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

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 three medial parallelograms, which will be 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. This leads to a conjecture as to how the formula extends to $n$-dimensional simplices for all $n > 3$. Remarkably, 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. 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 subset. The paper closes by noting that the algebraic structure of the zeros in the affine plane naturally defines the associated four-element, rank $3$ chirotope, aka affine oriented matroid.Comment: 51 pages, 6 sections, 5 appendices, 7 figures, 2 tables, 81 references; v7 clarifies the definitions made in the text leading up to Theorem 5.4, along with the usual miscellaneous minor corrections and improvement

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

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 extension 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 will be 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. This leads to a conjecture as to how the formula likely extends to $n$-dimensional simplices for all $n \u3e 3$. Remarkably, for $n = 3$ the zeros of the overall polynomial constitute a five-dimensional real semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices at infinite distances from one another. These unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to \mbb Z_2^4, wherein four-point configurations in the finite affine plane constitute a distinguished three-dimensional subset. The paper closes by noting that the algebraic structure of the zeros in the finite affine plane naturally defines the associated $4$-element, rank-$3$ chirotope, aka affine oriented matroid