Self-Intersection Points of Generalized Koch Curves

We present a two-parameter family of curves Ka,ө that are modifications of the Koch curve. For each ө, there is a corresponding pivotal value a(ө) of a, where Ka,ө is simple if a > a(ө), and Ka,ө is self-intersecting if a < a(ө). We find a(ө), for ө € (0, π/3], then show that the pivotal-valued curves comprise two classes of self-intersecting curves that we characterize by whether or not ө= π/n for some even positive integer n > 2.