Efficiency in Writing Systems: Testing Zipf's Law of Abbreviation Across Letters, N-Grams, and Words

This study investigates Zipf's Law of Abbreviation (ZLA) across 155 writing systems, analyzing how visual complexity optimizes with information content at three linguistic levels: letters, n-grams, and words. Using perimetric and skeleton-length complexity metrics, we demonstrate that letters exhibit the strongest correlation (� = 0.2–0.4 in most languages), confirming their role as primary units of efficiency optimization. Larger units (n-grams/words) show weaker effects due to structural constraints. While alphabetic scripts (e.g., Latin-based) align robustly with ZLA, logographic (e.g., Chinese) and abugida (e.g., Kannada) systems reveal exceptions—some with near-zero or negative correlations—highlighting script-specific pressures like distinctiveness or historical preservation. Our findings refine ZLA by emphasizing visual (not just length-based) effort minimization and underscore letters as the fundamental locus of abbreviation effects. Limitations in script diversity and complexity metrics suggest future directions, including phylogenetic controls and perceptual complexity measures. This work advances the cross-linguistic study of writing system evolution under efficiency pressures.