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Chaos and FractalsAn Elementary Introduction$
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David P. Feldman

Print publication date: 2012

Print ISBN-13: 9780199566433

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199566433.001.0001

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The Box‐Counting Dimension

The Box‐Counting Dimension

(p.187) 18 The Box‐Counting Dimension
Chaos and Fractals

David P. Feldman

Oxford University Press

There are several examples of fractals that are not exactly self-similar, as is the case with small parts of the random Koch curve which exhibit statistical self-similarity but not identicality. The dimension of a handful of fractal objects, including the Cantor set, the Sierpiński triangle and carpet, and the Koch curve, can be determined. This chapter considers the box-counting dimension, which extends the concept of dimension to objects that are not exactly self-similar. Instead of focusing on how many small copies of an object are contained in a large copy, it explains how the volume or size of the overall shape changes as measurement scales change.

Keywords:   fractals, self-similarity, box-counting dimension, Sierpiński triangle, Koch curve, Cantor set

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