# counting edges in tesselations of a torus

Tesselate a torus with finitely many simply connected polygons. Do not allow four or more of them to meet at a point. In counting the edges, don't count a "straight line" as just one edge if it's the boundary between polygons A and B until you reach a point after which it's the boundary between A and C; at that point one edge ends and the next starts. (You might say "straight line" means a geodesic, but we maybe don't need to be so sophisticated: just say there's a $180^\circ$ angle there, not $179^\circ$, etc.)

Then: The average number of edges of the tesselating polygons is exactly 6.

Proof: $V-E+F=0$, then massage.

The question: Is the statement after "then" in citable literature somewhere?

Later comment: It may seem odd to include the note about counting edges in a graph, since it's the only way anyone would count them, but when one thinks of counting edges of a polygon, it may seem odd to think of one of the four sides of a rectangle as two edges rather than one.

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There are graphs which embedded on the torus which are 4-valent and whose faces have exactly 4 sides. Perhaps I don't understand your rules? Look at: faculty.fortlewis.edu/Scull_L/math342/torus(1).jpg – Joseph Malkevitch Dec 29 '11 at 20:04
Joseph: the poster doesn't allow four edges to meet at a point. Under their rules (assuming I understand the last rule correctly), the statement is indeed true, as V = (average # of edges)*(# of faces)/3 and E = (average # of edges)*(# of faces)/2. – Lopsy Dec 29 '11 at 20:09
@Lopsy I did not read carefully enough. It clearly says 4 or more. My mistake. – Joseph Malkevitch Dec 29 '11 at 20:21
Really? Three new tags? – Asaf Karagila Dec 29 '11 at 23:38
The statement after "then" is "massage". Are you asking whether "massage" is in the citable literature? If not, could you be a little less elliptical in expressing the actual question? – Gerry Myerson Dec 30 '11 at 3:35