slicing the target surface $\mathbf{r}_t(\Psi=\pi/2)$ along the transverse direction at equidistant locations to get 2D curves. Fold vertices $\mathbf{r}^{i, j}$ on these curves, parameterized by ($l_x^{i,j}$, $l_z^{i,j}$) -- the length and height of cut of $i$-th popup unit of $j$-th slice, are obtained by minimizing $\mathcal{L}_j[\{l_x^{i,j},\,l_z^{i,j}\}]$ (see main text and SI Sec. \ref{['subsec:slice-opt']} for details). The cost function, $\mathcal{L}_j$ ensures neighboring popup units are similar in dimension and are all of approximately the same size. This minimization procedure is subject to isometric and topological constraints, resulting in a physically realizable popup structure. The solution $\{l_x^{i,j*},\,l_z^{i,j*}\}$ obtained from this procedure is converted to cut-fold pattern that is fed into cutting machine for prototyping. Figures on the right show experimental and simulation models of the target surface developed using this pipeline. (B) Physical prototypes (top row) and simulation results (bottom row) demonstrating popup structures with zero $(K=0)$, positive $(K>0)$, and negative $(K<0)$ Gaussian curvature (see SI Sec. \ref{['sec:branching']}, Table \ref{['tab:expdetails']} for details). (C) (Top): Normalized azimuthal error density $\mathcal{L}_\phi / \langle \mathcal{L}_\phi \rangle$ vs unit-cell index $i$, evaluated from the solution $\{l_x^{i,j*},\,l_z^{i,j*}\}$ for a quarter-circle, which is useful to capture all axi-symmetric target shapes. We see that minimum error occurs at the center fold, $i=(N+1)/2$ due to symmetry constraints that ensure match with a circular target and the error increases towards the edges due to the effect of curvature (see main text for details). (Bottom): Log-log plot of the loss function $\mathcal{L}_j$ vs number density $\Delta = L/N$, demonstrating convergence and homogeneous structure with increasing density of popup units.
• Figure 3: Non-uniform popup and multi-state popup with splayed units. (A) Popup structure containing all three gaussian curvature regimes ($K>0, K<0, K=0$) designed with popups whose widths vary along the longitudinal direction (see SI Sec. \ref{['subsec:width_variation']}, Table \ref{['tab:expdetails']} for details). The algorithm to compute the cut-fold pattern (described in the main text) accommodates complex surfaces with spatially varying curvature. (B) Schematic of a splayed unit-cell described by the vertex position, $\mathbf{r}(\Psi,\alpha)$ which is a function of the actuation angle, $\Psi$ and the slope, $\alpha$. (C) Contour plot of the orientation of the fold in the splayed unit, $\theta(\Psi,\alpha)$ in the $(\Psi,\alpha)$-plane (see SI Sec. \ref{['SIsplayed']} for details). The level set curve corresponding $\theta=\pi/2$ (black curve) separates configurations in which increasing $\alpha$/varying $\Psi$ results in higher/lower vertex positions. Crossing this boundary enables transitions in gaussian curvature during deployment as the actuation angle, $\Psi$ varies. (D) Gaussian curvature, $K$ vs deployment angle, $\Psi$ of the splayed popup structure undergoing transition from $K>0$ state at $\Psi = 0.16\pi$ to $K=0$ at $\Psi = 0.28\pi$ to $K<0$ at $\Psi = 0.47\pi$. Inset shows the shape of the structure in experiments.
• Figure 4: Potential applications of popup structures. (A) Sequence of snapshots showing the deployment of a popup structure attached to airfoil for possible drag reduction applications. (B) Popup structure made of a single sheet of Polypropylene sheet conforming around a sphere for packaging. (C) Snapshots showing a popup structure with splayed units acting as facade that controls the amount of light around the architecture.
• Figure S1: Illustration of curvature computation using fold vertices $\mathbf{r}^{i,j}$ in the five-unit-cell assembly. Left: Gaussian curvature $K$ is calculated using discrete Gauss--Bonnet: the angle defect $2\pi - \sum_{k=1}^4 \alpha_k$ in the star-shaped neighborhood (green central vertex, red boundary vertices) yields $K = 2\pi - \sum \alpha_k$. Right: Mean curvature $H$ is calculated using the cotangent Laplace operator: the mean-curvature normal $\mathbf{H}_0$ at the vertex is computed from edge vectors weighted by cotangents of opposite angles in adjacent triangles, then projected onto the surface normal $\mathbf{n}_i$ to obtain the signed scalar $H_0 = (\mathbf{H}_0 \cdot \mathbf{n}_i)/2$.