Group actions on the set of path codes in a combinatorial taxonomy of polygonal contours

Problem:The combinatorial enumeration of polygonal contours via path codes (P‑codes) leads to severe descriptive redundancy. The same geometric shape in different spatial orientations generates distinct, seemingly unrelated P‑codes, making the taxonomy unstructured and obscuring the invariant essence of the figures.Method:We perform a methodological shift: instead of applying symmetry groups to geometric objects, we transfer their action onto the space of their formal descriptions – the P‑codes. The cyclic group C₄ of 90° rotations acts on an 8‑directional anisotropic alphabet, transforming each symbol in a P‑code. P‑codes describing the same contour in different orientations are united into orbits of this group action. To bridge algebra and geometry, we introduce a dedicated visualization scheme: each P‑code is represented as a pattern of filled cells within a 3×3 block, bijectively mapped to the directional alphabet.Main Result:1. From the universe of 512 length‑3 P‑codes, we isolate 132 codes corresponding to non‑degenerate triangles.2. The action of C₄ partitions these 132 codes into 33 distinct orbits, each representing a unique geometric triangle type up to rotation.3. All stabilizers of P‑codes are trivial (Stab(p) = {e}), confirming the absence of rotational symmetry for generic triangles described by this discrete alphabet. This aligns with geometric intuition and provides a clean verification of the orbit‑stabilizer theorem (|Orb|·|Stab| = |C₄| = 4).4. We construct a two‑tiered taxonomic system: an absolute level of combinatorial encoding (P‑codes) and a relative level of geometric invariance (group orbits).Significance:1. Theoretical: Establishes a novel combinatorial‑algebraic framework for classifying asymmetric forms. It demonstrates that group theory can be a powerful tool for analyzing objects lacking symmetry by acting on their descriptions.2. Methodological: Introduces a canonical technique to eliminate redundancy in combinatorial taxonomies via group actions, elevating classification from a list of “instructions” to a structured system of “geometric archetypes.”3. Didactic & Visual: The 3×3‑block visualization serves as a “geometric realization of group theory,” making abstract concepts (orbits, stabilizers, group actions) visually accessible and operationally meaningful. It acts as both a research tool and a powerful teaching instrument.4. Foundational: This work lays the groundwork for the systematic study of more complex polygonal structures (n‑gons) and the analysis of larger symmetry groups (e.g., the dihedral group D₄) within a unified framework.