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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: – 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
up vote 1 down vote accepted

This doesn't exactly answer your reference request, but here is an example where the polygons are all hexagons (not just on average.)

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Another example where they're all hexagons is where they're all rectangles: four of them. Lay the torus on a horizontal surface and look at (1) the upper north quadrant, (2) the upper south quadrant, (3) the lower east quadrant, (4) the lower west quadrant. Each of these faces has four "straight" sides, but each of the horizontal sides gets interrupted halfway through, in just the way I described in my question, thus the sides get counted as six edges. So in that sense, they're hexagons. – Michael Hardy Dec 30 '11 at 4:54

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