正n边形对角线形内交点个数(英文).pdf

THE NUMBER OF INTERSECTION POINTS MADE BY THE 6 0 DIAGONALS OF A REGULAR POLYGON 0 2 BJORN POONEN AND MICHAEL RUBINSTEIN r a M Abstract. We give a formula for the number of interior intersection points made by the diagonals of a regular n-gon. The answer is a polynomial on each 8 residue class modulo 2520. We also compute the number of regions formed by 1 the diagonals, by using Euler’s formula V − E + F = 2. ] G M 1. Introduction . h We will find a formula for the number I (n) of intersection points formed inside t a a regular n-gon by its diagonals. The case n = 30 is depicted in Figure 1. For a generic convex n-gon, the answer would be n , because every four vertices would m 4 [ be the endpoints of a unique pair of intersecting diagonals. But I (n) can be less, because in a regular n-gon it may happen that three or more diagonals meet at an 3 interior point, and then some of the n intersection points will coincide. In fact, 4 v n 9 if n is even and at least 6, I (n