Flat origami is Turing Complete

2023-09-13

arXiv.org

AI Summary

This paper resolves the long-standing open question of whether planar origami is Turing-complete. The authors construct a class of planar crease patterns featuring optional creases—geometrically realizable folds whose inclusion or omission encodes computational choices—and rigorously simulate Rule 110, a known Turing-complete one-dimensional cellular automaton. Methodologically, they formalize the geometric constraints and layer-ordering relations inherent in flat-foldable crease patterns, encode the spacetime evolution of Rule 110 into a sequence of fold operations, and leverage optional creases to implement state selection and Boolean logic gates. This construction provides the first rigorous proof that planar origami constitutes a Turing-complete computational model; moreover, it establishes that the flat-foldability decision problem for such patterns is P-complete. The result not only characterizes the fundamental computational limits of physical origami but also furnishes a new formal foundation for programmable matter and geometric computation.

Abstract

"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 $f:Psubseteqmathbb{R}^2 omathbb{R}^2$ along with a layer ordering $lambda_f:P imes P o {-1,1}$ 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.

Problem

• Proving flat origami is Turing complete
• Simulating Rule 110 with flat origami
• Exploring computational complexity of flat origami

Innovation

• Flat origami as Turing complete
• Optional creases simulate Rule 110
• Mathematical mapping of layer ordering