A gasket construction of the Koch snowflake and variations

TL;DR

This paper presents a new gasket-based construction of the Koch snowflake that results in a different symmetrical decomposition and introduces a family of fractal curves with rectangular symmetry, expanding understanding of fractal geometry.

Contribution

It introduces a novel gasket construction method for the Koch snowflake, creating a continuous family of fractals with rectangular symmetry and analyzing their Hausdorff dimensions.

Findings

The new construction splits the snowflake into four self-similar curves.

Varying rhombus shapes generates a continuous family of fractals.

Hausdorff dimension peaks at the original Koch snowflake.

Abstract

We introduce a construction of the Koch snowflake that is not inherently six-way symmetrical, based on iteratively placing similar rhombi. This construction naturally splits the snowflake into four identical self-similar curves, in contrast to the typical decomposition into three Koch curves. Varying the shape of the rhombi creates a continuous family of new fractal curves with rectangular symmetry. We compute the Hausdorff dimension of the generalized curve and show that it attains a maximum at the original Koch snowflake.