"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 "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, or more specifically P-complete. We do this by showing that flat origami crease patterns with "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"指的是使用平坦、零曲率的纸张进行折叠，使得最终成品位于一个平面内。从数学上讲，平面折纸由一个连续的分片等距映射 $f:P\subseteq\mathbb{R}^2\to\mathbb{R}^2$ 以及一个层序函数 $\lambda_f:P\times P\to \{-1,1\}$ 构成，该函数追踪折叠时 $P$ 中哪些点位于其他点的上方或下方。平面折纸所产生的折痕线集合（即映射 $f$ 不可微的集合）被称为其"折痕图"。平面折纸映射及其层序函数可以具有惊人的复杂结构。例如，判断在 $P$ 上绘制的给定直线平面图是否是某个平面折纸的折痕图，已被证明是一个NP完全问题，这一1996年的成果引发了关于平面折纸计算特性的诸多探索。本文证明，当将平面折纸视为一种计算设备时，它是图灵完备的，更具体地说是P完全的。我们通过展示可以构造具有"可选折痕"（即根据其他折痕或输入施加的约束可能被折叠或保持不折）的平面折纸折痕图来模拟规则110，这一一维元胞自动机由Matthew Cook于2004年证明是图灵完备的。