A gasket construction of the Koch snowflake and variations

TL;DR

The paper presents a gasket-based construction of the Koch snowflake using rhombi of aspect ratio $a$, yielding a fourfold decomposition and rectangular symmetry that forms a continuous family of fractal curves $F^a$. It proves that each $F^a$ is a Jordan curve, and that for $a = 1/\sqrt{3}$ the gasket curve reproduces the standard Koch snowflake, establishing equivalence with the classical construction at this parameter value. The Hausdorff dimension of $F^a$ is derived from a self-similar relation and is maximized at $a = 1/\sqrt{3}$ with $\dim_H F^a = \log_3 4$, while the area of the region enclosed by $F^a$ is given by $\text{area}(F^a) = \dfrac{8a}{1+4a^2-a^4}$ and remains zero for the curve itself. The work provides a new perspective on Koch-type fractals by parameterizing symmetry, dimension, and area through the aspect ratio and demonstrates a precise correspondence to the classic snowflake at a specific limit case.

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.

Figures (14)

• Figure 1: The fractal created by the gasket construction at different values of $a$, where $a$ is the aspect ratio of the rhombi in the construction.
• Figure 2: The standard construction of the Koch snowflake.
• Figure 3: The Koch snowflake as the union of four congruent self-similar curves. Each of the four curves decomposes into three parts similar to the whole.
• Figure 4: The first two iterations of the gasket construction. The shaded regions in iterations 1 and 2 are the empty polygons.
• Figure 5: The next few iterations of the gasket construction and the final result.

Theorems & Definitions (19)

• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Theorem 2.5
• proof
• Definition 3.1
• Theorem 3.2