Title: n條線的平面切割
How many regions do n lines divide the plane
Authors: 蔡宛庭
Keywords: n條線的平面分割;Division of the Plane by Lines
Issue Date: 2017
Abstract: 這個問題的探討源自於V.I. Arnold 看了由A.P. Kiselev 所寫A.B. Givental 翻譯的一本教科書≪ Geometry ≫,在教科書裡有一個問題讓他 特別注意並且刺激他更進一步的思考―「How many lines must one take so as to divide the plane into five parts ? 」 他更深入的研究這個問題並把原本題目的討論範圍擴大延伸,從五個區 域改成任意的區域數,所以才發展出這個題目,也為喜歡數學的學生與學 者們開啟了一個新的領域。 已知在一個平面上畫n 條直線會將平面分出割的區域數最多是1+ (n+1 2 ) 最少是n + 1,但不是每個在n + 1 至1 + (n+1 2 ) 之間的數都能用n 條直線 分割出來。比如說當n 3 的時候,無法用n 線分割出n + 2 區域。想要 給這些區域數一個確定的答案,到目前為止,還是一個非常困難的問題。 在這篇論文,我們將使用尤拉公式在平面圖上的性質來計算區域數,也 就是說,我們可以用找「平行族」與「頂點度數」,這種比較有效率的方式 來找出我們的區域數。
It is well-known that if we draw n lines on a plane, then we can obtain at most Cn+1 2 +1 regions and at least n+1 regions. For convenience, we use Rn to denote the set of possible number of region. Therefore, Rn [n+1;Cn+1 2 +1]. Clearly, the containment is proper since there are quite a few elements in [n + 1;Cn+1 2 + 1] which are not in Rn, for example, n + 2 if n 3. To determine the exact answer for Rn remains an open problem so far. In this thesis, we shall use the Euler’s formula on planer graph to estimate the number of regions. It turns out that we can count the number of regions with a more efficient way by using the number of ”pencils” and ”stars”.
URI: http://etd.lib.nctu.edu.tw/cdrfb3/record/nctu/#GT070252231
Appears in Collections:Thesis