Re: [math-fun] big sunflowers
That's not exactly what I meant. Rather, one will see approximate short white line segments (in the complement of the dots) almost everywhere, since the dots are lined up parallel to two local axes. That's all I meant. The *details* of how these axes might line up, and why in some places they fit together to make apparently circular curves, would take a careful analysis of the specifics of the equation defining the locations of the dots, (r,theta) = (sqrt(n), n*(sqrt(5)-1)), n = 1,2,3,.... --Dan Allan wrote: << I think we are all agreed that most of the visible aliasing we see in the 50,000-dot image is due to pixel quantization. But Dan Asimov argues that even if we could get rid of pixel quantization, we would still see some moire-like effects. If I understand his argument, he is saying that in each area of the phi-based sunflower, the array of dots approximates the vertices of a lattice of parallelograms; the axis vectors of this lattice distort slowly as you move to nearby areas, and occasionally snap to a different set of axes. He anticipates "phase transitions" between domains governed by different axis vectors, and expects that these transitions will appear as visible discontinuities. I agree (again, hedging that I might not be following Dan's thoughts correctly) that different regions have different natural coordinate systems, but I disagree that the transitions will be abrupt or visible. Instead, I expect them to shade into each other imperceptibly; along the borders between these domains there will be regions that appear ambiguous, where one will be able to choose semiconsciouly (as in the Necker illusion) which lattice one sees.
________________________________________________________________________________________ It goes without saying that .
In that case, I'm back in complete agreement with you. On Thu, Mar 22, 2012 at 5:07 PM, Dan Asimov <dasimov@earthlink.net> wrote:
That's not exactly what I meant. Rather, one will see approximate short white line segments (in the complement of the dots) almost everywhere, since the dots are lined up parallel to two local axes. That's all I meant.
The *details* of how these axes might line up, and why in some places they fit together to make apparently circular curves, would take a careful analysis of the specifics of the equation defining the locations of the dots, (r,theta) = (sqrt(n), n*(sqrt(5)-1)), n = 1,2,3,....
--Dan
Allan wrote: << I think we are all agreed that most of the visible aliasing we see in the 50,000-dot image is due to pixel quantization. But Dan Asimov argues that even if we could get rid of pixel quantization, we would still see some moire-like effects.
If I understand his argument, he is saying that in each area of the phi-based sunflower, the array of dots approximates the vertices of a lattice of parallelograms; the axis vectors of this lattice distort slowly as you move to nearby areas, and occasionally snap to a different set of axes. He anticipates "phase transitions" between domains governed by different axis vectors, and expects that these transitions will appear as visible discontinuities.
I agree (again, hedging that I might not be following Dan's thoughts correctly) that different regions have different natural coordinate systems, but I disagree that the transitions will be abrupt or visible. Instead, I expect them to shade into each other imperceptibly; along the borders between these domains there will be regions that appear ambiguous, where one will be able to choose semiconsciouly (as in the Necker illusion) which lattice one sees.
________________________________________________________________________________________ It goes without saying that .
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Most responses to my post have focussed on the frequency-domain, but what about the specific patterns (circles, hyperbolas, etc.) that we see in the spatial-domain? At some point these need to be explained. In fact, even leaving aside sunflowers, I'd like to see a nice mathematical explanation of Moire patterns of the conventional kind. Has such a thing ever been written? You'd think so, but I wasn't able to find one anywhere on the web last time I checked! Jim Propp
This looks good http://diwww.epfl.ch/w3lsp/books/moire/prefaceKluwer.html Previewable on Amazon and if you dig, all the figures seem to be online On Fri, Mar 23, 2012 at 12:00 PM, James Propp <jamespropp@gmail.com> wrote:
Most responses to my post have focussed on the frequency-domain, but what about the specific patterns (circles, hyperbolas, etc.) that we see in the spatial-domain? At some point these need to be explained.
In fact, even leaving aside sunflowers, I'd like to see a nice mathematical explanation of Moire patterns of the conventional kind. Has such a thing ever been written? You'd think so, but I wasn't able to find one anywhere on the web last time I checked!
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
Looks great! Thanks. Jim On Fri, Mar 23, 2012 at 12:07 PM, Thane Plambeck <tplambeck@gmail.com>wrote:
This looks good
http://diwww.epfl.ch/w3lsp/books/moire/prefaceKluwer.html
Previewable on Amazon and if you dig, all the figures seem to be online
On Fri, Mar 23, 2012 at 12:00 PM, James Propp <jamespropp@gmail.com> wrote:
Most responses to my post have focussed on the frequency-domain, but what about the specific patterns (circles, hyperbolas, etc.) that we see in the spatial-domain? At some point these need to be explained.
In fact, even leaving aside sunflowers, I'd like to see a nice mathematical explanation of Moire patterns of the conventional kind. Has such a thing ever been written? You'd think so, but I wasn't able to find one anywhere on the web last time I checked!
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
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participants (4)
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Allan Wechsler -
Dan Asimov -
James Propp -
Thane Plambeck