Conjectured optimal Tile(p) parameters near sqrt(2)−1 and 2−sqrt(2)

Establish whether Tile(p) arrays with tiling parameter p equal to sqrt(2) − 1 and p equal to 2 − sqrt(2) indeed correspond to the two enhanced-performance geometries observed near p ≈ 0.42 and p ≈ 0.58, and determine precisely their beamforming signal-to-noise ratio advantages relative to regular triangular arrays and other aperiodic tilings.

Background

The paper studies seismic array beamforming using the Hat family of aperiodic monotile tilings parameterized by p, with notable members including the Hat (p = 1/(1+√3)), the Specter (p = 1/2), and the Turtle. Performance is evaluated via the array response function and synthetic beamforming (single and distributed sources), showing that arrays based on Tile(p) can beat aliasing limits and outperform regular arrays.

In single-tile and large-N scenarios, the Specter (p = 1/2) consistently displays superior performance. The authors also identify two additional p ranges (approximately 0.42 and 0.58) with strong beamforming performance peaks. They conjecture these correspond to exact values p = √2 − 1 and p = 2 − √2, suggesting specific algebraic parameters may underlie optimal array designs within the Hat family.