A method for evaluating the fractal dimension in the plane, using coverings with crosses

Claude Tricot

Fundamenta Mathematicae (2002)

  • Volume: 172, Issue: 2, page 181-199
  • ISSN: 0016-2736

Abstract

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Various methods may be used to define the Minkowski-Bouligand dimension of a compact subset E in the plane. The best known is the box method. After introducing the notion of ε-connected set E ε , we consider a new method based upon coverings of E ε with crosses of diameter 2ε. To prove that this cross method gives the fractal dimension for all E, the main argument consists in constructing a special pavement of the complementary set with squares. This method gives rise to a dimension formula using integrals, which generalizes the well known variation method for graphs of continuous functions.

How to cite

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Claude Tricot. "A method for evaluating the fractal dimension in the plane, using coverings with crosses." Fundamenta Mathematicae 172.2 (2002): 181-199. <http://eudml.org/doc/282752>.

@article{ClaudeTricot2002,
abstract = {Various methods may be used to define the Minkowski-Bouligand dimension of a compact subset E in the plane. The best known is the box method. After introducing the notion of ε-connected set $E_\{ε\}$, we consider a new method based upon coverings of $E_\{ε\}$ with crosses of diameter 2ε. To prove that this cross method gives the fractal dimension for all E, the main argument consists in constructing a special pavement of the complementary set with squares. This method gives rise to a dimension formula using integrals, which generalizes the well known variation method for graphs of continuous functions.},
author = {Claude Tricot},
journal = {Fundamenta Mathematicae},
keywords = {plane fractals; box-counting dimension; plane compact sets},
language = {eng},
number = {2},
pages = {181-199},
title = {A method for evaluating the fractal dimension in the plane, using coverings with crosses},
url = {http://eudml.org/doc/282752},
volume = {172},
year = {2002},
}

TY - JOUR
AU - Claude Tricot
TI - A method for evaluating the fractal dimension in the plane, using coverings with crosses
JO - Fundamenta Mathematicae
PY - 2002
VL - 172
IS - 2
SP - 181
EP - 199
AB - Various methods may be used to define the Minkowski-Bouligand dimension of a compact subset E in the plane. The best known is the box method. After introducing the notion of ε-connected set $E_{ε}$, we consider a new method based upon coverings of $E_{ε}$ with crosses of diameter 2ε. To prove that this cross method gives the fractal dimension for all E, the main argument consists in constructing a special pavement of the complementary set with squares. This method gives rise to a dimension formula using integrals, which generalizes the well known variation method for graphs of continuous functions.
LA - eng
KW - plane fractals; box-counting dimension; plane compact sets
UR - http://eudml.org/doc/282752
ER -

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