A directed tree of 24 triangle types
Vadim Khaikov · Zenodo (CERN European Organization for Nuclear Research) · 2026
We present a complete hierarchical graph model that classifies all planar non-degenerate triangles. The model uses a discrete alphabet of 8 directions and encodes each triangle as a path code (P-code) — an ordered triple of symbols. Of the 8×8×8 = 512 possible triples, only 132 correspond to real triangles. These 132 codes collapse into 24 equivalence classes under the action of the permutation group S₃. In this paper, we analyze the graph structure of these 24 classes. The classification is visualized as a directed rooted tree with 4 levels, 41 nodes, and 40 edges. The tree has 24 leaves, each representing a unique triangle type. The branching is notably uneven: the three main branches contain 12, 4, and 8 classes respectively. This graph is officially registered in the "House of Graphs" project under number 57185 (https://houseofgraphs.org/graphs/57185). It reveals star-shaped patterns and demonstrates the hierarchical organization of triangle types. The model is algorithmically implementable and directly applicable in computational geometry, computer vision, and pattern recognition. This preprint is part of the "Not Just Triangles" (NJT) project. Bilingual Publication Structure: This work is presented in a bilingual format (English/Russian), offering complete parallel texts of the abstract, introduction, core methodology, results, and conclusions. This approach ensures broad accessibility and facilitates dissemination within the international community as well as among Russian-speaking researchers.