How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension

@article{Mandelbrot1967HowLI,
  title={How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension},
  author={Benoit B. Mandelbrot},
  journal={Science},
  year={1967},
  volume={156},
  pages={636 - 638}
}
Geographical curves are so involved in their detail that their lengths are often infinite or, rather, undefinable. However, many are statistically "selfsimilar," meaning that each portion can be considered a reduced-scale image of the whole. In that case, the degree of complication can be described by a quantity D that has many properties of a "dimension," though it is fractional; that is, it exceeds the value unity associated with the ordinary, rectifiable, curves. 
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References

SHOWING 1-6 OF 6 REFERENCES

Self-Similar Error Clusters in Communication Systems and the Concept of Conditional Stationarity

A model of certain random perturbations that appear to come in clusters, or bursts is presented by introducing the concept of "self-similar stochastic point process in continuous time" and presents fascinating peculiarities from the mathematical viewpoint.

General systems : yearbook of the Society for General Systems Research

Very similar considerations apply in turbulence, where the characteristic sizes of the "features

  • IEEE Inst. Elect. Electroni. Enig. Trans. Informn. Theory
  • 1965

or in The Randomn Character of Stock Market Prices

  • Business
  • 1963

The concept of "dimension" is elusive and very

    Elect. Electron. Eng. Trans. Commun. Technol

    • Elect. Electron. Eng. Trans. Commun. Technol
    • 1965