A Tube Formula for the Koch Snowflake Curve, with Applications to Complex Dimensions

A formula for the interior e-neighbourhood of the classical von Koch snowflake curve is computed in detail. This function of e is shown to match quite closely with earlier predictions from [La-vF1] 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.

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