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：平面折纸是指将平坦、零曲率的纸张进行折叠，使得最终物体位于平面上的过程。数学上，平面折纸由一个连续的、分段等距映射 $f:P\subseteq\mathbb{R}^2\to\mathbb{R}^2$ 与一个层次顺序 $\lambda_f:P\times P\to \{-1,1\}$ 组成，后者记录在折叠时 $P$ 中各点的上下叠放关系。平面折纸形成的折痕线集合（即映射 $f$ 不可微的点集）称为其\textit{折痕图案}。平面折纸映射及其层次顺序可能具有惊人复杂的结构。例如，判断绘制在 $P$ 上的给定直线平面图是否为某个平面折纸的折痕图案已被证明是NP完全问题，这一1996年的结论引发了众多关于平面折纸计算方面的探索。本文中我们证明，将平面折纸视为计算设备时，它是图灵完备的。我们通过构造具有\textit{可选折痕}（即根据其他折痕或输入施加的约束而可折叠或保持展开的折痕）的平面折纸折痕图案来模拟Rule 110——一个由Matthew Cook于2004年证明为图灵完备的一维元胞自动机。