Classification of triangles via path codes and group actions: a complete table of 24 archetypes
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2026
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| _version_ | 1866901968262266880 |
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| author | Khaikov, Vadim |
| author_facet | Khaikov, Vadim |
| contents | <p>This corrected version (v2) fixes erroneous path codes (P-codes) in Table 3 (24×6 matrix) identified during independent re-verification. The core conceptual framework and the classification of 24 archetypes remain unchanged. This version supersedes v1 (DOI: 10.5281/zenodo.18367714).<br>> ===== <<br><em><strong>Problem</strong></em>: In discrete geometry and computer modeling, there is a lack of a systematic classification of plane triangles by their contours.<br><em><strong>Method</strong></em>: An approach is developed based on representing shape as a cyclic path (P-code) using an anisotropic alphabet of 8 directions. To classify all possible contours, a factorization method by the action of the full symmetry group G = D₄ ⋉ (S₃ × C₂) is applied. This group combines global lattice transformations (D₄) and local vertex relabeling and traversal reversal (S₃ × C₂).<br><em><strong>Main result</strong></em>: A complete 24×6 table is obtained, representing 24 archetypal equivalence classes of plane non-degenerate triangles. Each archetype is described by six synonymous P-codes (PEX-forms), corresponding to descriptive synonymy. A fundamentally new phenomenon of form homonymy is discovered and explained — the coincidence of the full sets of names (PEX-forms) for two distinct archetypes, forming so-called PIN-forms. This phenomenon marks a fundamental limit of the discriminability of geometric shapes within this discrete-linguistic representation framework.<br>Significance: The proposed model reduces the infinite variety of triangles to a finite system of 24 archetypes, creating a compact and rigorous language for describing discrete forms. The results open prospects for creating similar classifications for other polygons and for applications in computer graphics and digital image processing.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18376297 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Classification of triangles via path codes and group actions: a complete table of 24 archetypes Khaikov, Vadim discrete geometry combinatorial geometry triangle classification path code (P-code) equivalence class group action anisotropic direction alphabet "Not Just Triangles" project <p>This corrected version (v2) fixes erroneous path codes (P-codes) in Table 3 (24×6 matrix) identified during independent re-verification. The core conceptual framework and the classification of 24 archetypes remain unchanged. This version supersedes v1 (DOI: 10.5281/zenodo.18367714).<br>> ===== <<br><em><strong>Problem</strong></em>: In discrete geometry and computer modeling, there is a lack of a systematic classification of plane triangles by their contours.<br><em><strong>Method</strong></em>: An approach is developed based on representing shape as a cyclic path (P-code) using an anisotropic alphabet of 8 directions. To classify all possible contours, a factorization method by the action of the full symmetry group G = D₄ ⋉ (S₃ × C₂) is applied. This group combines global lattice transformations (D₄) and local vertex relabeling and traversal reversal (S₃ × C₂).<br><em><strong>Main result</strong></em>: A complete 24×6 table is obtained, representing 24 archetypal equivalence classes of plane non-degenerate triangles. Each archetype is described by six synonymous P-codes (PEX-forms), corresponding to descriptive synonymy. A fundamentally new phenomenon of form homonymy is discovered and explained — the coincidence of the full sets of names (PEX-forms) for two distinct archetypes, forming so-called PIN-forms. This phenomenon marks a fundamental limit of the discriminability of geometric shapes within this discrete-linguistic representation framework.<br>Significance: The proposed model reduces the infinite variety of triangles to a finite system of 24 archetypes, creating a compact and rigorous language for describing discrete forms. The results open prospects for creating similar classifications for other polygons and for applications in computer graphics and digital image processing.</p> |
| title | Classification of triangles via path codes and group actions: a complete table of 24 archetypes |
| topic | discrete geometry combinatorial geometry triangle classification path code (P-code) equivalence class group action anisotropic direction alphabet "Not Just Triangles" project |
| url | https://doi.org/10.5281/zenodo.18376297 |