La coupe du roi and other methods to halve objects

Abstract

The symmetry theory of halving objects into two identical fragments is developed and illustrated by examples. La coupe du roi is a fascinating way to slice an apple into chiral fragments with the same handedness. It may be applied to any object with point group symmetry 2 or 222 and their supergroups, e.g. to cones, cylinders or spheres. Variants of coupes du roi are presented and applied to a cube. An object is divided into halves by a number of cuts. The important cuts extend to the centre of the object, and these must form a single closed loop. The symmetry of the division into two fragments is given by the point group of the decorated object, i.e. the object including all the cuts. Point groups comprising only rotations result in chiral fragments with the same handedness. Such divisions may illustrate the reaction path for changing the sense of chirality of molecules via a dimeric achiral transition state rather than by deformation. Non-trivial divisions of objects into achiral fragments or into racemic chiral fragments are obtained with point groups, again of the decorated object, comprising roto-inversions X̅, excepting X = 4n + 2. All other point groups either do not lead to half-objects, or they result in a trivial single planar cut parallel to a mirror plane.