Making the Cut: Lattice Kirigami Rules

In this Letter we explore and develop a simple set of rules that apply to cutting, pasting, and folding honeycomb lattices. We consider origami-like structures that are extrinsically flat away from zero-dimensional sources of Gaussian curvature and one-dimensional sources of mean curvature, and our cutting and pasting rules maintain the intrinsic bond lengths on both the lattice and its dual lattice. We find that a small set of rules is allowed providing a framework for exploring and building kirigami—folding, cutting, and pasting the edges of paper.

Figure 1

The essence of kirigami. Top: We remove the hatched region and make mountain ($M$) and valley ($V$) folds along the indicated lines. Bottom: Final state with the edges of the cut identified by pasting.

Figure 2

The two lattices $\mathbf{\Lambda }$ (red dots) and $\tilde{\mathbf{\Lambda }}$ (green dots) offset by $\mathbf{d}$. Yellow edges are on $\mathbf{\Lambda }$. We show the basis vectors of the lattices $\{{\mathbf{e}}_i\}$ and the unit cells $\{{\mathit{\delta }}_i\}$. A 2–4 pair on the honeycomb is a standard 5–7 pair on $\mathbf{\Lambda }$. The cut is absorbed by the $\tilde{5}–\tilde{7}$ pair on $\tilde{\mathbf{\Lambda }}$ creating a partial climb. The plateaus of the 2–4 and $\tilde{5}–\tilde{7}$ pairs are different heights upon folding.

Figure 3

The glide cut. We remove the hatched region, make mountain ($M$) and valley ($V$) folds along the indicated lines, and identify the sides of the triangles as indicated. The inset shows the folded geometry. Note the cut along the line indicated by the knife.

Figure 4

The “sixon.” (a) Basic template. (b) This structure is degenerate and can be split three different ways into matching 2–4 pairs with pop up and pop down configurations. (c) An intermediate state connecting (b) to the final state (d). The hexagon in (e) can be made with with mountain and valley folds made along the red and blue lines, respectively, and then inserted into the threefold intersection in (d).