On the geometry of generalised Koch snowflakes

TL;DR

This paper studies a broad class of generalized Koch snowflakes, analyzing their geometric properties such as area, measure, length, and dimension, and classifying when they form quasicircles, using deterministic methods and fractal theory.

Contribution

It introduces a deterministic framework for generalized Koch snowflakes with variable triangle orientations, providing exact geometric and measure-theoretic results.

Findings

The set of all enclosed areas forms a closed interval.

The union of all generalized snowflakes has zero Lebesgue measure.

Each curve has infinite length and the same Hausdorff dimension as the classical Koch snowflake.

Abstract

We consider the geometry of a class of fractal sets in $\mathbb{R}^{2}$ that generalise the famous Koch curve and Koch snowflake. While the classical Koch curve is defined by an iterative process that divides a line segment into three parts and replaces the middle part by the legs of an isosceles triangle 'above' the line segment, in this more general setting, a choice can be made at each iteration as to whether to place this triangle 'above' or 'below' the line segment. The resulting fractals bear a striking visual resemblance to curves appearing in nature, such as coastlines and snowflakes. While these fractals can be generated by a random process that flips a coin each time to decide the orientation of the triangle, leading to 'almost sure' results for their geometrical properties, we define and study them deterministically to provide exact results. In particular, we show, using the theory of non-integer expansions, that the set of all possible values for the area enclosed by these generalised Koch curves is a closed interval. Moreover, we prove that the union of all these generalised snowflakes does not contain an open set, and has zero $2$-dimensional Lebesgue measure. Complementing these results, using arguments from calculus and fractal geometry, namely properties of geometric series and Frostman's Lemma, we show that each generalised Koch curve has infinite length and the same Hausdorff dimension as its classical counterpart. Further, we also give a classification for when a generalised Koch curve is a quasicircle.