Correlations of Symmetry and Frequency in Alphabets and in Natural Systems

We investigate the relationship between the graphical symmetry of symbols and their frequency of occurrence in culturally evolved symbolic systems and compare the observed tendencies with those found in physical structures. Capital letters of the English and Russian alphabets were classified according to their invariance under vertical reflection, horizontal reflection, and rotation through 180°. The graphical symmetry of numerical figures was analyzed across four numeral systems. Symmetry was examined for 1000 Japanese Kanji arranged into frequency-ranked groups. Spearman's and Kendall's rank-correlation tests revealed positive symmetry-frequency tendencies in both alphabets, although the associations did not reach conventional statistical significance. A strong and statistically significant positive association was obtained for the average symmetry of the numerical figures and their leading-digit frequencies predicted by Benford's law. The Japanese Kanji produced the strongest statistical result. Despite their different cultural origins and graphical organization, the investigated alphabets and numeral systems exhibited surprisingly similar symmetry-class entropies, suggesting a possible common balance between graphical regularity and symbol distinguishability. For comparison, seven-bond and eight-bond colloidal clusters demonstrated the opposite tendency, with asymmetric configurations occurring more frequently and dominating their symmetry-class distributions.