The shapes of a random sequence of triangles

Abstract

A triangle with vertices z 1, z 2 , z 3 in the complex plane may be denoted by a vector Z , Z = [z 1, z 2, z 3]t . From a sequence of independent and identically distributed 3×3 circulants { C j }∞ 1, we may generate from Z 1 the sequence of vectors or triangles { Z j }∞ 1, by the rule Z j = C j Z j–1 (j> 1), Z 1 = Z . The ‘shape’ of a set of points, the simplest case being three points in the plane has been defined by Kendall (1984). We give several alternative, ab initio discussions of the shape of a triangle, and proofs of a limit theorem for shape of the triangles in the sequence { Z j }∞ 1. In Appendix A, the shape concept is applied to the zeros of a cubic polynomial. Appendix B contains some further remarks about shape. Appendix C uses the methods of this paper to give proofs of generalizations of two old theorems on triangles.