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

Bound smoothing under chirality constraints

SIGLETIB Hannover: RO 8278(90-017) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Bound Smoothing Under Chirality Constraints

Dress A, Havel TF. Bound Smoothing Under Chirality Constraints. SIAM Journal on Discrete Mathematics. 1991;4(4):535-549.Procedures for determining the feasibility of lower and upper bounds on Euclidean distances of fixed dimension play a central role in the analysis of many kinds of scientific data. Shown in this paper is how results from graph optimization theory can be used to solve the feasibility problem in one dimension, subject to the condition that the order of the points along the real line is known. The solution is used to derive a PSPACE, O(n3 . n!)-time sequential algorithm for finding one-dimensional representations subject to arbitrary distance (and order) constraints. The wider applicability of these results in measurement theory is discussed, in particular, Roy's elegant proofs of the classical representation theorems for interval orders and semiorders, and they are used to obtain a new representation theorem for a ternary relation called epsilon-collinearity