triangulating a sphere

Question

Can there be a triangulation of the sphere with $3$ triangles? We can start with the sphere and three vertices fixed. This divides the sphere into two triangles, which is illegal as the triangles would share more than two sides. Then if we add another vertex, we get four triangles, which is possible. Is there no way of obtaining three?

Answer 1

There is no possible way to triangulate the sphere using 3 vertices. Just like you cannot have two faces with the same three edges surrounding them, you cannot have two edges whose endpoints are the same two vertices. Said differently, a pair of vertices determine a unique edge in a triangulation. So with 3 vertices, the maximum number of edges possible is 3, which is not enough to triangulate the sphere.

Answer 2

A triangulation is a special case of a regular CW complex decomposition. Your decomposition with $2$ triangles is not a triangulation but it is a perfectly good regular CW complex. However there is no such decomposition of the sphere with an odd number of triangles joined edge-to-edge, as the edges would have to be joined in pairs (because the sphere is a surface without boundary). In particular, as Adam Lowrance points out you cannot find a triangulation of the sphere with $3$ triangles.