math.MG — Metric Geometry

Regular Morley tetrahedra from $p$-secting planes

Contributed by Deep Bhattacharjee

Morley's theorem says that the adjacent angle trisectors of a triangle meet in the vertices of an equilateral triangle. Tran carried the construction to a tetrahedron by trisecting its six dihedral angles and, more generally, by $p$-secting planes, with one parameter $e,f,g>1$ for each pair of opposite edges; the resulting tetrahedron is denoted $\mathcal M_{e,f,g}(ABCD)$. Tran showed that a regular Morley tetrahedron does not force the original tetrahedron to be regular, observed that in his examples some pair of opposite edges has four equal cross edges, and conjectured that this happens whenever $\mathcal M_{e,f,g}(ABCD)$ is regular. We show that the conjecture is false. We place an isosceles tetrahedron in the box $[-a,a]\times[-b,b]\times[-c,c]$ and prove that $\mathcal M_{e,f,g}(ABCD)$ is regular exactly when its vertex $A'$ differs from the centre of the box and lies on one of the four lines through the centre in the directions $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$ and $(-1,-1,1)$. Every interior point of these lines other than the centre occurs for exactly one triple of parameters. When $a$, $b$ and $c$ are pairwise different, no pair of opposite edges has four equal cross edges, so each of these tetrahedra is a counterexample. Two examples are given with the parameters in closed form. Finally, for quadrisecting planes, $e=f=g=4$, an isosceles tetrahedron has a regular Morley tetrahedron only if it is regular.