This paper shows how geometric algebra can be used to derive a novel generalization of Heron’s classical formula for the area of a triangle in the plane to higher dimensions. It begins by illustrating some of the many ways in which the conformal model of three-dimensional Euclidean space yields provocative insights into some of our most basic intuitive notions of solid geometry. It then uses this conceptual framework to elucidate the geometric meaning of Heron’s formula in the plane, and explains in detail how it extends naturally to the volumes of tetrahedra in space. The paper closes by outlining a proof of a previously conjectured extension of the formula to the hyper-volumes of simplices in all dimensions.

本文展示了如何使用几何代数将平面上三角形面积的海伦经典公式推导到更高维度。它首先说明了三维欧几里得空间的共形模型对我们一些最基本的立体几何直观概念产生启发性见解的多种方式中的一些方式。然后，它使用这个概念框架来阐明海伦公式在平面中的几何意义，并详细解释它如何自然地扩展到空间中的四面体体积。论文最后概述了先前推测的公式扩展到所有维度的超体积单纯形的证明。