The Question of Co unting for Intersection Po ints of Inner Diagonal Lines of Regular N Polygon

Lunbo Xu Hainan Senior High School

S.-T. Yau High School Science Awarded Papers mathscidoc:1608.35016

2008
In the early 1980s of twenty century, Professor Zhang Zhongfu, an expert in graph theory, raised a question in his research: how many points of intersection of diagonal lines are there inside a regular N polygon, hoping to find out a formula of count. It has been more than 20 years since the question was raised, which has aroused the interest of quite a lot of experts, scholars and those who love mathematics, but still remains unsolved. When N is an odd number, proposition 1 can be derived from formula of counting based on proposition 2, which can also be verified by the programme the author has made. Proposition 1: when N is an odd number, there is no concurrence of 3 or more than 3 diagonal lines of regular N polygon. Proposition 2: when N is an odd number, the number of intersection points of inner diagonal lines of regular N polygon is: &a_n=c^{4}_{n}=frac{1}{24}(n-1)(n-2)(n-3)$ But when N is an even number, it becomes quite complicated. When N are some special even numbers referred to by Chang Jianming, there are some laws: the research is made from the perspective of prime factorization of figures. The writer primarily studies the issue by making use of geometrical drawing and imitation by computer programme, so that the programme is improved and the imitating solutions of the numbers of intersection points of inner diagonal lines of regular N polygon with specific programmes can be solved. The solution is popularizable. The writer has worked out the numbers of intersection points of inner diagonal lines of regular 2N polygon when N 50, if conditions of computers in schools allow. By comparing the counting between drawing and computer imitation, the writer has also found out some interesting laws about the numbers of intersection points of inner diagonal lines of regular N polygon, and has put forward some hypotheses.
Regular N polygon, intersection points of diagonal line, formula of count for intersection points, imitation programme
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@inproceedings{lunbo2008the,
  title={The Question of Co unting for Intersection Po ints of Inner Diagonal Lines of Regular N Polygon},
  author={Lunbo Xu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160813215154799158064},
  year={2008},
}
Lunbo Xu. The Question of Co unting for Intersection Po ints of Inner Diagonal Lines of Regular N Polygon. 2008. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160813215154799158064.
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