從球面幾何的角度探討五種正多面體

Abstract

古代用立體幾何的方式來計算五種正多面體的幾何量，西元前300年歐幾里得幾何原本（Elements）第11~13卷有介紹立體幾何，在第13卷中特別研究正多面體的作圖，以及五種正多面體的存在性。17世紀解析幾何（Analytic geometry）出現之後，我們可以藉由向量空間、球面坐標系統、對稱性等工具，比古代更容易地可以得到五種正多邊形幾何量的結論。

Ancient mathematicians derive the geometry quantities of the five Platonic solids using the methods from the so-called 'solid geometry'. In 300 B.C., Euclid introduced solid geometry in book XI to XIII of his work, Elements. The compass-and-straightedge constructions of the Platonic solids were investigated in book XIII, as well as the existence of the five solids. On the other hand, in seventeenth century, these geometry quantities can be more easily computed using tools in analytic geometry, for example vector space, polar coordinate, symmetry, etc.