Polygonal figures and their form spaces II. Angle permutations in equilateral polygons

Abstract

In a closed nonregular equilateral polygonal curve, angles cannot be permuted arbitrarily without opening the curve. Here we investigate the possible angle permutations that leave a polygonal curve closed. To this end we introduce angle sequences, i.e., symbol sequences describing the number and arrangement of like angle values in classes of forms. We find that there exist “ universal angle sequences” giving closed polygons for arbitrary angle values provided that the angle sum is kept constant at one of certain permissible values. The universal angle sequences are periodic: they consist of at least two identical series of symbols. Within a series, the symbols can be permuted freely on condition that in all other series the very same permutation is applied. All these correlated permutations, when applied to the angles of a corresponding polygonal curve, leave this curve closed. Polygons belonging to a given angle sequence (form class) constitute a submanifold of the manifold M I of equilateral polygons within the form space. These submanifolds structure M I in a hierarchical manner so that the form class concept gives a valuable complementary classification scheme for the forms of M I . Furthermore, we find that polygons with constant angle composition (the term “composition” is used here in its chemical sense) show a wide variety of possible shapes (from quasi-circular to wormlike and even self-intersecting ones) corresponding to different degrees of “segregation” of like angles within the polygonal curve.