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Journal of the London Mathematical Society 2006 74(2):397-414; doi:10.1112/S0024610706022988
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© The London Mathematical Society 2006

A TUBE FORMULA FOR THE KOCH SNOWFLAKE CURVE, WITH APPLICATIONS TO COMPLEX DIMENSIONS

Michel L. Lapidus and Erin P. J. Pearse

Department of Mathematics, University of California Riverside, CA 92521-0135, USA lapidus{at}math.ucr.edu
Department of Mathematics, University of California Riverside, CA 92521-0135, USA erin{at}math.ucr.edu

Received 22 November 2004. Revision received 4 November 2005.

A formula for the interior {varepsilon}-neighborhood of the classical von Koch snowflake curve is computed in detail. This function of {varepsilon} 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 analog 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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