Undecidability of Tiling the Plane with a Fixed Set of 29 Wang Bars

Core Concepts

The tiling problem for a set of 29 Wang bars is undecidable, implying the undecidability of tiling the plane with Wang tiles having a fixed color deficiency of 25.

Abstract

The paper proves that the tiling problem for a set of 29 Wang bars is undecidable, improving upon the previous result of 44 Wang bars by Jeandel and Rolin.
The key steps in the proof are:

Constructing 6 groups of Wang bars - encoder, selector, aligner, linkers, and two groups of fillers. These groups work together to simulate the tiling of a set of Wang tiles.

Showing that to tile the plane with this set of 29 Wang bars, the tiling must follow a specific pattern. This pattern is equivalent to tiling the plane with a set of Wang tiles.

Proving that the set of 29 Wang bars can tile the plane if and only if the corresponding set of Wang tiles can be tiled.

As a consequence, the paper also shows that the tiling problem for Wang tiles with a color deficiency of 25 is undecidable.

Stats

The paper constructs a set of 29 Wang bars to prove the undecidability result.

Quotes

"If the tiling problem for k Wang bars is undecidable, then the tiling problem for Wang tiles with color deficiency k-1 is undecidable."
"The tiling problem for 29 Wang bars is undecidable."

Key Insights Distilled From

Undecidability of tiling the plane with a fixed number of Wang bars

by Chao Yang,Zh... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04504.pdf

Deeper Inquiries

How can the techniques developed in this paper be applied to investigate the undecidability of tiling problems with even smaller fixed parameters

The techniques developed in this paper can be further applied to investigate the undecidability of tiling problems with even smaller fixed parameters by adapting the construction of Wang bars and their relationships. One approach could involve reducing the number of Wang bars in the initial set while maintaining the complexity of the encoding and linking mechanisms. By carefully adjusting the construction of the Wang bars and their interactions, researchers can explore the undecidability of tiling problems with reduced fixed parameters. This iterative process of refinement and analysis can help uncover the threshold at which tiling problems become decidable or undecidable for smaller fixed parameters.

What are the implications of this result on the computational complexity of practical tiling problems in areas such as computer graphics, materials science, or architecture

The implications of the undecidability of tiling problems on the computational complexity of practical applications in computer graphics, materials science, and architecture are significant. In computer graphics, where tiling is essential for rendering textures and patterns, the undecidability of certain tiling problems can pose challenges in generating complex and unique designs efficiently. Similarly, in materials science, where tiling patterns influence material properties, the undecidability of tiling can impact the development of novel materials with specific characteristics. In architecture, where tiling is used for aesthetic and structural purposes, the undecidability of tiling problems can influence the design and construction of buildings with intricate patterns and shapes. Understanding the computational complexity of tiling problems can guide researchers and practitioners in developing algorithms and tools to address these challenges effectively.

Are there any connections between the undecidability of tiling problems and the emergence of complex patterns and structures in natural or artificial systems

The undecidability of tiling problems has intriguing connections to the emergence of complex patterns and structures in natural and artificial systems. In natural systems, such as biological organisms and geological formations, complex patterns often arise from the interactions of simple components following specific rules, akin to the rules governing tiling configurations. The undecidability of tiling problems highlights the inherent complexity and richness of patterns that can emerge from seemingly simple rules, mirroring the intricate patterns found in nature. In artificial systems, such as algorithmic art and generative design, the exploration of undecidable tiling problems can inspire the creation of novel patterns and structures that push the boundaries of creativity and innovation. By studying the undecidability of tiling problems, researchers can gain insights into the fundamental principles underlying pattern formation and complexity in diverse systems.