This work introduces a geometry-driven reformulation of Fourier transform computation based on hexagonal lattice structures and recursive spatial decomposition. In contrast to classical Fast Fourier Transform (FFT) algorithms, which achieve efficiency through algebraic factorization of index sets, the proposed framework derives computational structure from planar symmetry and congruent partitioning of the transform domain. The method operates on a hexagonal sampling lattice expressed in axial coordinates, enabling a natural representation of data aligned with isotropic spatial structures. The hexagonal domain is recursively decomposed into three congruent rhombic subdomains, each of which can be further subdivided into triangular primitives. This hierarchical partitioning induces a structured computation in which a global transform is replaced by coordinated operations over smaller, geometrically related regions. A key feature of the framework is symmetry-based reuse. Transform evaluations computed on a primary subdomain can be mapped to others via rotational and reflective operators, reducing redundant computation while preserving structural consistency. This shifts the computational paradigm from purely arithmetic factorization toward a hybrid model in which integer-indexed lattice representations and geometric symmetries jointly define efficiency. Beyond its computational implications, the proposed approach aligns with physical and engineering systems that naturally exhibit hexagonal organization, including sampling theory, mesh-based simulation, and cellular material structures. The work therefore contributes both a conceptual generalization of transform design and a practical pathway toward computation on non-Cartesian domains.
Gyavira Ayebare.B (Mon,) studied this question.