Mathematical Foundations of Classical Arabic Prosody: A Group-Theoretic Analysis of the Tawil Meter’s Cognitive and Cultural Dominance

Mathematical Foundations of Classical Arabic Prosody

Resumen

This study provides a thorough mathematical analysis of Arabic poetry meters, focusing on the Tawil meter, and clarifies significant connections between abstract algebra, information theory, and classical Arabic prosody. By establishing a prosodic transformation group $\mathcal{G} \cong D_8 \rtimes \mathbb{Z}_2$, we demonstrate that Al-Khalil's system of 16 taf'ilat constitutes a comprehensive group-theoretic framework wherein meter transformations are non-commutative operations that preserve poetic validity. The research establishes an entropy-based complexity metric \( C(B) = \alpha H(P_t) + \beta \log_2|\text{Aut}(B)| + \gamma R(B) \), illustrating that traditional metrics attain an optimal balance between predictability and information density (\( 0.4 \leq R \leq 0.6 \)). Our examination of the Tawil meter $(10110, 1011010)^2$ reveals its distinct mathematical characteristics. Information-theoretic analysis reveal that classical meters operate within a constrained complexity spectrum ($0 \leq H(B) \leq 3$ bits), with Tawil positioned at the core of the ``poetic sweet spot." Historical data indicates that Tawil was culturally preeminent, with an annual growth rate of 0.023, with 38\% of classical Arabic poetry originating from this region. Neuroaesthetic study correlates its mathematical structure with a 30\% increase in alpha-wave brain responses compared to non-phi metrics. The research develops a metric space $(V_T,d_T)$ for meter variations, defines a linear operator $\mathcal{T}$ with eigenvalues $\phi$ and $-1/\phi$, and provides constraints for composition stability $\|\mathcal{T}^n v_0\| \leq \phi^n \|v_0\|$. These mathematical structures provide new tools for the computational analysis of poetic tradition, clarifying the enduring cognitive and cultural appeal of ancient Arabic meters through their inherent mathematical harmony.