Representations of classes of plane triangles identified based on the fixed direction of traversal. 2. Representations of classes by graphs and diagrams
Year: 2025
Type: article
Abstract: : This study advances the hierarchical classification of plane triangles by formalizing the orientation of side traversal relative to a designated starting vertex through three-component tuples of direction indicators. Building upon the identification of twenty-four non-overlapping triangle classes, the second part systematically develops graph-theoretic and diagrammatic representations that complement the earlier tabular and direct graphical treatments. The complete set of classes is first modeled as a star graph S₂₄, a planar, connected, acyclic structure with twenty-five vertices, diameter two, chromatic number two, and an adjacency matrix of 625 elements, which conveys only the external structure of the classification. A claw subgraph S₃ is shown to represent conventional length- and angle-based classifications uniformly, underscoring the specificity of the proposed scheme. A rooted tree graph is then constructed to encode hierarchical ordering, in which paths from root to leaves correspond to ordered three-component tuples of side directions, establishing a bijection between triangle classes and direction codes. This tree comprises forty-one vertices with an adjacency matrix of 1,681 elements and reproduces the color groupings introduced in the first part. Euler–Venn diagrams are employed to analyze intersections among tuple components, yielding eight three-petal and two five-petal configurations that reveal ten cases of partial directional coincidence and permit merging into an eight-petal flower. A bipartite graph is subsequently introduced, with four abscissa and nine ordinate tuple types as the two vertex parts, thirteen vertices in total, and twenty-four edges corresponding to the triangle classes. Its symmetric, sparse 13 × 13 adjacency matrix contains only forty-eight ones, and matrix multiplication verifies the edge count, while the graph's bichromaticity and maximum matchings of four edges are characterized. A circular diagram arranged as a twenty-four-hour clock face provides an intuitive visualization linked to color groupings and prior class notations. A dendrogram based on Euclidean distance and complete linkage clusters the classes into three groups of eight, twelve, and four members, reproducing the color structure through an independent methodological route. Keywords: plane triangle class, class table, triangle class association, tree graph, bipartite graph, pie chart, dendrogram. Contents:1. Introduction2. Star graph3. Tree graph4. Bigraph (bipartite / bichromatic graph)5. Pie chart6. Dendrogram of triangle classes and their clusters7. ConclusionReferences Suggested citation: Khaikov, V.L. Representations of classes of plane triangles identified based on the fixed direction of traversal. 2. Representations of classes by graphs and diagrams. // Electronic Information Systems, No. 4 (47), 2025, pp. 100–115. EDN: RGDVWS. (In Russian)
Author: Khaikov, Vadim
Language: Russian
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208837890