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.