Title: Pythagorean Quadruples
Post by: BradC on December 15, 2009, 09:06:29 AM
Here's a plot of a bunch of points (x,y,z) (normalized) which are integer solutions to the equation 
. I plotted all solutions for 
.
Does anyone know why the resulting pattern has a flavor so similar to an Apollonian sphere packing? Is there a connection?
(http://upload.wikimedia.org/wikipedia/commons/thumb/9/95/Apollonian_spheres.jpg/180px-Apollonian_spheres.jpg)
(http://www.fractalforums.com/gallery/1/938_15_12_09_8_49_36.jpg)
http://www.fractalforums.com/gallery/?sa=view;id=1187Does anyone know why the resulting pattern has a flavor so similar to an Apollonian sphere packing? Is there a connection?
(http://upload.wikimedia.org/wikipedia/commons/thumb/9/95/Apollonian_spheres.jpg/180px-Apollonian_spheres.jpg)
Title: Re: Pythagorean Quadruples
Post by: Tglad on December 15, 2009, 12:50:58 PM
I can't comment on it technically, but my gut reaction is that's a pretty amazing thing, I wouldn't have expected that result.
Do the circles get smaller as d range increases? or just get higher resolution?
Looks like your sphere has 6 main vertices in a cube, then 24 after that, in a truncated cube.
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?
Do the circles get smaller as d range increases? or just get higher resolution?
Looks like your sphere has 6 main vertices in a cube, then 24 after that, in a truncated cube.
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?
Title: Re: Pythagorean Quadruples
Post by: kram1032 on December 15, 2009, 02:34:46 PM
my guess is because a sphere is made of circles. So it's not surprising that integer solutions give you a sphere of circles.
Also the hexahedric pattern isn't that surprising either as the pattern had to be symmetric over the euclidean axes....
Such patterns amaze me but they do not surprise me.
A way odder result would have been less likely to me...
Also the hexahedric pattern isn't that surprising either as the pattern had to be symmetric over the euclidean axes....
Such patterns amaze me but they do not surprise me.
A way odder result would have been less likely to me...
Title: Re: Pythagorean Quadruples
Post by: Tglad on December 16, 2009, 12:30:09 AM
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?
It certainly leads to interesting questions...
Why the little dot in each circle?
Why the black outline of each circle?
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)?
It certainly leads to interesting questions...
Why the little dot in each circle?
Why the black outline of each circle?
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)?
Title: Re: Pythagorean Quadruples
Post by: BradC on December 16, 2009, 03:26:21 AM
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 (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 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.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)?
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/1/938_16_12_09_2_51_18.jpg)
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.
Title: Re: Pythagorean Quadruples
Post by: kram1032 on December 16, 2009, 04:54:48 PM
well, here a density-like colouring might give nicer results.
simplest case would be to colour never-hit points black and multiple times hit points more and more white, with the maximum number of hits being white ;)
Also I guess what he meant with x³ is the 3-norm
^{1 \over 3 })
and the 2.3-norm
the n-norm is
^{1 \over n })
;)
simplest case would be to colour never-hit points black and multiple times hit points more and more white, with the maximum number of hits being white ;)
Also I guess what he meant with x³ is the 3-norm
and the 2.3-norm
the n-norm is
;)
Title: Re: Pythagorean Quadruples
Post by: BradC on December 16, 2009, 08:03:11 PM
well, here a density-like colouring might give nicer results.
simplest case would be to colour never-hit points black and multiple times hit points more and more white, with the maximum number of hits being white ;)
That's actually how I colored it. It's implemented as a big histogram, and as the number of points in a bin (pixel) increases, the coloring goes along a coloring gradient from black -> blue -> cyan -> white. Most of the points hadn't gotten past blue yet.simplest case would be to colour never-hit points black and multiple times hit points more and more white, with the maximum number of hits being white ;)
Title: Re: Pythagorean Quadruples
Post by: David Makin on December 16, 2009, 08:14:03 PM
well, here a density-like colouring might give nicer results.
simplest case would be to colour never-hit points black and multiple times hit points more and more white, with the maximum number of hits being white ;)
That's actually how I colored it. It's implemented as a big histogram, and as the number of points in a bin (pixel) increases, the coloring goes along a coloring gradient from black -> blue -> cyan -> white. Most of the points hadn't gotten past blue yet.simplest case would be to colour never-hit points black and multiple times hit points more and more white, with the maximum number of hits being white ;)
If IFS is any guide then using log(#hits) may work better ?
Title: Re: Pythagorean Quadruples
Post by: kram1032 on December 16, 2009, 08:19:14 PM
probably, yeah.
the hit-rate most likely is exponential-ish... :)
the hit-rate most likely is exponential-ish... :)
Title: Re: Pythagorean Quadruples
Post by: Tglad on December 16, 2009, 11:41:03 PM
Fascinating.
Of course, the reducing circle size is exactly what you'd expect on the surface of a sphere with Apollonian sphere packing as you up the resolution.
The size of the internal sphere represented by a circle could be inferred by the size of the surrounding small circles... but my guess is that these spheres are exactly the size needed to pack the unit sphere.
By the x^3 case, I meant, does the same thing happen if you plot all x^3 + y^3 + z^3 = d^3 solutions (normalised onto sphere).
>> What happens to the pattern if you normalise each point to (a,b,c) instead of (0,0,0)?
I meant, for every solution x,y,z, you plot the point (x-a, y-b, z-c)/magnitude(x-a, y-b, z-c) for some vector a,b,c. However, I think that the answer to this must be that the pattern doesn't change much.
I bet there's a way to render this as an actual sphere packing... probably something to do with plotting each solution divided by the greatest common divisor of x,y,z and d.
Of course, the reducing circle size is exactly what you'd expect on the surface of a sphere with Apollonian sphere packing as you up the resolution.
The size of the internal sphere represented by a circle could be inferred by the size of the surrounding small circles... but my guess is that these spheres are exactly the size needed to pack the unit sphere.
By the x^3 case, I meant, does the same thing happen if you plot all x^3 + y^3 + z^3 = d^3 solutions (normalised onto sphere).
>> What happens to the pattern if you normalise each point to (a,b,c) instead of (0,0,0)?
I meant, for every solution x,y,z, you plot the point (x-a, y-b, z-c)/magnitude(x-a, y-b, z-c) for some vector a,b,c. However, I think that the answer to this must be that the pattern doesn't change much.
I bet there's a way to render this as an actual sphere packing... probably something to do with plotting each solution divided by the greatest common divisor of x,y,z and d.