Growth and Form

In the study of cutting Generalized Möbius-Listing bodies with polygons as cross section, it is well known that the Möbius phenomenon, whereby the cutting process yields only one body, occurs only in even polygons with an even number of vertices and sides, and only in the specific when the knife cuts through the center of the polygon. This knife cuts from vertex to vertex, vertex to side or side to side, cutting exactly two points on the boundary of the polygon. This is called a chordal knife, in connection to the chord cutting a circle. If the knife is a radial knife, i.e. it cuts only one point of the boundary, the Möbius phenomenon can occur both in odd and even polygons, but only when the radial knife cuts the center of the polygon. One finding is the reduction of a problem in 3D (with internal geometry) to a planar problem and the concomitant reduction of the analytic representation with multiple parameters to a few only. The shape of the cross section and number of twisting in the 3D representation suffice and reduce the problem to cutting of regular polygons and cyclic permutations.