I should have also added: With modern digital computer screens, the individual dots are "too" sharp, so you can actually see the square edges. Back in the days of CRT's (only available today in Chinese remainder bins) you could adjust the dots to overlap, to perform part of the task of the "reconstruction" filter. So with today's too-sharp computer screens, your own eyes form the last part of the "reconstruction" chain. The blurring of the dots can be performed by simply holding the screen far enough away so that the individual dots can no longer be distinguished. At 07:27 AM 3/22/2012, Henry Baker wrote:

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