Moiré patterns
Overlay two periodic structures — grids of lines, concentric circles, arrays of dots — and tilt one by a fraction of a degree. Large-scale interference patterns bloom from the overlap. The patterns are not in either layer. They exist only in the relationship between the two.
What are Moiré patterns?
When two periodic structures — grids, line arrays, meshes — are overlaid with a slight mismatch in angle, spacing, or position, a new pattern emerges at a much larger scale. This is a Moiré pattern (from the French moiré, a type of watered silk whose folds produce similar effects). The interference is not physical in the way light-wave interference is: it is purely geometric. Each layer is transparent and regular. The apparent large-scale structure exists only in the overlap. Your visual system is detecting the beat frequency of two spatial periodicities — the same phenomenon that produces beats in sound when two nearly equal frequencies are played together.
The mathematics
The basic principle is the beat frequency. When two gratings with spatial frequencies f1 and f2 are superimposed, the Moiré pattern has frequency |f1 − f2|. For two identical gratings rotated by a small angle θ, the Moiré period is d / (2 sin(θ/2)), where d is the grating period. At small angles, this simplifies to approximately d / θ (in radians). A 1° rotation of a 5-pixel-spaced grid produces a Moiré with a period of about 286 pixels — the pattern is magnified by the reciprocal of the angle. This is why Moiré fringes are so sensitive to misalignment: a tiny angular change produces a dramatic visual change. The effect is essentially a spatial analog of heterodyning in radio engineering.
Where they appear
Moiré patterns appear whenever periodic structures overlap in everyday life. Photograph a screen with a digital camera and you see Moiré fringes — the camera’s pixel grid interfering with the screen’s pixel grid. Fold sheer fabric and the overlapping weaves create undulating patterns. Security printing on banknotes and identity documents uses Moiré deliberately: microprinted line patterns that produce visible symbols when photocopied (the copier’s scan resolution creates the second grid). In engineering, Moiré interferometry measures surface strain to sub-micron precision by overlaying a deformed grating with a reference grating. In art, the Op Art movement of the 1960s — Bridget Riley, Victor Vasarely — exploited these interference effects for their hypnotic visual energy.
Aliasing and sampling theory
Moiré patterns are a manifestation of aliasing — the same phenomenon that makes wagon wheels appear to spin backward in film. When you sample a signal (spatial or temporal) at a rate too close to the signal’s own frequency, you get a false low-frequency component: the alias. The Nyquist–Shannon sampling theorem sets the boundary: you need at least two samples per period to faithfully represent a frequency. Below that rate, the high-frequency signal masquerades as a low-frequency one. The overlapping grid is effectively sampling one periodic pattern with another, and the Moiré fringe is the alias. This is why cameras have anti-aliasing filters — low-pass filters placed before the sensor to blur away frequencies that would alias against the pixel grid and create Moiré artifacts.
The Moiré magnifier
A Moiré magnifier uses this interference principle to make tiny patterns visible at large scale. Print an array of micro-images and overlay a lens array with a slightly different pitch. Each lens shows a slightly shifted view of the micro-image beneath it, and the composite creates a magnified virtual image floating above the surface. The magnification equals the ratio of the lens pitch to the pitch difference — the same beat-frequency formula as before, now applied to produce useful magnification. Security holograms on credit cards and passports use this principle: the holographic micro-lens array and a printed micro-image array combine to show a pattern visible only at the correct viewing angle, making counterfeiting extremely difficult.