Moire pattern (1st of January 2011)
Moire, moire, moire ... Yes, but moire what? Moire net? Moire effect? What? And then I've seen a weird movie
>> Pi by Darren Aronofsky and it came to me -
Moire pattern! Because that is what it is about, it is about finding patterns in complexity. Obsessed mathematician
from the Pi movie searches for the pattern in wrong places, using wrong methods, but the eye and the part of the brain
dedicated to processing visual information discover patterns in complexity really easy because they evolved to do
just that. In fact, human brain is so "obsessed" with search for visual patterns that it discovers them even in
places where there are none - this is the basis of visual trickery.
Moire pattern appears when two periodic and slightly different visual information (nets) are shown simultaneously, one
over the other. The nets can differ either in small relative rotation or in the periodicity parameter, i.e. the size of
the elementary, unit cell. In such cases, the eye registers periodicities in the composite information, i.e. places
where the information repeats locally in both nets. On the image above two hexagonal nets are shown (red and blue).
In the red net, the side of a hexagon has the length of 1, and in the blue net, the hexagon side is 1.07 long. If
you analyze the image in some detail, it will seem to you that you can see a new periodicity in the composite information
that is in horizontal direction 42 blue units i.e. 45 red units long. Calculating the length in question, we find
42 * 1.07 = 44.94 (blue net)
We thus conclude that after 45 units of length, the red and the blue net approximately overlap, i.e. starting from a hexagon center, both nets have a center of another hexagon after that distance. The same holds if we examine repetitions in the direction angled at 30 degrees from the horizontal. We note that the patterns in the composite information "repeat" after 15 * 31/2 = 25.981 units (red net) and after 14 * 31/2 * 1.07 = 25.946 (blue net).
An astute reader could have concluded that the eye indeed discovered a pattern, but only an approximate one, i.e. that the overlapping of the nets is not exact but only approximate. This means that after several repetitions of this approximate periodicity (e.g. on the scale 10 or 50 times larger than the one we see in the images), the error would amplify so that the centers of blue and red hexagons would not repeat their overlapping even in an approximate sense. This also means that the pattern in the composite information is limited in its extent/radius around the point where the both nets exactly overlap, which is in our case in the centers of the images. To me, it is interesting that the eye is sensitive to the discussed "approximateness" and that it sees it clearly in the complex pattern. Another Moire structure is shown on the image above, and the nets this time have characteristic lengths of 1 (red) and 1.12345367 (blue). Try to relate these two numbers with the periodicity in the composite information you see/receive, it shouldn't be difficult.
But, why would we stop on only two nets, why not consider what happens in case of three nets whose sizes differ a bit? One such experiments is shown in the image above, and characteristic sizes of the nets are 1 (red), 1.12345367 (blue) and 1.07349367 (green). Interesting! And much more complex than in the case of only two nets. The reasons for that are clear: the eye now has a chance to note more overlapping, three times more: red-blue, red-green and blue-green overlapping.
Above I show three hexagonal nets with parameters 1 (red), 1.07345367 (blue) and 1.03349367 (green). The composite pattern is visibly different than in the previous case.
Above I show three hexagonal nets with parameters 1 (red), 1.07345367 (blue) and 1.11349367 (green).
And in the end, there is nothing special in hexagonal nets. Moire pattern can also be seen on other periodic nets. Above I show three square nets with characteristic sizes of 1.1 (red), 1.2945367 (blue) and 1.37349367 (green), and below are three triangular nets with parameters 1.0 (magenta), 1.054 (blue) and 1.107 (green).
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Last updated on 1st of January 2011.