Superior Koch Curve

Preliminaries

Begin with a straight line (the blue segment in Figure 1). Divide it into three equal segments and replace the middle segment by the two sides of an equilateral triangle of the same length as the segment being removed (the two red segments in the middle figure). Now repeat, taking each of the four resulting segments, dividing them into three equal parts and replacing each of the middle segments by two sides of an equilateral triangle (the red segments in the bottom figure). Continue this construction.

Figure 1.

Koch curve

The Koch curve is the limiting curve obtained by applying this construction an infinite number of times. For a proof that this construction does produce a “limit” that is an actual curve, i.e., the continuous image of the unit interval, see the text by Edgar.

The first iteration for the Koch curve (Figure 2) consists of taking four copies of the original line segment, each scaled by r = 1/3. Two segments must be rotated by 60°, one counterclockwise and one clockwise. Along with the required translations, this yields the following Iterated Function System

Figure 2.

First iteration of Koch curve

The fixed invariant set of this IFS (Iterated Function System) is same as the Koch curve.

Similarity Dimension

We have hyperbolic IFS (Iterated Function System) with each map being a similitude of ratio r < 1. Therefore the similarity dimension, d, of the unique invariant set of the IFS is the solution to