Platonic geodesic trees in the $K-$Space

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 $V\cup Q.$

arXiv comment9 pages