The Domino Problem of the Hyperbolic Plane Is Undecidable

In this paper, we prove that the general tiling problem of the hyperbolic plane is undecidable by proving a slightly stronger version using only a regular polygon as the basic shape of the tiles. The problem was raised by a paper of Raphael Robinson in 1971, in his famous simplified proof that the general tiling problem is undecidable for the Euclidean plane, initially proved by Robert Berger in 1966.

💡 Research Summary

The paper establishes the undecidability of the general tiling problem (GTP) on the hyperbolic plane by constructing a reduction from the halting problem of Turing machines. The authors begin by recalling the classic proofs of undecidability for the Euclidean plane due to Berger (1966) and Robinson (1971), which rely on “origin‑constrained” tilings that simulate the space‑time diagram of a Turing machine. They then move to the hyperbolic setting, choosing the regular tessellation {7,3} (the ternary heptagrid) as the underlying lattice. This tessellation is generated by reflecting a regular heptagon whose interior angle is 2π/3, producing a uniform, infinite hyperbolic tiling.

The core of the construction is a set of 4 “α‑tiles” (centers) and 17 “β‑tiles” (petals) that enforce a local rule: every α‑tile must be surrounded exclusively by β‑tiles. When these rules are applied across the heptagrid, they give rise to a global pattern called the “mantilla”. The mantilla partitions the plane into regions called “flowers”, each consisting of a central α‑tile surrounded by β‑tiles. Within each flower, the mid‑point lines of adjacent heptagons trace out a Fibonacci tree: black nodes have two children (black and white), white nodes have three children (one black, two white). The root of each tree is white.

The boundaries of these trees are infinite curves termed “iso‑clines”. An iso‑cline separates the hyperbolic plane into two unbounded components and is constructed so that black tiles always appear on it, guaranteeing a consistent coloring scheme that will later encode binary data.

To embed computation, the authors introduce a “bracket” system. Generation 0 consists of a periodic sequence of labeled points R, M, B, M along a line. Intervals between an R and the next B are called “active”, while intervals between a B and the next R are “silent”. For each subsequent generation, the points still labeled M are examined; a randomly chosen M that lies at the midpoint of an active interval is relabeled either R or B, thereby flipping the activity status of the surrounding intervals. This iterative process yields an infinite model where each generation alternates the colors of active and silent intervals.

The infinite model is lifted into the Euclidean plane as a family of interwoven isosceles triangles. Active intervals become colored triangles, while silent intervals become “phantoms”, transparent triangles that can be stacked in two layers with alternating colors. Triangles of the same color never overlap; phantoms may be nested but remain disjoint from colored triangles except at shared edges. The vertices of triangles and phantoms lie on a distinguished horizontal line called the “axis”.

Finally, the Euclidean construction is mapped back onto the hyperbolic mantilla. The mid‑point lines of the Fibonacci trees coincide with the legs of the triangles, and the iso‑clines correspond to the bases of the triangles. Consequently, each node of a Fibonacci tree represents a cell of a Turing machine: the color (black/white) encodes the tape symbol (0/1), the branching structure encodes the transition function, and movement along an iso‑cline corresponds to the passage of time. The presence of a valid tiling of the hyperbolic plane with the prescribed tile set is therefore equivalent to the existence of an infinite, non‑halting computation of the encoded Turing machine. Since the halting problem is undecidable, the hyperbolic domino problem is also undecidable.

The paper also mentions an alternative combinatorial proof by Jarkko Kari, noting that Kari’s approach uses non‑effective arguments, whereas the present construction is explicit and geometric. The authors conclude by suggesting directions for future work, such as simplifying the mantilla construction, extending the technique to other non‑Euclidean geometries, and exploring connections with higher‑dimensional tiling problems.