### Abstract

Richardson's empirical equation relating length to measuring scale has been used extensively to assign fractal dimensions to rugged curves. The inconsistency in units, length on one side of the equation vs. fractional units of length on the other has led to considerable debate. This letter presents a derivation of Richardson's equation based upon measure theory. In this context, the constant appearing in Richardson's equation is shown to have a precise meaning; it is the Hausdorff measure of the curve.

Original language | English (US) |
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Journal | Materials Science and Engineering A |

Volume | A186 |

Issue number | 1-2 |

State | Published - Jan 1 1994 |

### All Science Journal Classification (ASJC) codes

- Materials Science(all)
- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

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## Cite this

Mackin, T. J., Chen, C. T., Mecholsky, J. J., & Runt, J. P. (1994). Measure theoretic derivation of Richardson's equation.

*Materials Science and Engineering A*,*A186*(1-2).