Two new versions of Heron's Formula

Abstract

Recollect that Heron's formula for the area of a triangle given its sides has a counterpart given the medians instead, which carries an extra factor of $\frac{4}{3}$. On the one hand, we formulate the pair of these in Linear Algebra terms, showing that they are related by a sides-to-medians involution $J$, which we find to furthermore commute with the `Heron map' $H$ as visible in the expanded version of Heron. Upon further casting the pair of these in terms of mass-weighted Jacobi coordinates, we find moreover that they are placed on an exactly equal footing, the factor of $\frac{4}{3}$ having now cancelled out. This motivates the `Heron--Jacobi' version of Heron's formulae, for mass-weighted area in terms of mass-weighted sides and mass-weighted medians respectively. On the other hand, we show that diagonalizing the Heron map $H$ provides new derivations of, firstly, the famous Hopf map, and, secondly, of Kendall's Theorem that the space of triangles is a sphere. This occurs by the `Heron--Hopf' version of Heron's formula simplifying down to none other than the on-sphere condition. Thus we establish that -- both an important fibre bundle model, and a foundational theorem of Shape Theory: a widely applicable Differential Geometry and Topology topic -- arise together as consequences of just Heron's formula, Stewart's Theorem, and some elementary Linear Algebra manipulations. This working also accounts for the extra factor of 4 in the Hopf coordinate that is elsewise equal to the mass-weighted area in the 3-body problem context. It finally offers a new interpretation of the shape-theoretic ellipticity and anisoscelesness which realize the other two Hopf quantities: as eigenvectors shared by the Heron map $H$ and the sides-to-medians involution $J$.

arXiv comment20 pages and 9 figures

References in corpus (8)

• The Problem of Time and Quantum Cosmology in the Relational Particle Mechanics Arena
• Nonparametric statistics on manifolds with applications to shape spaces
• Foundations of Relational Particle Dynamics
• The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line
• The Smallest Shape Spaces. II. 4 Points in 1-d Suffices to have a Complex Background-Independent Theory of Inhomogeneity
• The Smallest Shape Spaces. III. Triangles in 2- and 3-d
• Relational mechanics of shape and scale
• Alice in Triangleland: Lewis Carroll's Pillow Problem and Variants Solved on Shape Space of Triangles

Cited by in corpus (11)

• The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line
• The Smallest Shape Spaces. III. Triangles in 2- and 3-d
• Maximal Angle Flow on the Shape Sphere of Triangles
• $N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities
• Monopoles of Twelve Types in 3-Body Problems
• Shape (In)dependent Inequalities for Triangleland's Jacobi and Democratic-Linear Ellipticity Quantitities
• Background Independence: ${\mathbb{S}}^1$ and $\mathbb{R}$ absolute spaces differ greatly in Shape-and-Scale Theory
• Topological Shape Theory
• Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces
• Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations
• Isotropy Groups and Kinematic Orbits for 1 and 2-$d$ $N$-Body Problems