Editing Fractal (section)

==Introduction==
[[File:Simple Fractals.png|thumb|right| A simple fractal tree|200x200px]]
[[File:FractalTree.gif|thumb|A fractal "tree" to eleven iterations]]

The word "fractal" often has different connotations for mathematicians and the general public, where the public is more likely to be familiar with [[fractal art]] than the mathematical concept. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background.

The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in the [[infinite regress]] in parallel mirrors or the [[homunculus]], the little man inside the head of the little man inside the head ...). The difference for fractals is that the pattern reproduced must be detailed.<ref name="Mandelbrot1983" />{{rp|166; 18}}<ref name="Falconer" /><ref name="Mandelbrot quote" />

This idea of being detailed relates to another feature that can be understood without much mathematical background: Having a [[fractal dimension]] greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric [[shapes]] are usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure is [[rep-tile]]d into pieces each 1/3 the length of the original, then there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled into pieces each scaled down by a factor of 1/3 in both dimensions, there are a total of 3<sup>2</sup> = 9 pieces.

We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/''r'', there are a total of ''r''<sup>''n''</sup> pieces. Now, consider the [[Koch curve]]. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So, strictly by analogy, we can consider the "dimension" of the Koch curve as being the unique real number ''D'' that satisfies 3<sup>''D''</sup> = 4. This number is called the ''fractal dimension'' of the Koch curve; it is not the conventionally perceived dimension of a curve. In general, a key property of fractals is that the fractal dimension differs from the ''conventionally understood'' dimension (formally called the topological dimension).

[[File:3D Computer Generated Fractal.png|thumb|3D computer-generated fractal]]
This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere [[Differentiable function|differentiable]]". In a concrete sense, this means fractals cannot be measured in traditional ways.<ref name="Mandelbrot1983" /><ref name="vicsek" /><ref name="Gordon" /> To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of [[rectifiable curve|measuring]] with a tape measure. But in measuring an infinitely "wiggly" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e. the snowflake has an infinite perimeter.<ref name="Mandelbrot1983" />

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