Any Scalene Triangle Is the Most Chiral Triangle

Abstract

AbstractThe shape space of all possible triangles is represented by a triangular diagram, an analogue of the phase diagram for ternary mixtures. Each point of its interior corresponds to the angle set of a pair of (2D) enantiomeric physical triangles and, with appropriate conventions, to just one member of the pair. Points on median lines represent achiral triangles, those on sides represent degenerate chiral triangles, and those on vertices achiral degenerate linear triangles. A chirality index for triangles must vanish on these lines, but nowhere else within the six compartments of the diagram, and should alternate in sign between them. The archetype is the lowest A2‐symmetric eigenfunction of the Schrödinger equation for the particle confined to an equilateral triangular box. Within the constraints, the extrema of an acceptable function may be pushed onto any D3h‐symmetric hexagonal set of points in the diagram, thereby verifying a conjecture of Dunitz that any scalene triangle is the most‐chiral triangle for some legal 2D‐chirality index.