Generalizations of the Koch curve

We present an iterative method to de. ne a two-parameter family of continuous functions f(a,theta) : I -> C such that f(1/3),(pi/3) is the Koch curve. We consider the two-cases theta = pi/3 and theta = pi/4 of these generalized Koch curves f(a,theta)(I). In each case we determine the pivotal value of a, the largest value of a for which the corresponding curve is not simple. We give characterizations of the double points of the curve (points on the curve that have two inverse images). In the case where theta = pi/3 double points are vertices of equilateral triangles. When theta = pi/4 the double points form Cantor sets in the plane. We conclude with a more general result that proves that if the fixed set (attractor) of an iterated function system is connected, then it is a curve.