Tiling with Three Polygons is Undecidable

TL;DR

The paper proves that tiling the plane with three simple polygons is $co ext{-}RE$-complete (undecidable) by reducing from Wang tiling via a three-tile construction (Wheel, Shuriken, Staple) that encodes any $n$-tile Wang set. It simultaneously establishes a $co ext{-}RE$ membership result for tiling and tiling completion within a broad model of computable polygons, using an Extension Theorem and the notion of neat carpets to connect finite patches to plane tilings. The work tightens the constant-prototile bound from five to three and shows undecidability extends to periodic-target variants and fixed-tile completion problems, while also clarifying the decidability landscape in specialized algebraic representations. Overall, it significantly advances the understanding of when tiling problems are algorithmically intractable and under which representations one can regain decidability, with far-reaching implications for logic, computability, and tiling theory.

Abstract

We prove that the following problem is co-RE-complete and thus undecidable: given three simple polygons, is there a tiling of the plane where every tile is an isometry of one of the three polygons (either allowing or forbidding reflections)? This result improves on the best previous construction which requires five polygons.

Figures (5)

• Figure 1: The three tiles in our construction, to scale; Figure \ref{['fig:three tiles zoomed']} shows zoomed details of the construction. The wheel is just an example; it depends on the $n$ Wang tiles being simulated. The shuriken depends (only) on $n$.
• Figure 2: Zoomed views of portions of the three tiles in our construction ($15\times$ scale compared to Figure \ref{['fig:three tiles']}).
• Figure 3: Matching a glue (top) and its negative (bottom) between two wheels.
• Figure 4: Example tiling with the wheel, shuriken, and staple.
• Figure 5: A neat patch within $<r$, and how it can interact with the smaller disk $D_{r/2}$. From left to right: no chords, one chord, and two chords.

Theorems & Definitions (51)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem 1.5
• Corollary 1.6: Stronger form of Theorem \ref{['thm:intro:main']}
• Theorem 2.1: Berger1966
• Lemma 2.2
• proof
• Lemma 3.1