A Moiré pattern is the interference pattern that emerges when you superimpose two grids that nearly line up. For instance, you could have two identical metal sheets with circular holes. If you put one in front of the other, and slightly rotate it, then you get a Moiré pattern.
A reader expressed curiosity about what would happen if you replaced the circles with a different shape. As it happens, I am a former condensed matter physicist, which makes me a sort of expert in lattices. So I already know the answer, but let me walk you through with illustrations.
Here’s a second example of a Moiré pattern. This one uses two hexagonal grids of square holes. One grid is 10% larger than the other. (In general Moiré patterns can be formed by all sorts of distortions, but in this article, I stick to slight rotations and slight size adjustments). Something you might notice is that the Moiré pattern has a definite square-ness to it.
So far we have…
Circular hole -> Circular Moiré pattern
Square hole -> Square Moiré pattern
Can we extrapolate, and say that a cat-shaped hole will produce a cat-shaped Moiré pattern, or that an Eiffel-Tower-shaped hole will produce an Eiffel-Tower-shaped Moiré pattern? Unfortunately not. I will illustrate this with the simplest shape possible, an L. Here’s what the Moiré pattern looks like.
The Moiré pattern definitely has a shape, something like a diagonal barbell. But it is definitely not the same L-shape as the holes.
What you’re looking at is the autocorrelation function of the L shape. The autocorrelation function is equal to the area of the overlap between two L shapes, when one is shifted relative to the other. The center of the Moiré pattern corresponds to a shift of zero. If one grid is made larger than the other grid, then the left and right sides of the Moiré pattern correspond to a horizontal shift. The top and bottom of the Moiré pattern correspondes to a vertical shift, and so on. I illustrate this in the image below.
If instead of making one grid larger than the other, we rotate the grids relative to each other, then the Moiré pattern consists of the autocorrelation function rotated by 90 degrees.
Unfortunately, autocorrelation functions cannot be used to make arbitrary shapes. For one thing, autocorrelation functions have 180 degree rotational symmetry. So, no cat-shaped patterns.
Unless! You use differently shaped holes for the two grids. For instance, one grid can be made from circular holes, and the other from cat-shaped holes. The Moiré pattern will not look like the autocorrelation between a shape and itself, but instead look like the correlation between the two shapes. I tried it out, and it works!