I currently have code that draws a sphere as a series of quad strips for each latitude. There is a hole at the pole which could be filled with an end cap made of a triangle fan.

Instead, I would like to cover a sphere in a single strip of triangles. This would be efficient, as it is a single operation, but it is harder to visualize.
I created a paper model of an icosahedron and verified that I can start with one face, and number the faces sequentially hitting every one once until all are covered. My question is whether this would work for any arbitrary triangle covered shape with no holes.

Let's start with an icosahedron that is recursively split so that each face turns into 4 triangles. This is done by divide each edge in two. Can I number one face and go all through the faces? I have no idea and this one is not as easy to cut out and try.

Last, if it is possible, how could I determine the points and the order?