PACKING TRIANGLES ON A SPHERE

Abstract. The question of how many regular unit tetrahedra with a vertex at the origin can be packed into the unit sphere is a well-known and difficult problem. The answer is between 20 and 22, and these results are reproduced-here. The classical method for calculating such bounds involves projecting the tetrahedra onto the surface of the sphere, yielding equilateral spherical triangles. Motivated by this, we search for upper and lower bounds on how many equilateral spherical triangles may be arranged on the unit sphere as a function of the spherical angle α between two sides. For α ≥ 2pi 5, we describe some possible arrangements of equilateral spherical triangles on the sphere, but the primary focus of this paper is on pi 3 < α ≤ 2p