A tube formula for the Koch snowflake curve, with applications to complex dimensions

TL;DR

This paper derives a precise tube formula for the Koch snowflake's interior neighborhoods, linking it to complex dimensions and Fourier analysis, enhancing understanding of fractal geometry and self-similarity.

Contribution

It provides an explicit, detailed tube formula for the Koch snowflake, connecting geometric properties with complex dimensions and Fourier coefficients, advancing fractal analysis methods.

Findings

The tube formula closely matches earlier predictions.

Explicit computation of the Koch snowflake's complex dimensions.

The formula reflects the self-similar structure of the curve.

Abstract

A formula for the interior epsilon-neighborhood of the classical von Koch snowflake curve is computed in detail. This function of epsilon is shown to match quite closely with earlier predictions of what it should be, but is also much more precise. The resulting `tube formula' is expressed in terms of the Fourier coefficients of a suitable nonlinear and periodic analogue of the standard Cantor staircase function and reflects the self-similarity of the Koch curve. As a consequence, the possible complex dimensions of the Koch snowflake are computed explicitly.