The product

Two stacked transparencies pass the product of their transmittances. For two sinusoidal gratings the product is a sum and a difference:

The sum is finer than either grating and the difference coarser. Anything that cannot resolve the gratings, an eye at a distance or a lens slightly out of focus, removes both of them and the sum, and what is left is the difference. The moiré is not in either layer. It is the low-frequency part of their product.

Period and direction

For line gratings of pitches $a$ and $b$ at angle $\theta$, the difference vector has length

At $\theta = 0$ this is the beat, $ab/|a-b|$; at $a = b$ it is $a/(2\sin(\theta/2))$. The fringes run perpendicular to the difference vector, which for two identical gratings means almost perpendicular to the lines themselves, bisecting the obtuse angle between them.

Magnification

Translating layer $B$ by $\delta$ shifts its phase by $\delta/b$ of a period, and the moiré, being a difference of phases, shifts by the same fraction of its own period. The fringes move $\Lambda/b$ times as far as the layer. For pitches of 10 and 11 that is ten: the principle of the vernier, and of moiré strain measurement, where a deformation too small to see moves fringes large enough to count.

Beyond the first order

Sharp lines have harmonics, and every product of a harmonic of one layer with a harmonic of the other lands somewhere. When the pitches are near a ratio of two, $\mathbf{k}_A - 2\mathbf{k}_B$ lands near zero and is the moiré you see; the first-order formula predicts something finer than the lines. Sweep the pitch ratio and the visible period diverges at every simple fraction.

A line of duty $d$ has harmonics proportional to $\sin(\pi n d)/(\pi n)$, which vanish whenever $nd$ is an integer. At a duty of one half every even harmonic is zero, and every moiré that needs one disappears.

Addition makes none

Two projectors add light instead of multiplying it. A sum of two Fourier series has no cross terms, so no difference frequency. The bands visible at full sharpness are a beat in local contrast, and blurring past the pitch leaves a uniform field. Measured, the coarse structure that survives is one ten-thousandth of what survives when the same layers are stacked.

Lattices, and graphene

Two identical triangular lattices twisted by $\theta$ form a triangular superlattice of constant $a/(2\sin(\theta/2))$. With carbon's $a = 0.246$ nm and $\theta = 1.1^\circ$ that is 12.8 nm, the superlattice of magic-angle graphene, whose flat electronic bands produce correlated insulating and superconducting states. The playground reproduces the geometry and nothing more; the magic angle itself depends on the coupling between the layers.

The screen is a grating

A display samples on a lattice of pixels, and a pattern with frequencies above half the pixel frequency folds back below it. A single grating of pitch 3.1 px drawn without antialiasing shows a 31 px pattern: its third harmonic beating against the pixels. Drawing each pixel with the ink it actually covers makes the pixel a box filter, whose zeros fall at multiples of the pixel frequency, exactly where the frequencies that would fold to near zero sit. The pattern falls by a factor of about thirty. The same folding is why a camera pointed at a screen records moiré the eye does not see.

Limits

Geometric optics only: gratings near the wavelength of light diffract, and none of that is here. The viewer is a Gaussian blur, where the eye is a band-pass filter whose peak moves with viewing distance. Ring moirés are predicted locally and fail within about a pitch of a centre. Each of these is listed, with what would falsify it, in the assumptions panel.

Model