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3
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'fractal'
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| Title of object: |
fractal |
| Canonical Name: |
Fractal |
| Type: |
Definition |
| Created on: |
2002-05-31 16:34:19-04 |
| Modified on: |
2002-05-31 16:36:22-04 |
| Classification: |
msc:28A80 |
| Defines: |
Hausdorff dimension, fractal |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
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%\usepackage{xypic}
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\newcommand{\R}{\mathbb R} |
Content:
Option 1: Some equvialence class of subsets of a manifold. Example: The area colored black in the canonical "happy face", $\mathbb{Q} \equiv \R$. (Usually sets are equivalent if some generalised "distance", like
\[d(F,G) := \inf_{f\in F} \sup_{g\in G} d(f,g) + \inf_{g\in G} \sup_{f\in F} d(f,g)\] is zero.)
Option 2: A subset of a manifold with non-integral Hausdorff dimension. Example: (we think) the coast of Britain, a Koch snowflake.
Option 3: A "self-similar object". That is, one which can be covered by copies of itself using a set of (usually at least two) transformation mappings. This isn't much different from option 1 because of something called the collage theorem. Example: A square region, a Koch curve, a fern frond.
Definition of Hausdorff dimension: let $N(\epsilon)$ be the minimum number of discs of radius $\epsilon$ required to cover some set $\Theta \subset M$ where $M$ is a manifold. Then define the Hausdorf dimension $d_H$ to be
\[ d_H := - lim_{\epsilon \rightarrow 0} \frac{\log N(\epsilon)}{\log \epsilon}.\]
For $\Theta$ good for Option 3, with all the relevant transformation mappings effecting the same scaling factor $\rho$ and if there are $n$ of them, $d_H = - \frac{log n}{log \rho}$ |
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