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

TL;DR

This work extends Heron's formula from triangles to tetrahedra by introducing six area-based natural parameters, together with their inverse counterparts, and shows the fourth power of the tetrahedron’s volume is given by $t^4 = s^2 oldsymbol{\Omega}(u,v,w,x,y,z)$. It develops a rich, areal-geometry framework built on cross-product areal vectors, exterior/interior face areas, and Minkowski-type relations, culminating in a determinant-based expression that can be factorized into four quadratic components in square roots of the natural parameters. The zeros of this extension (i.e., degenerate tetrahedra) possess a deep projective structure: they correspond to points on the Klein quadric modulo a $oldsymbol{\mathbb{Z}}_2^4$ action, linking Euclidean degeneracies to inversive geometry and affine planar configurations via Yetter’s identity. The paper further establishes a systematic stratification of zeros, relates them to four-point configurations in the affine plane, and discusses higher-dimensional generalizations, potential connections to hyperbolic geometry, and avenues for future exploration in geometry and physics.

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.

Figures (7)

• Figure 1: The geometric interpretation of the Heron parameters $u, v, w$ as: (left) the distances between the vertices of the triangle $\overline{\mathsf{ABC}}$ and the in-touch points $\overline{\mathsf{J}}$, $\overline{\mathsf{K}}$, $\overline{\mathsf{L}}$ of its in-circle, with in-center $\overline{\mathsf{\,I\,}}$ and in-radius $r$; (right) the distances between the vertices of $\overline{\mathsf{ABC}}$ and the ex-touch points at which its ex-circles intersect its edges, as well as the lines spanned by those edges. (NB: these and all the other color figures in this series of papers were made using the GeoGebra dynamic geometry software GeoGebra:2021 unless otherwise specified.)
• Figure 1: Two perspectives on a generic tetrahedron $\overline{\mathsf{ABCD}}$, constructed starting from its in-sphere (green) as described in the main text, with its in-center $\overline{\mathsf{\,I\,}}$ and in-touch points $\overline{\mathsf{J}} \in \overline{\mathsf{BCD}}$, $\overline{\mathsf{K}} \in \overline{\mathsf{ACD}}$, $\overline{\mathsf{L}} \in \overline{\mathsf{ABD}}$ & $\overline{\mathsf{N}} \in \overline{\mathsf{ABC}}$ all labeled accordingly. The congruent line segments connecting each vertex to its three adjacent in-touch points are drawn using the same color as the vertex. The pair of congruent triangles $\overline{\mathsf{ABN}}$ & $\overline{\mathsf{ABL}}$ can clearly be seen on the right, and it is evident that $\overline{\mathsf{AB}} \perp \overline{\mathsf{LN}}$. (This figure was rendered using the Processing graphics library Reas:2014.)
• Figure 1: Hasse diagram of the lattice of inclusions in $\mathbb R_{\ge0}^{\,6}$ of the solutions to $\Omega(u,\,\ldots,\,z) = 0$ and all possible simultaneous subsets of the equations $u\tilde{u} = 0\,$, $\ldots\,$, $z\tilde{z} = 0$ (see text). The ranks of the areal Gram matrix $\mathbf G_\mathsf A$ (which may differ from the lattice rank) at the levels separated by dashed lines are shown on the left, while the generic dimensions of the sets of solutions defined by the equations at each level of the lattice are shown on the right. Black circles indicate points at which lines of the diagram merge rather than cross.
• Figure 1: The 16 classes of rank $1$ zeros from Fig. III :III:fig:regions, as described in the caption to that figure, now labeled by pool ball icons for easy reference.
• Figure 2: The medial octahedron of a tetrahedron $\overline{\mathsf{ABCD}}$, with parallel line segments in space all having the same color. Opposite pairs of edges of the octahedron $\overline{\mathsf{UVWXYZ}}$ have lengths equal to half that of the parallel edge of the tetrahedron, and its volume is half that of the tetrahedron itself. The medial parallelograms formed by pairs of parallel and congruent edges are $\overline{\mathsf{UVZY}}$, $\overline{\mathsf{UWZX}}$ (heavy lines) & $\overline{\mathsf{VXYW}}$; their diagonals $\overline{\mathsf{WX}}$, $\overline{\mathsf{VY}}$ & $\overline{\mathsf{UZ}}$, which are known as the bimedians of the tetrahedron, were not drawn to reduce clutter. (This figure was rendered using the Processing graphics library Reas:2014.)

Theorems & Definitions (82)

• Proposition 1
• proof
• Remark 1
• Lemma 2: Minkowski's Identity
• proof
• Proposition 3
• proof
• Remark 2
• Lemma 4: The Areal Law of Cosines
• proof