The shapes of a random sequence of triangles

Abstract

A triangle with vertices z1, z2, z3 in the complex plane may be denoted by a vector Z, Z = [z1, z2, z3]t. From a sequence of independent and identically distributed 3×3 circulants {Cj}∞1, we may generate from Z1 the sequence of vectors or triangles {Zj}∞1, by the rule Zj = CjZj–1 (j> 1), Z1=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 {Zj}∞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.