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There are several ways of defining a fractal, and a reader will need to reference their source to see which definition is being used. Perhaps the simplest definition is to define a \emph{fractal} to be a subset of $\mathbb{R}^n$ with non-integral Hausdorff dimension. See, for example, the Koch snowflake. |
| There are several ways of defining a fractal, and a reader will need to reference their source to see which definition is being used. |
Alternatively, we can slightly more pedantically (but more rigorously) define fractals through an equivalence relation on subsets of $\R^n$ by defining a distance relation: Let $F,G\subset \R^n$, and let $d(x,y)$ be the usual distance on $\R^n$. Define the distance $D$ between $F$ |
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and $G$ as |
| Perhaps the simplest definition is to define a \emph{fractal} to be a subset of $\mathbb{R}^n$ with Hausdorff dimension greater than its topological dimension. It is worth noting that typically (but not always), fractals have non-integer Hausdorff dimension. See, for example, the Koch snowflake and the Mandelbrot set (named after Benoit Mandelbrot, who also coined the term ``fractal" for these objects). |
\[ D(F,G) := \sup_{f\in F} \inf_{g\in G} d(f,g) + \sup_{g\in G} \inf_{f\in F} d(f,g)\]. |
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Two subsets $F,G$ of $\R^n$ are said to be equivalent as fractals if $D(F,G)=0$, and a \emph{fractal} is defined as an equivalence class of subsets under this equivalence. |
| A looser definition of a \emph{fractal} is a ``self-similar object". That is, a subset or $\R^n$ which can be covered by copies of itself using a set of (usually two or more) transformation mappings. Another way to say this would be ``an object with a discrete approximate scaling symmetry." |
For example, $\Q$ and $\R$ are equivalent as fractals. |
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Finally, a looser definition of a \emph{fractal} is a ``self-similar object". That is, a subset or $\R^n$ which can be covered by copies of itself using a set of (usually two or more) transformation mappings. |
| See also the discussion near the end of the entry \PMlinkname{Hausdorff dimension}{HausdorffDimension}. |
Another way to say this would be ``an object with a discrete approximate scaling symmetry." |
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For example, Square regions, Koch curves, and fern fronds fit this definition. |
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A cursory description of some relationships between some of these definitions is given towards the end of the entry on \PMlinkname{Hausdorff dimension}{HausdorffDimension}. |
