In one dimension, the phenomenon is called "aliasing". Basically, the sampling theorem _requires_ that all frequencies above 1/2 the sampling frequency be removed _completely_ prior to sampling, else you get aliasing of these higher frequencies down to lower frequencies during the process of reconstruction. In particular, frequencies just above 1/2 the sampling frequency become very low frequencies on reconstruction. In 2D these low frequencies are called Moire patterns. The reconstruction filter also has to be perfect (allow no frequencies above 1/2 the sampling frequency), else it may induce artifacts, as well. In practise, it is essentially impossible to achieve perfect pre-sampling and reconstruction filters, so there will always be small amounts of aliasing in practise. Also, human eyes & ears seem to tolerate very small amounts of aliasing in order to get slightly sharper results in the higher frequencies (but still below the filter cutoff frequency). http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem At 11:24 PM 3/21/2012, James Propp wrote:

Is there a good place to read about the mathematics of Moiré patterns?

Jim

On Wed, Mar 21, 2012 at 11:18 PM, Simon Plouffe <simon.plouffe@gmail.com>wrote:

...

Hello,

Yes, this is the pixel problem, it is caused by the square grid of the image made of pixels.

For example, when you draw lines that are close together on a pixel screen you get also Moiré patterns,

http://www.plouffe.fr/simon/**distributions%20modulo%201/** imagepages/image50.html<http://www.plouffe.fr/simon/distributions%20modulo%201/imagepages/image50.html>

Now the image of the sunflower and the Moiré is a mixed effect between the natural occuring spirals and the 'closeness' of the points which causes the pattern,

and of course : what is the formula or name of this pattern is a good question,

Best regards, Simon Plouffe

Le 22/03/2012 07:08, James Propp a écrit :

...

Back in September my friend Joshua Burton sent me this email. He and I and our mutual friend Michael Larsen exchanged a few emails about the phenomenon, but we never followed up:

http://www.cs.uml.edu/~jpropp/**sunflowers.html<http://www.cs.uml.edu/~jpropp/sunflowers.html>

(I'm sending the URL because one of the imbedded images is quite large.)

I asked Josh if it was okay to share this problem with others, and he replied: "By all means, share! The first to-do, I think, is to redo the work independently of the Mma engine, to confirm that the moire patterns are real, and not artifacts of some tool-specific rounding issue. If it's real, I guess the next thing is to come up with some numerical measure of the anomalous behavior of a big sunflower, as a function of N. That, or an actual clue what's going on."

Jim Propp