TY - JOUR

T1 - Napoleonic Triangles on the Sphere

AU - Dipierro, Serena

AU - Noakes, Lyle

AU - Valdinoci, Enrico

N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Brazilian Mathematical Society 2024.

PY - 2024/6

Y1 - 2024/6

N2 - 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.

AB - 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.

KW - Examples and counterexamples

KW - Napoleon’s Theorem

KW - Spherical geometry

UR - http://www.scopus.com/inward/record.url?scp=85189437943&partnerID=8YFLogxK

U2 - 10.1007/s00574-024-00393-9

DO - 10.1007/s00574-024-00393-9

M3 - Article

AN - SCOPUS:85189437943

SN - 1678-7544

VL - 55

JO - Bulletin of the Brazilian Mathematical Society

JF - Bulletin of the Brazilian Mathematical Society

IS - 2

M1 - 19

ER -