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

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

2004-12-09

arxiv.org/abs/math-ph/0412029v2

J. London Math. Soc. (2) 74 (2006), 397-414

Physics

Mathematical Physics

Updated version: new method of calculation drastically reduces length of proof. 18 pages, 10 figures

Scientific paper

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.

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