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

Published online by Cambridge University Press:  25 October 2006

MICHEL L. LAPIDUS  and
ERIN P. J. PEARSE
MICHEL L. LAPIDUS
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521-0135, USAlapidus@math.ucr.edu
ERIN P. J. PEARSE
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521-0135, USAerin@math.ucr.edu
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Abstract

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.

Keywords

26A3028A1228A7528A80 (primary)11K5526A2728A7828D2042A16 (secondary)
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 74 , Issue 2 , October 2006 , pp. 397 - 414
Copyright
The London Mathematical Society 2006

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