An Extension of Heron's Formula to Tetrahedra, and the Projective Nature of Its Zeros

TL;DR

This paper extends Heron's formula to tetrahedra, expressing volume as a polynomial in face areas and medial parallelograms, and explores the geometric and algebraic structure of its zeros.

Contribution

It introduces a novel polynomial formula for tetrahedral volume based on face and medial areas, and analyzes the geometric and algebraic properties of its zeros, including connections to projective geometry and matroids.

Findings

Volume polynomial relates to face and medial areas

Zeros form a semi-algebraic variety of collinear tetrahedra

Algebraic zeros define an affine oriented matroid

Abstract

A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four faces and of its three medial parallelograms, which are accordingly referred to herein as "interior faces." Geometrically, these rational functions are the areas of the triangles into which the exterior faces are divided by the points at which the tetrahedron's in-sphere touches those faces. Part I presents an overview of these results and some necessary but little-known background in areal geometry. Part II derives the promised extension, and ends with a conjecture as to how the formula extends to $n$-dimensional simplices for all $n > 3$. Part III explains how, for $n = 3$, the zeros of the polynomial constitute a five-dimensional semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices separated by infinite distances, but with generically well-defined distance ratios; it further proves that these unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to $\mathbb Z_2^4$, wherein four-point configurations in the affine plane constitute a distinguished three-dimensional subvariety. Part IV consists of five appendices which show, among other things, that the algebraic structure of the zeros in the affine plane naturally defines the associated four-element, rank $3$ chirotope, aka affine oriented matroid.