Which surfaces up to homeomorphism can be formed by taking a

Which surfaces (up to homeomorphism) can be formed by taking a hexagon and identifying pairs of edges? For each surface, give an example of a choice of edge identifications. Hint: To narrow down the list of possibilities, consider: what can the Euler characteristic of such surface be?

Solution

if we observe this hexagon

One explanation comes from Euler’s this is formula that relates the number of vertices, edges, and faces in a polyhedron. This formula says, quite simply, that if there are v vertices, e edges, and f faces in any convex polyhedron (not necessarily a fullerene), then ve+f=2,

always! For example, a cube has 8 vertices, 12 edges, and 6 faces, and indeed 812+6=2.

and this for hexagon also it depends on vertices, edges, and faces.