<story><title> A new kind of fractal?</title><url>http://www.gibney.de/does_anybody_know_this_fractal</url><text>I stumbled across this and wonder what it is.</text></story><parent_chain><item><author>madhadron</author><text>It&apos;s not a fractal, but it is something familiar.&lt;p&gt;Multiply two complex numbers z and c is equivalent to taking z and applying a rotation and dilation to it, the rotation through arg(c) and the dilation through |c|. Division is the inverse of both, so z/c is z rotated by -arg(c) and dilated by 1/|c|.&lt;p&gt;What you&apos;re looking at, then, is taking the operation defined by c (rotate by -arg(c) and dilate by 1/|c|) and asking, if you take the Gaussian integers as the vertexes of a directed graph, what fraction of the vertexes are the source of an edge.&lt;p&gt;Consider the 1 dimensional analogy using real numbers. Given some real number c, take all the integers as the vertexes of a graph, and if z/c (for some integer z) is also an integer, I put a directed edge from z to z/c. When are these connected? Well, if c is irrational, never. If c is rational, then there will be an infinite number of connections, but how infinite? When c is 2, there will be twice as many edges on average in any subset of the source vertexes as when c is 4. If we can write c as p/q, then the smaller p is, the more edges we&apos;ll get, and the brighter the pixel in your image.&lt;p&gt;The 1 dimensional analogy will have a spike at 1/2, smaller spikes at 1/3 and 2/3, yet smaller spikes at 1/4 and 3/4, smaller ones yet at 1/5, 2/5, 3/5, and 4/5, etc. The spikes will all be distinct (because between any two rationals there is an irrational), but will be infinitely close (because the rationals are dense in the reals). As you keep zooming in, you will get more and more edges like this.&lt;p&gt;What you&apos;re seeing is a variation on the classical structure of the rationals dense within the reals.&lt;p&gt;Now, a fractal is a set with a fractional Hausdorff dimension. We have to extract a set from your function of c in order to talk about fractal dimension. We could take the support of the function (everywhere it&apos;s not zero). In the one dimensional case, that&apos;s the rationals. We could take level sets farther up (the set of c such that f(c) = k, for a constant k). Those are subsets of the rationals. However, the rationals, while dense in the reals, are of measure zero in the reals, and have Hausdorff dimension zero, and so do all the level sets. So it&apos;s not a fractal.&lt;p&gt;Doesn&apos;t make it any less pretty though.</text></item></parent_chain><comment><author>ChristianMarks</author><text>I don&apos;t think this has anything to do with directed graphs. The operation looks like this. For nonzero &lt;i&gt;c&lt;/i&gt; in &lt;i&gt;C&lt;/i&gt;,  compute the distance &lt;i&gt;d(a(i+1)/c, Z+iZ)&lt;/i&gt; to the nearest Gaussian integer, where &lt;i&gt;a&lt;/i&gt; is an integer.&lt;p&gt;The max distance is &lt;i&gt;A = 1/sqrt(2)&lt;/i&gt; so color the point &lt;i&gt;a(1+i)/c&lt;/i&gt;  proportionally by&lt;p&gt;&lt;pre&gt;&lt;code&gt;  (A - d(a(i+1)/c, Z+iZ))/A
&lt;/code&gt;&lt;/pre&gt;
I&apos;m taking a liberty with the description, since computing &quot;the percentage of &lt;i&gt;a(i+1)/c&lt;/i&gt;&apos;s that are Gaussian integers&quot; is  open to interpretation, and if taken literally could mean something else entirely, such as computation of the ratio&lt;p&gt;&lt;pre&gt;&lt;code&gt;  #(A(c,n)\cap Z+iZ) / n
&lt;/code&gt;&lt;/pre&gt;
where&lt;p&gt;&lt;pre&gt;&lt;code&gt;  A(c, n) = { a(i+1)/ c : 1 &amp;#60;= a &amp;#60;= n }
&lt;/code&gt;&lt;/pre&gt;
which is asking for a proportion products that become Gaussian integers--this is a divisibility test. However, this doesn&apos;t show how to vary the brightness of the images.</text></comment>
<story><title> A new kind of fractal?</title><url>http://www.gibney.de/does_anybody_know_this_fractal</url><text>I stumbled across this and wonder what it is.</text></story><parent_chain><item><author>madhadron</author><text>It&apos;s not a fractal, but it is something familiar.&lt;p&gt;Multiply two complex numbers z and c is equivalent to taking z and applying a rotation and dilation to it, the rotation through arg(c) and the dilation through |c|. Division is the inverse of both, so z/c is z rotated by -arg(c) and dilated by 1/|c|.&lt;p&gt;What you&apos;re looking at, then, is taking the operation defined by c (rotate by -arg(c) and dilate by 1/|c|) and asking, if you take the Gaussian integers as the vertexes of a directed graph, what fraction of the vertexes are the source of an edge.&lt;p&gt;Consider the 1 dimensional analogy using real numbers. Given some real number c, take all the integers as the vertexes of a graph, and if z/c (for some integer z) is also an integer, I put a directed edge from z to z/c. When are these connected? Well, if c is irrational, never. If c is rational, then there will be an infinite number of connections, but how infinite? When c is 2, there will be twice as many edges on average in any subset of the source vertexes as when c is 4. If we can write c as p/q, then the smaller p is, the more edges we&apos;ll get, and the brighter the pixel in your image.&lt;p&gt;The 1 dimensional analogy will have a spike at 1/2, smaller spikes at 1/3 and 2/3, yet smaller spikes at 1/4 and 3/4, smaller ones yet at 1/5, 2/5, 3/5, and 4/5, etc. The spikes will all be distinct (because between any two rationals there is an irrational), but will be infinitely close (because the rationals are dense in the reals). As you keep zooming in, you will get more and more edges like this.&lt;p&gt;What you&apos;re seeing is a variation on the classical structure of the rationals dense within the reals.&lt;p&gt;Now, a fractal is a set with a fractional Hausdorff dimension. We have to extract a set from your function of c in order to talk about fractal dimension. We could take the support of the function (everywhere it&apos;s not zero). In the one dimensional case, that&apos;s the rationals. We could take level sets farther up (the set of c such that f(c) = k, for a constant k). Those are subsets of the rationals. However, the rationals, while dense in the reals, are of measure zero in the reals, and have Hausdorff dimension zero, and so do all the level sets. So it&apos;s not a fractal.&lt;p&gt;Doesn&apos;t make it any less pretty though.</text></item></parent_chain><comment><author>beagle3</author><text>&amp;#62; Now, a fractal is a set with a fractional Hausdorff dimension.&lt;p&gt;Is there an authoritative definition of a fractal? The one you use rules out structures like Hilbert curves, which are generally considered fractals.</text></comment>