Heron’s Formula from a 4-Dimensional Perspective

The purpose of this paper is to demonstrate a proof of Heron’s formula using scissors congruences in dimension 4. Our motivations come from various conversations in various places. So we will relate the personal details before the technical details. In [4], Carter and Champanerkar indicated a higher dimensional interpretation of ∫ 1 0 x n dx as the (n+ 1)-dimensional volume of a pyramidal structure whose base is the n-cube. They indicated that (n+ 1) such pyramids filled the (n+ 1)-dimensional unit cube. Thus the fact that ∫ 1 0 x n dx = 1/(n+ 1) can be seen as a scissors decomposition of the (n+ 1)-dimensional cube. This description had also been discovered by [1], and so that manuscript languishes. Nevertheless, the proof is a nice dinner-time or tea-time story that can be related to the mathematician who is higher-dimensionally inclined. During one such conversation in Hanoi, Dylan Thurston asked Carter and Champanerkar if we knew of a 4-dimensional proof that