Moiré pattern

Beautiful two-dimensional moiré patterns in the Rotterdam metro. The pictures were taken from different viewpoints to demonstrate how the patterns change with distance.

Once you become aware of these patterns you see them everywhere. Of course Marcel Minnaert has noticed this phenomenon and has devoted a long paragraph to it. But he describes the one-dimensional case only:

55. Beats between Two Sets of Railings 
Whenever one ean see the posts of one set of railings between the posts of another set, one perceives broad light and dark bands in the intensity of the light, which move when one moves.

These are due to the fact that the apparent distance between the posts of the two sets of railings differs more or less, either because the one has wider spaces than the other, or because they are at different distances from our eye. In certain directions the posts seem to coincide, and in others the posts of the first railing fill exactly the space between the posts of the second, so that a difference arises in the average brightness. We can say that they are 'in step' or 'out of step.'

When one has once noticed these beats, one sees them in all sorts of places. Every bridge with a parapet in the form of a railing on both sides shows these undulations in intensity when seen from a certain distance. They appear, too, when one sees the shadow of a railing between its own posts, in which case the period is the same, but the distance to our eye is different.

In some stations a goods-lift is surrounded by wire-netting, and the combination of the side nearest to us and the side farthest away forms a kind of moiré, such as one sees when one lays two pieces of wire-gauze on one another, or two combs with unequal distances between the teeth.

Marcel Minnaert then proceeds to analyze the mathematics of the situation as follows:

One interference wave (one beat) between the two patterns occurs when the differences between the two patterns add up to the distance between two posts:

n = g1 / (g1 - g2)
and:
g2 = L / x2
g1 = L / x1
so:
n = x2 / (x2 - x1)
and:
x2 - x1 is constant

If you walk further away from the pattern one beat / wave will contain more units, but the angle that is covered by one beat will remain constant: n * g2 =  L / (x2 - x1).


Wikipedia - Moiré pattern has also derived the mathematics for us. But their derivation is different from the one used by Marcel Minnaert.