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<div><h2 id="ex:mandelbrot">1.12.2 Colour Scale Bars</h2>
<p>By default, plots with colour maps with single-parameter colour mappings are accompanied by colour scale bars, which appear by default on the right-hand side of the plot. Such scale bars may be configured using the <tt class="tt">set colourkey</tt> command<a name="a0000000850" id="a0000000850"></a>. Issuing the command </p><pre>
set colourkey
</pre><p>by itself causes such a scale to be drawn on graphs in the default position, usually along the right-hand edge of the graphs. The converse action is achieved by: </p><pre>
set nocolourkey
</pre><p>The command </p><pre>
unset colourkey
</pre><p>causes PyXPlot to revert to its default behaviour, as specified in a configuration file, if present. A position for the key may optionally be specified after the <tt class="tt">set colourkey</tt> command, as in the example: </p><pre>
set colourkey bottom
</pre><p>Recognised positions are <tt class="tt">top</tt>, <tt class="tt">bottom</tt>, <tt class="tt">left</tt> and <tt class="tt">right</tt>. <tt class="tt">above</tt> is an alias for <tt class="tt">top</tt>; <tt class="tt">below</tt> is an alias for <tt class="tt">bottom</tt> and <tt class="tt">outside</tt> is an alias for <tt class="tt">right</tt>. </p><p>The format of the ticks along such scale bars may be set using the <tt class="tt">set c1format</tt> command<a name="a0000000851" id="a0000000851"></a> command, which is similar in syntax to the <tt class="tt">set xformat</tt> command (see Section <a href="sec-set_xformat.html">1.8.8</a>), but which uses <tt class="tt">c</tt> as its dummy variable. </p><p> <span class="upshape"><span class="mdseries"><span class="rm">An image of the Mandelbrot Set.</span></span></span></p><div>
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<td style="border-top-style:solid; border-left:1px solid black; border-right:1px solid black; border-top-color:black; border-top-width:1px; text-align:left"><p> The Mandelbrot set is a set of points in the complex plane whose boundary forms a fractal with a Hausdorff dimension of two. A point <img src="images/img-0040.png" alt="$c$" style="vertical-align:0px;
width:8px;
height:8px" class="math gen" /> in the complex plane is defined to lie within the Mandelbrot set if the complex sequence of numbers </p><table id="a0000000852" class="equation" width="100%" cellspacing="0" cellpadding="7">
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<td style="width:40%"> </td>
<td><img src="images/img-0526.png" alt="\[ z_{n+1} = z_ n^2 + c, \]" style="width:113px;
height:21px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
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</table><p> subject to the starting condition <img src="images/img-0527.png" alt="$z_0=0$" style="vertical-align:-2px;
width:49px;
height:14px" class="math gen" />, remains bounded. </p></td>
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<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p>The map of this set of points has become a widely-used image of the power of chaos theory to produce complicated structure out of simple algorithms. To produce a more pleasing image, points in the complex plane are often coloured differently, depending upon how many iterations <img src="images/img-0014.png" alt="$n$" style="vertical-align:0px;
width:11px;
height:8px" class="math gen" /> of the above series are required for <img src="images/img-0528.png" alt="$|z_ n|$" style="vertical-align:-5px;
width:24px;
height:18px" class="math gen" /> to exceed 2. This is the point of no return, beyond which it can be shown that <img src="images/img-0529.png" alt="$|z_{n+1}|>|z_ n|$" style="vertical-align:-5px;
width:94px;
height:18px" class="math gen" /> and that divergence is guaranteed. In numerical implementations of the above iteration, in the absence of any better way to prove that the iteration remains bounded for a certain value of <img src="images/img-0040.png" alt="$c$" style="vertical-align:0px;
width:8px;
height:8px" class="math gen" />, some maximum number of iterations <img src="images/img-0312.png" alt="$m$" style="vertical-align:0px;
width:16px;
height:8px" class="math gen" /> is chosen, and the series is deemed to have remained bounded if <img src="images/img-0530.png" alt="$|z_ m|<2$" style="vertical-align:-5px;
width:63px;
height:18px" class="math gen" />. This is implemented in PyXPlot by the built-in mathematical function <tt class="tt">fractal_mandelbrot(z,m)</tt><a name="a0000000853" id="a0000000853"></a>, which returns an integer in the range <img src="images/img-0531.png" alt="$0\leq i\leq m$" style="vertical-align:-3px;
width:79px;
height:15px" class="math gen" />. </p></td>
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<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p><tt class="tt">set numerics complex</tt><br /><tt class="tt">set sample grid 500x500</tt><br /><tt class="tt">set size square</tt><br /><tt class="tt">set nokey</tt><br /><tt class="tt">set log c1</tt><br /><tt class="tt">plot [-2:2][-2:2] fractal_mandelbrot(x+i*y,70)+1 with colourmap</tt> </p></td>
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<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p>The resulting image is shown below: </p></td>
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<td style="border-bottom-style:solid; border-bottom-width:1px; border-left:1px solid black; border-right:1px solid black; text-align:left; border-bottom-color:black"><center>
<img src="images/img-0533.png" alt="\includegraphics[width=8cm]{examples/eps/ex_mandelbrot}" style="width:8cm" /> </center></td>
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