How may edges on a sphere?

Answer 1

A sphere is a geometric object, an edge is a topological one, so I will assume you're talking about creating a covering of the sphere with faces, edges and vertices like a volleyball. The patches of volleyball are what I'm calling "faces" the seams between the faces are called "edges" and the connections between the seams (where they meet) are called "vertices."

If the topology is manifold and covers the sphere, then the number of faces minus the number of edges plus the number of vertices = 2! (f - e + v = 2) This remarkable result was first remarked on by Euler. Said differently, the Euler number for a sphere is two.

Therefore the answer to the question is e = f + v - 2.

The Euler number for a cube is also 2. You can morph a cube into a sphere by geometric changes, e.g. centering the sphere and the cube at the origin and projecting the vertex positions of the cube onto the sphere, and setting the geometry of the edges to geodesics between the new vertices.

To understand why the Euler number doesn't change, consider the Euler operation of splitting a face. To do so you introduce a new edge to the model. If the new edge connects points on the interior of the boundary edges of the face, then you have 1 new face added, 3 new edges added (since two edges were split and a new one introduced) and two new vertices added. So the new Euler number is (f + 1) - (e + 3) + (v + 2) = 2. The Euler number is not changed! Of course there are other ways to split faces, but if it is done "legally" (in that the resulting topology is manifold) the answer is always 2 for a sphere. Similar comments can be applied to merging faces, splitting edges, etc.

Note that not all topologies have Euler number 2. For example, a torus has an Euler number of 0, a two-handled trophy cup has Euler number -2!