2024/03/05 by Serena Dipierro, Dipierro, Serena, Lyle Noakes +3
Engineering · Mathematics · #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Analysis of PDEs (math.AP) #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2403.02830
openalex publication_date 2024/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
As is well-known, numerical experiments show that Napoleon's Theorem for planar triangles does not extend to a similar statement for triangles on the unit sphere S2. Spherical triangles for which an extension of Napoleon's Theorem holds are called ``Napoleonic'', and until now the only known examples have been equilateral. In this paper we determine all Napoleonic spherical triangles, including a class corresponding to points on a 2-dimensional ellipsoid, whose Napoleonisations are all congruent. Other new classes of examples are also found, according to different versions of Napoleon's Theorem for the sphere. The classification follows from successive simplifications of a complicated original algebraic condition, exploiting geometric symmetries and algebraic factorisations.