Identidades de Área e Volume que Caracterizam a Trissecção de Morley em um Cubo Triangulado
Year: 2026
Type: preprint
Abstract: Triangulate each square face of a cube by a diagonal. In every right isosceles triangle so obtained and for every edge of it, intersect the two rays from the endpoints that make angles 2θ at the right angle and θ at the other vertices; for θ = π/(4n) these are the first n-sectors, and θ = π/12 is the facial Morley construction. Through each of the 36 facial points draw the line normal to its square face. Exactly two triangulations make the two normals along every cube edge meet, and for 0 < θ < π/8 the 36 normals have exactly 56 intersection points, in shells of 24, 24 and 8. Four convex hulls Q, T, R, I built from these shells have areas and volumes given by explicit rational functions of t = tan θ. At trisection one has, for instance, 6 vol(Q) = vol(cube) + vol(I), and each of three volume identities and one area identity holds only at θ = π/12, with a linear stability estimate. The mechanism is arithmetic: tan(π/12) is the only value tan(π/(4n)), n ≥ 3, of degree two, and at n = 3 every normalized volume lies in Q(√3), which forces a rational linear relation in each such triple. The record contains the English version (main file), a faithful Portuguese version, and the supplementary archive (Python scripts, CSV/JSON data, figures, and a Lean 4 file, checked with the AXLE verifier, formalizing the factorizations, the four characterizations and the stability inequality from the closed formulas). Paper 2 of 3 in a series on three-dimensional Morley constructions.
Author: Soares de Melo, Fábio Dayan
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225598130