Flat origami refers to the folding of flat, zero-curvature paper such that the finished object lies in a plane. Mathematically, flat origami consists of a continuous, piecewise isometric map $f:P\subseteq\mathbb{R}^2\to\mathbb{R}^2$ along with a layer ordering $\lambda_f:P\times P\to \{-1,1\}$ that tracks which points of $P$ are above/below others when folded. The set of crease lines that a flat origami makes (i.e., the set on which the mapping $f$ is non-differentiable) is called its \textit{crease pattern}. Flat origami mappings and their layer orderings can possess surprisingly intricate structure. For instance, determining whether or not a given straight-line planar graph drawn on $P$ is the crease pattern for some flat origami has been shown to be an NP-complete problem, and this result from 1996 led to numerous explorations in computational aspects of flat origami. In this paper we prove that flat origami, when viewed as a computational device, is Turing complete. We do this by showing that flat origami crease patterns with \textit{optional creases} (creases that might be folded or remain unfolded depending on constraints imposed by other creases or inputs) can be constructed to simulate Rule 110, a one-dimensional cellular automaton that was proven to be Turing complete by Matthew Cook in 2004.

翻译：平面折纸指将平坦、零曲率的纸张折叠后使成品位于同一平面的过程。数学上，平面折纸由连续分段等距映射$f:P\subseteq\mathbb{R}^2\to\mathbb{R}^2$与层序关系$\lambda_f:P\times P\to \{-1,1\}$共同构成，其中层序关系记录折叠时$P$中各点的上下相对位置。平面折纸产生的折痕线集合（即映射$f$不可微的点集）称为其\textit{折痕图}。平面折纸映射及其层序关系可呈现惊人的复杂结构。例如，1996年的一项研究证明，判断$P$上给定的直线平面图是否为某平面折纸的折痕图属于NP完全问题，这一结论推动了平面折纸计算特性的广泛探索。本文证明，将平面折纸视为计算设备时，其计算能力达到图灵完备。我们通过构造包含\textit{可选折痕}（根据其他折痕或输入约束决定是否折叠的折痕）的平面折纸折痕图来模拟Rule 110（马修·库克于2004年证明的图灵完备一维元胞自动机），从而完成该证明。