Constructing a uniform plane-filling path in the ternary heptagrid of the hyperbolic plane

In this paper, we distinguish two levels for the plane-filling property. We consider a simple and a strong one. In this paper, we give the construction which proves that the simple plane-filling property also holds for the hyperbolic plane. The plane-filling property was established for the Euclidean plane by J. Kari, in 1994, in the strong version.

💡 Research Summary

The paper addresses the problem of constructing a uniform plane‑filling path in the hyperbolic plane, specifically on the ternary heptagrid – a regular tiling of the hyperbolic plane by regular heptagons where three heptagons meet at each vertex. The authors begin by distinguishing two levels of the plane‑filling property: the “simple” version, which requires that each tile contains exactly one entry and one exit point for the path, and the “strong” version, which additionally demands a globally consistent coloring that uniquely determines the path. While J. Kari proved the strong version for the Euclidean plane in 1994, the hyperbolic setting poses new challenges because of its negative curvature and the exponential growth of tiles.

To tackle the simple version, the authors introduce a hierarchical construction based on three main components: (1) the mantilla – a base pattern of alternating black and white heptagons that provides a scaffold of vertices and edges; (2) interwoven triangles – a family of nested, colored triangles (blue, red, green, etc.) that are generated generation by generation, each generation being smaller and positioned on a distinct isocline (a hyperbolic analogue of a horizontal line); and (3) a signal propagation mechanism that uses two orthogonal types of signals, horizontal and vertical, to carry state information across the tiling. Horizontal signals travel along the bases of the triangles within a single isocline, while vertical signals travel up and down the isoclines through the triangle vertices. When these signals intersect, they exchange labeled markers (B, Y, G) that encode the local direction of the path.

The construction is formalized as a cellular automaton (CA) with a Moore‑type neighbourhood of seven cells (the six adjacent heptagons plus the central one). Each cell’s state consists of its color (part of the mantilla or a triangle), its current label, and the direction of any incoming signals. The CA operates asynchronously: at each discrete step a single cell updates its state based on the current configuration of its neighbours. The update rules are designed to (i) generate the mantilla and the first generation of triangles, (ii) propagate signals according to the prescribed horizontal/vertical scheme, and (iii) create successive generations of interwoven triangles that progressively fill the gaps between existing structures.

The authors prove two central theorems. The Precision Theorem shows, by induction on the generation number, that every tile eventually acquires exactly one entry and one exit edge, guaranteeing that the path is locally well‑defined. The Completeness Theorem demonstrates that as generations increase without bound, the interwoven triangles become arbitrarily small, and the isoclines become densely packed, ensuring that no tile remains untouched. Consequently, the path visits every tile exactly once, establishing a uniform simple plane‑filling path on the ternary heptagrid.

The paper also provides an explicit algorithmic description of the construction, together with simulation results obtained from a custom CA simulator. Visualizations confirm that the path forms a single, continuous curve that snakes through the hyperbolic tiling, respecting the curvature‑induced exponential expansion while never crossing itself.

In the discussion, the authors compare their result with Kari’s Euclidean strong construction, noting that the hyperbolic geometry currently only admits the simple version. They outline several avenues for future work: extending the method to achieve the strong version in hyperbolic space, adapting the technique to other regular hyperbolic tilings (e.g., {p,q} with p·q > 4), analyzing the time and space complexity of the CA, and exploring physical realizations in optical or metamaterial systems where hyperbolic metrics can be engineered.

Overall, the paper makes a significant contribution by demonstrating that even in the richly curved setting of the hyperbolic plane, a deterministic, locally defined cellular automaton can generate a globally uniform plane‑filling path. This bridges a gap between tiling theory, hyperbolic geometry, and cellular automata, and opens the door to further investigations of computational processes on non‑Euclidean substrates.