Penny Graphs in Hyperbolic Plane: Bounds on Touching Circle Pairs
Published:Dec 31, 2025 12:53
•1 min read
•ArXiv
Analysis
This paper investigates the maximum number of touching pairs in a packing of congruent circles in the hyperbolic plane. It provides upper and lower bounds for this number, extending previous work on Euclidean and specific hyperbolic tilings. The results are relevant to understanding the geometric properties of circle packings in non-Euclidean spaces and have implications for optimization problems in these spaces.
Key Takeaways
- •Provides upper and lower bounds for the number of touching pairs in a packing of congruent circles in the hyperbolic plane.
- •Extends previous results from Euclidean and specific hyperbolic tilings.
- •Identifies a potential extremal construction based on a spiral pattern.
- •Offers a lower bound showing a linear growth rate with a constant greater than 2.
Reference
“The paper proves that for certain values of the circle diameter, the number of touching pairs is less than that from a specific spiral construction, which is conjectured to be extremal.”