Year: 2026
Type: preprint
Abstract: In this paper, We prove that: (A) If $m\ge 4,$ there exists a Platonic geodesic tree having $m$ edges (geodesic segments), which form $m$ platonic angles with measure $120^{o}$ and $(\frac{m(m-1)}{2}-m)$ platonic angles with measure $90^{o}.$ (Existence of a Fermat Coordinate system) (B) If $m=2,$ there exists a Platonic geodesic tree with three vertices and 2 edges, which meet at $120^{\circ}$ or $150^{\circ}$ or $180^{\circ}$ (Minimal spanning geodesic tree). (C) If $m=3,$ there exists a Platonic tree and the Steiner point is identical with the Fermat point, having three edges, which meet at $120^{\circ}$ (Hexagonal coordinate system). (D) If $m$ is even and $m \ge 4,$ there exists a Platonic geodesic tree having $m$ edges , which form platonic angles with only two distinct measures $90^{o}$ and $180^{\circ}.$ (Existence of a Cartesian coordinate system). (E) If $m\ge 4$, there does not exist a Platonic geodesic tree having $m$ edges , which form only platonic angles with measure $120^{o}$ (Non existence of a Fermat coordinate system). (F) If $m\ge 3,$ there does not exist a Platonic geodesic tree having $m$ edges, which form platonic angles with only one measure $150^{o},$ or two distinct measures $120^{\circ}$ and $150^{\circ}$ or two distinct measures $90^{\circ}$ and $150^{\circ}$ (Non existence of a Fermat coordinate system). A classification of platonic geodesic trees with boundary vertex set $V$ and added vertex set $Q$ in the $K-$Space is given by conditions (A), (B), (C), (D), (E), (F), which characterize each vertex in $V\cup Q.$
Source: arXiv (Cornell University)
Author: Zachos, Anastasios N.
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221816618