Editing Fractal dimension (section)

== Mathematical definition ==
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[[Image:Fractaldimensionexample-2.png|right|thumb|alt=Lines, squares, and cubes.|Figure 4. Traditional notions of geometry for defining scaling and dimension.<br/>
<math>1</math>, <math>1^2 = 1</math>, <math>1^3 = 1;</math><br/>
<math>2</math>, <math>2^2 = 4</math>, <math>2^3 = 8;</math><br/>
<math>3</math>, <math>3^2 = 9</math>, <math>3^3 = 27.</math><ref>Appignanesi, Richard; ed. (2006). ''Introducing Fractal Geometry'', p.&nbsp;28<!--shows division by 2 and 3-->. Icon. {{ISBN|978-1840467-13-0}}.</ref>]]The mathematical definition of fractal dimension can be derived by observing and then generalizing the effect of traditional dimension on measurement-changes under scaling.<ref name = fil>{{cite book |author=Iannaccone, Khokha |year=1996 |title=Fractal Geometry in Biological Systems |publisher=CRC Press |isbn=978-0-8493-7636-8}}</ref> For example, say you have a line and a measuring-stick of equal length. Now shrink the stick to 1/3 its size; you can now fit 3&nbsp;sticks into the line. Similarly, in two dimensions, say you have a square and an identical "measuring-square". Now shrink the measuring-square's side to 1/3 its length; you can now fit 3^2 = 9&nbsp;measuring-squares into the square. Such familiar scaling relationships obey equation&nbsp;{{EqNote|1}}, where  <math>\varepsilon</math> is the scaling factor, <math>D</math> the dimension, and <math>N</math> the resulting number of units (sticks, squares, etc.) in the measured object:{{NumBlk|:|<math>N = \varepsilon^{-D}.</math>|{{EquationRef|1}}}}

In the line example, the dimension <math>D = 1</math> because there are <math>N = 3</math> units when the scaling factor <math>\varepsilon = 1/3</math>. In the square example, <math>D = 2</math> because <math>N = 9</math> when <math>\varepsilon = 1/3</math>.

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[[Image:KochFlake.svg|right|thumb|alt=A fractal contour of a koch snowflake|Figure 5. The first four [[iteration]]s of the [[Koch snowflake]], which has a [[Hausdorff dimension]] of approximately 1.2619.]]

Fractal dimension generalizes traditional dimension in that it can be fractional, but it has exactly the same relationship with scaling that traditional dimension does; in fact, it is derived by simply rearranging equation {{EqNote|1}}:

{{NumBlk|:|<math>D = -\log_\varepsilon N = -\frac{\log N}{\log \varepsilon}.</math>|{{EquationRef|2}}}}

<math>D</math> can be thought of as the power of the scaling factor of an object's measure given some scaling of its "radius". 

For example, the [[Koch snowflake]] has <math>D = 1.26185\ldots</math>, indicating that lengthening its radius grows its measure faster than if it were a one-dimensional shape (such as a polygon), but slower than if it were a two-dimensional shape (such as a filled polygon).<ref name="vicsek" />

Of note, images shown in this page are not true fractals because the scaling described by <math>D</math> cannot continue past the point of their smallest component, a pixel. However, the theoretical patterns that the images represent have no discrete pixel-like pieces, but rather are composed of an [[Infinity|infinite]] number of infinitely scaled segments and do indeed have the claimed fractal dimensions.<ref name="Mandelbrot1983" /><ref name="fil" />