I wanted to know if I could start at one point on an icosahedron and traverse to all the others sequentially without visiting any one twice, which I assume I could model as a Hamiltonian path in a graph if I correctly mapped the polyhedron to a graph.
I didn't try because I was unclear on the graph. Each face is connected to three adjacent faces, and I found keeping track of the intersections to be confusing so instead I just built an icosahedron and numbered the sides and could see that it worked: I can start at one face and end at another face on the opposite side, having only visited adjacent faces and numbered them all from 1 to 20.
What I would like to know is if I did this to an arbitrary polyhedron where all faces are triangles, Is there always a path that starts at one face, and continues using only adjacent faces, convering every one?