[Paper Review] Constructing a uniform plane-filling path in the ternary heptagrid of the hyperbolic plane

Maurice Margenstern|ArXiv.org|Oct 1, 2007

Cellular Automata and Applications6 references3 citations

TL;DR

This paper constructs a uniform plane-filling path in the ternary heptagrid (hyperbolic {7,3} tiling) using a recursive, hierarchical tiling method based on decreasing trapezoidal regions and path-guiding mauve triangles. The construction proves the simple plane-filling property holds in the hyperbolic plane, with all solutions generating a single infinite path except one exceptional case involving two disjoint infinite paths that can be joined at infinity.

ABSTRACT

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.

Motivation & Objective

• To establish the existence of a uniform plane-filling path in the hyperbolic plane, specifically in the ternary heptagrid {7,3}.
• To investigate whether the simple or strong plane-filling property holds in hyperbolic tilings, given that it is known in the Euclidean plane.
• To provide a constructive method for generating a single infinite path that visits every tile exactly once in the hyperbolic tiling.
• To demonstrate the existence of a cellular automaton on the {7,3} tiling that constructs such a path in infinite time.
• To lay foundational groundwork for algorithmic constructions of Peano curves and undecidability results in hyperbolic cellular automata.

Proposed method

• Uses a recursive, inward-deepening construction based on dividing each heptagonal tile into smaller trapezoidal regions of increasing order (k), starting from the boundary and moving toward the center.
• Defines path elements via midpoints of radiuses and edges, forming segments that connect across trapezoidal regions to form a continuous path.
• Introduces 'mauve triangles' as guiding tiles to control path direction and ensure connectivity across adjacent trapezoids at each recursive level.
• Applies a two-dimensional dichotomic process to subdivide trapezoids into four smaller trapezoids at each step, preserving path continuity through defined entry/exit points.
• Relies on the local Euclidean-like behavior of infinitesimal neighborhoods in the hyperbolic plane to ensure convergence of the path construction to a limit curve.
• Constructs a cellular automaton that simulates the path generation process step-by-step across the tiling, operating in infinite time to produce the full path.

Experimental results

Research questions

• RQ1 Does the simple plane-filling property hold in the hyperbolic plane, specifically in the ternary heptagrid {7,3}?
• RQ2 Can a uniform plane-filling path be constructed in the hyperbolic plane such that all tilings with a given tile set generate a single infinite path?
• RQ3 What is the nature of the exceptional solution in which the path splits into two disjoint infinite paths?
• RQ4 Can a cellular automaton on the {7,3} tiling simulate the construction of a uniform plane-filling path?
• RQ5 How does the recursive, inward-traveling path construction in the hyperbolic plane converge to a continuous, space-filling curve?

Key findings

• A uniform plane-filling path exists in the ternary heptagrid of the hyperbolic plane, proving the simple plane-filling property holds in this setting.
• All solutions to the tiling problem generate a single infinite path, except for one exceptional case involving two disjoint infinite paths that can be topologically joined at infinity.
• The exceptional solution is interpreted as a limit case, where the two paths π₁ and π₂ can be connected at infinity, forming a single continuous trace.
• The path construction converges to a limit curve P∞ that visits every point of the hyperbolic plane exactly once, analogous to a Peano curve.
• A cellular automaton on the {7,3} tiling can construct the uniform plane-filling path in infinite time, demonstrating effective algorithmic realizability.
• The construction relies on recursive subdivision of trapezoidal regions and path-guiding tiles (mauve triangles), with path continuity ensured via defined entry/exit points at each level.