The “form” of a triangle

Abstract

Heron’s formula, maybe due to Archimedes, expresses the squared area of a triangle as a polynomial in the squared lengths of its sides. A true understanding of this formula comes from an adequate coding of the shape of a triangle, i.e. of the triangle up to translations, rotations and reflexions. One can then build a “space of triangle shapes” which, if orientation is added becomes the “shape sphere” and possesses very nice symmetries which were recently used in researches on the 3-body problem. The invariant interpretation of the shape sphere in turn sheds light on the nature of Heron’s formula.

Keywords

• heron’s formula
• volumes
• inertias
• dispositions
• invariants