Abstract
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 \(S^2\). 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.
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Notes
The condition \(\langle P_{i+1}P_{i+2}\rangle \geqslant -1/2\) is needed to ensure that there exist equilateral spherical triangles \(Q_iP_{i+1}P_{i+2}\), and \(\langle P_{i+1},P_{i+2}\rangle \not = -1/2\) ensures that the interiors of the equilateral triangles are well-defined (otherwise \(Q_i,P_{i+1},P_{i+2}\) would be cogeodesic).
The conditions \(d_i\in (0,\sqrt{3})\) must still be satisfied.
When \(\langle P_0,P_1\rangle =-1/2\) the equilateral spherical triangle \(P_0P_1Q_2\) is the great circle through \(P_0,P_1\).
Because of these choices, if \(\langle P_2,P_0\rangle =\langle P_1,P_0\rangle \) then \(\langle R_2,R_0\rangle =\langle R_1,R_0\rangle \). Similarly, if \(\langle P_0,P_1\rangle =\langle P_2,P_1\rangle \) then \(\langle R_0,R_1\rangle =\langle R_2,R_1\rangle \), and if \(\langle P_1,P_2\rangle =\langle P_0,P_2\rangle \) then \(\langle R_1,R_2\rangle =\langle R_0,R_2\rangle \). At this stage however, our attention is not limited to the case where \(P_0P_1P_2\) is isosceles.
Indeed, one can suppose that a rewarding strategy in signal transmission consists in “covering the largest possible surface with the least possible mutual distances among the antennas”: the formulation of this isoperimetric problem in the class of triangular patterns of antennas is optimised by equilateral configurations (in turn, the isoperimetric problem for triangles can be explicitly solved, for example, by writing the area of a general triangle using Heron’s formula and using the AM-GM inequality to see that the optimiser is equilateral; another proof can be performed by purely geometrical arguments by considering an ellipse with foci centred at two vertices and passing through the third one).
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Dipierro, S., Noakes, L. & Valdinoci, E. Napoleonic Triangles on the Sphere. Bull Braz Math Soc, New Series 55, 19 (2024). https://doi.org/10.1007/s00574-024-00393-9
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DOI: https://doi.org/10.1007/s00574-024-00393-9