Do the circles get smaller as d range increases? or just get higher resolution?
More and more points fill in, so the circles appear to get smaller, and there are also more of them. See the new image below.
Does the pattern only show up when you normalise the x,y,z? or do the actual x,y,z form a pattern? like spheres?
The raw (x,y,z) probably form some kind of nice pattern, but I'm not sure how to plot that in a way that would be clear. I think it would be a 3D version of this 2D picture
http://en.wikipedia.org/wiki/File:Pythagorean_triple_scatterplot2.png. Imagine all those points normalized onto the unit circle, and that's analogous to what I did.
Am I correct in saying that you find every integer solution x^2 + y^2 + z^2 = d^2 and normalise the vector then colour that point blue on the unit sphere?
Yeah, that's right, except I actually colored them black above, not blue. So because the magnitude of (x,y,z) is d, the points that actually get plotted are (x/d,y/d,z/d). First I just drew a blank shaded sphere for background, then I plotted all the points over it.
Why the little dot in each circle?
The little dots in each circle are solutions to the equation that don't have other solution points nearby. This tends to happen for solutions where d is small. These "isolated" points also tend to get plotted multiple times. For example, consider the solution x=1, y=2, z=2, d=3. This is a solution because
. This gets plotted as a point at (1/3, 2/3, 2/3), which is a unit vector. Then the following would also be solutions:
etc., and all these solutions would get plotted as a point at (1/3, 2/3, 2/3). For some reason, other solutions tend to avoid the area around points with smaller denominators. You can see this effect in the 2D-version Wikipedia image I linked above too. The rays emanating from the origin tend to have extra white space around them.
Why the black outline of each circle?
The black outlines are where lots of solution points cluster together enough to look black.
Would there be a visual pattern if the points were coloured by the value of d?
What effect with x^3 etc?
Does it work with fractional exponent? x^2.3 + ...
What happens to the pattern if you normalise each point to (a,b,c) instead of (0,0,0)?
I haven't looked into any of these. I'm not sure if I understand the last question.
Here's a new picture, this time it's all solutions for -6144 <= x,y,z <= 6144. Lots more points are filling in the gaps, so it ends up looking kinda boring
http://www.fractalforums.com/gallery/?sa=view;id=1189The thin crosshair-like lines on the x and y axes are there because I accidentally used a rounding function that has a glitch at zero. Those lines shouldn't be there.