[math-fun] Hilbert function as images
I wrote up a little program to generate .xpm images of iterations of Hilbert where pixels are of this form: 32 01 Where 0-3 are either shades of gray or R,Y,G,B. Sources, binaries and PNG images here, with both images in their original (mostly tiny) sizes and scaled to a uniform 2048x2048 size: http://lunkwill.org/src/hilbert/ So, for instance, you might go here: http://lunkwill.org/src/hilbert/color/ Then look at these: http://lunkwill.org/src/hilbert/color/scaled-1.xpm.png http://lunkwill.org/src/hilbert/color/scaled-2.xpm.png ...
It might be interesting if the pixels were computed additively, picking up some color rotation or darkness at each level of iteration. We might see a path that traced out the square, while going from white to black or red to blue. Rich -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Jason Sent: Sunday, April 13, 2008 6:23 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Hilbert function as images I wrote up a little program to generate .xpm images of iterations of Hilbert where pixels are of this form: 32 01 Where 0-3 are either shades of gray or R,Y,G,B. Sources, binaries and PNG images here, with both images in their original (mostly tiny) sizes and scaled to a uniform 2048x2048 size: http://lunkwill.org/src/hilbert/ So, for instance, you might go here: http://lunkwill.org/src/hilbert/color/ Then look at these: http://lunkwill.org/src/hilbert/color/scaled-1.xpm.png http://lunkwill.org/src/hilbert/color/scaled-2.xpm.png ... _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I tried Rich's suggestion of accumulating color with the recursion, and... it looks amazing! (9-11 are prettiest.) This would make a really neat quilt or stained-glass window, if you could get the materials. Actually, the window might be doable given an accurate ability to add tint to the glass -- just follow the recursions down. http://lunkwill.org/src/hilbert/additive/ Source: http://lunkwill.org/src/hilbert/hilbert-additive.c
Printing shops should be able to render the picture on transparent plastic. With your choice of colors the result should look quite pretty! Note that printing on paper is not likely to result in a neat result. * Jason <jason@lunkwill.org> [Apr 17. 2008 15:20]:
I tried Rich's suggestion of accumulating color with the recursion, and... it looks amazing! (9-11 are prettiest.) This would make a really neat quilt or stained-glass window, if you could get the materials. Actually, the window might be doable given an accurate ability to add tint to the glass -- just follow the recursions down.
http://lunkwill.org/src/hilbert/additive/
Source: http://lunkwill.org/src/hilbert/hilbert-additive.c
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It already looks like it IS a quilt -- strange 3d effect. Also, there seems to be a bit of an optical illusion in that certain lines that must be parallel look like they are not parallel. On Wed, Apr 16, 2008 at 11:32 PM, Jason <jason@lunkwill.org> wrote:
I tried Rich's suggestion of accumulating color with the recursion, and... it looks amazing! (9-11 are prettiest.) This would make a really neat quilt or stained-glass window, if you could get the materials. Actually, the window might be doable given an accurate ability to add tint to the glass -- just follow the recursions down.
This is related to the square pyramid thread, since it appears that the number of colors in your squares 3.xpm, 4.xpm, 5.xpm ... goes 20,35,56,84,120,165,220,286,364 ... so it appears that the color-count of n.xmp equals the number of balls in a triangular pyramid with edge n+1 balls, namely C(n+3,3) = (n+1)*(n+2)*(n+3)/6. It appears that your squares will have a square number of colors only when n=2 or n=47. But here the pyramid is triangular, not square. This is also an integer points on an elliptic curve problem. Is it in some since the same curve, or is proving this a different problem? On Thu, Apr 17, 2008 at 8:52 AM, James Buddenhagen <jbuddenh@gmail.com> wrote:
It already looks like it IS a quilt -- strange 3d effect. Also, there seems to be a bit of an optical illusion in that certain lines that must be parallel look like they are not parallel.
On Wed, Apr 16, 2008 at 11:32 PM, Jason <jason@lunkwill.org> wrote:
I tried Rich's suggestion of accumulating color with the recursion,
and... it
looks amazing! (9-11 are prettiest.) This would make a really neat quilt or stained-glass window, if you could get the materials. Actually, the window might be doable given an accurate ability to add tint to the glass -- just follow the recursions down.
It's much easier to see the structure in the colored ones than in the monochrome ones because of a happy coincidence: yellow and green are brighter than blue and red and they're adjacent, respectively, which means that at iteration 2, there's a dark C-shaped curve on a pale background. This echoes the C shape of the underlying unit, but makes it stand out from the background. If you re-render your monochrome ones but use black for 0,3 and white for 1,2, you'll get much clearer structure. On Sun, Apr 13, 2008 at 5:23 PM, Jason <jason@lunkwill.org> wrote:
I wrote up a little program to generate .xpm images of iterations of Hilbert where pixels are of this form:
32 01
Where 0-3 are either shades of gray or R,Y,G,B. Sources, binaries and PNG images here, with both images in their original (mostly tiny) sizes and scaled to a uniform 2048x2048 size:
http://lunkwill.org/src/hilbert/
So, for instance, you might go here: http://lunkwill.org/src/hilbert/color/
Then look at these: http://lunkwill.org/src/hilbert/color/scaled-1.xpm.png http://lunkwill.org/src/hilbert/color/scaled-2.xpm.png ...
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
participants (5)
-
James Buddenhagen -
Jason -
Joerg Arndt -
Mike Stay -
Schroeppel, Richard