Platonic geodesic trees in the K−Space
Abstract
In this paper, We prove that: (A) If m≥ 4, there exists a Platonic geodesic tree having m edges (geodesic segments), which form m platonic angles with measure 120o and ((m(m−1))/2−m) platonic angles with measure 90o. (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∘ or 150∘ or 180∘ (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∘ (Hexagonal coordinate system). (D) If m is even and m ≥ 4, there exists a Platonic geodesic tree having m edges , which form platonic angles with only two distinct measures 90o and 180∘. (Existence of a Cartesian coordinate system). (E) If m≥ 4, there does not exist a Platonic geodesic tree having m edges , which form only platonic angles with measure 120o (Non existence of a Fermat coordinate system). (F) If m≥ 3, there does not exist a Platonic geodesic tree having m edges, which form platonic angles with only one measure 150o, or two distinct measures 120∘ and 150∘ or two distinct measures 90∘ and 150∘ (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∪ Q.
Related
- On universal elements for doubling geodesic trees
- Geodesic simplices of pseudo-hyperbolic space
- Totally Geodesic Submanifolds of Teichmüller Space in Complex Dimension at Least Three
- On certain mapping properties of a weighted totally geodesic k-plane transform on the real hyperbolic space
- Steiner Minimal Trees in the Gromov--Hausdorff Space and in Gromov--Hausdorff Clouds
BibTeX
@misc{zachos2026platonic,
title = {{Platonic geodesic trees in the $K-$Space}},
author = {Anastasios N. Zachos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13950},
eprint = {2610.13950},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13950},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Platonic geodesic trees in the K−Space
AU - Anastasios N. Zachos
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.13950
UR - https://arxiv.org/abs/2610.13950
AB - In this paper, We prove that: (A) If m≥ 4, there exists a Platonic geodesic tree having m edges (geodesic segments), which form m platonic angles with measure 120^o and ((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^∘ or 150^∘ or 180^∘ (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^∘ (Hexagonal coordinate system). (D) If m is even and m ≥ 4, there exists a Platonic geodesic tree having m edges , which form platonic angles with only two distinct measures 90^o and 180^∘. (Existence of a Cartesian coordinate system). (E) If m≥ 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≥ 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^∘ and 150^∘ or two distinct measures 90^∘ and 150^∘ (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∪ Q.
ER -