一個點線面的切割問題

Abstract

在這篇論文中，我們希望用不同角度來重新探討一個古典的數學問題；點、線、面切割最多區域問題，雖然這個問題已經經由許多方法得到公式，例如：遞迴關係、差分方程式、歐拉公式、標準n維空間切割系統等等，並延伸出其他方面的問題，可以運用在很多地方，所以我們希望可以再找到更簡單易懂的論證方式，可以讓國中學生也能理解。 \n 思考學生現有的數學觀念，我們發現利用不等式的數學觀念，藉由定義出一套有規則的系統以及數學歸納法，可以以更直接，簡單的理論驗證出此數學公式，最後我們更希望能將這理論推廣至n維度空間。

In this research, we will discuss a classical mathematical question from different aspects. The question of maximizing the number of regions made up by points, lines and planes has been proved and developed many formulas, using Recurrence Relations, Difference Equations, and Euler`s Formula etc., which can extend to other questions and apply to many areas. Therefore, we hope to find an easier way to prove it which may help middle school students to understand better.\n We find that we can use the concept of inequality from what the students learn so far. By defining a logical system and using Induction, we can prove this mathematical formula in an easier and more direct way. Finally we hope it can be generalized to n-dimensional space.