Flat origami is Turing Complete (2309.07932v2)

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 $f:P\subseteq\mathbb{R}^2\to\mathbb{R}^2$ 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.

• The paper demonstrates flat origami's Turing completeness by simulating Rule 110 via crease patterns, revealing its capacity for universal computation.
• It employs geometric manipulations and logical gate constructions to map computational processes onto crease configurations.
• The findings open pathways for unconventional computation with potential applications in materials science and programmable matter.

Analyzing the Computational Capacities of Flat Origami

The paper "Flat Origami is Turing Complete" by Thomas C. Hull and Inna Zakharevich explores the notion of computational origami, specifically investigating the computational power of flat-fold origami by proving its equivalence to a Turing complete system. Hull and Zakharevich achieve this by demonstrating the ability of a flat origami crease pattern to simulate Rule 110, a well-known Turing complete cellular automaton.

Core Concepts and Findings

Flat origami comprises folding a two-dimensional sheet into a flat plane without any tearing or stretching. The intrinsic complexity of flat origami is linked to determining whether a given crease pattern is globally flat-foldable, a task acknowledged as NP-hard. In this paper, the authors extend this line of inquiry by posing flat origami as a computational device and derive its equivalence to universal computation.

The fundamental breakthrough achieved by Hull and Zakharevich lies in their construction of optional crease patterns capable of simulating Rule 110. Rule 110 is an elementary one-dimensional cellular automaton proven to be capable of universal computation as it can emulate a Turing machine. The authors exploit this property by mapping logical operations such as AND, OR, and NOT gates using structures and conventions inherent to flat origami.

Methodology and Implementation

To simulate Rule 110, the authors introduce a technique using wires—comprising pairs of parallel mountain and valley folds—to transmit Boolean values. These geometric manipulations within the crease patterns dictate the folding paths available and control the output. By integrating logical gates and functional gadgets like intersectors, twists, and eaters, the crease pattern can propagate logical information and generate an output reflecting the computational rules of Rule 110.

Central to their demonstrated simulation are two specific twists: the triangle and hexagon twists that inherently rotate their central polygons to enact computational logic rotation based on the orientation and selection of optional creases. Ultimately, they construct a tessellation system spanning an infinite plane such that, with a finite set of localized operations or perturbations, the system effectively processes inputs and generates outputs tantamount to those specified by Rule 110.

Implications and Theoretical Speculations

The implications of this paper stretch beyond the purview of theoretical origami, suggesting frameworks for unconventional computation. Being Turing complete, flat origami manifests theoretical potential for solving computational problems given sufficient input complexity. This ties into broader themes in materials science and architectural design, addressing dynamic structural transformations and programmable matter.

Moreover, the realization of flat origami's ability to perform universal computation propels innovative intersections with emerging technologies such as mechanized origami-based devices. The paper suggests that future explorations of computational origami may involve combinatorial designs that actively leverage its folding properties based on initial conditions, analogous to energy functions in physical computations.

Future Directions in AI and Computational Theory

Several avenues appear promising for further research. Exploring computational efficiency, optimality, and the lowest bound complexity for performing specific functions using flat origami are natural extensions. Moreover, integrating principles from this paper could redefine constructs within machine learning, facilitating novel algorithmic applications or AI systems encoded on flexible substrates like soft robotics.

In conclusion, flat origami's established Turing completeness not only furthers our understanding of the computational capacities inherent in simple geometrical constructs, but also paves the way for a new field of computation that could eventually merge with practical implementations in artificial intelligence, robotics, and beyond.