The 64×8 matrix as the core of a complete combinatorial taxonomy of triangles

Problem: Classical triangle classifications are fragmentary and fail to reveal the structure of the entire manifold of forms.Method: A discrete-combinatorial approach based on a path code (P‑code) — a three‑letter word over an anisotropic alphabet of directions. On this basis, a geometric figure — a planar non‑degenerate triangle — is redefined and considered not as a set of three vertices and sides with lengths/angles, but as a class of combinatorial side‑traversal trajectories in the form of a P‑code.Main Result: It is established that the universe of all planar non degenerate triangles is discretely partitioned into exactly 132 combinatorial types based on their P codes. This result is presented in the form of a complete 64×8 taxonomic matrix.Significance: A computable invariant is created, translating a continuous geometric manifold into a discrete symbolic space and establishing a link between geometry, combinatorics, and formal language theory. The method yields an exhaustive “periodic table” of triangles and opens prospects for applications in computer vision and computational geometry.