Face-Normal Morley Geometry with n-Sectors: Orthocentric, Midsphere and Rigid Regimes
Year: 2026
Type: preprint
Abstract: On each face of a Euclidean tetrahedron, take the point where the first n-sectors of the two angles adjacent to a given edge meet, and draw through it the line normal to the face. The tetrahedron is admissible if the two lines attached to every edge meet. For n = 3 this is the face-normal Morley construction, whose admissible locus is a curve with seven branches. We show that n = 1 and n = 2 behave differently: the admissible tetrahedra are exactly the orthocentric ones for n = 1, and exactly those with a sphere tangent to the six edges for n = 2, and both loci are three-dimensional. For n ∈ {3, 4, 5, 6} and for the limiting angle-ratio rule n = ∞, the admissible locus consists of the regular tetrahedron, a 3+1 family independent of n and a 2+2 family depending on n. It has the same seven-branch singularity and the same tangent cone as for n = 3. The dichotomy is governed by a mixed curvature L_xz, for which we give a closed formula valid for every n. It vanishes identically for n = 1, 2, where the relevant edge-ratio function is separable. It is positive for n = 3, …, 6, by a Bernstein certificate in ball arithmetic that needs a blow-up at a degenerate corner of the square. For n = ∞ it is an explicit difference ψ(A₂) − ψ(A₁) of a function of two face angles, whose sign reduces to sin φ < φ; this last lemma was formalized in Lean 4 and checked with the AXLE verifier. Numerically, L_xz has the sign of (n − 1)(n − 2) for every real n > 1, and we conjecture that the rigid regime persists for every real n > 2. The record contains the English version (main file), a faithful Portuguese version, and the supplementary archive (Python scripts for the closed formula, the certificates and the numerical search, the certificate log, figures, and the Lean 4 files with the AXLE reports). Paper 4 in a series on three-dimensional Morley constructions.
Author: de Melo, Fábio Dayan Soares
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225559339