Composing the first frame…
The Order Beneath · No.02 · Live plates
Fourier analysis
Plates on waves and the frequencies they are made of, each one computed in the browser as you watch: a curve drawn by circles turning on circles, the figures that sand draws on a sounding plate, the crystals and quasicrystals made by adding plane waves, the moiré of two gratings laid one over the other, and white light spread by a hole into the hole's own Fourier transform. Every plate has its spectrum in the corner, the same picture seen as a list of frequencies; in the last plate the roles swap, and the corner shows the hole.
Plate I · 4 movements · 1:00 loop
Each term cne2πins of a Fourier series is a circle of radius |cn| that turns n times while the pen goes round once. Put each circle on the rim of the one before and the pen, on the rim of the last, draws the sum. The spectrum in the corner shows which circles are turning and how large each one is.
The term cne2πins is a point going round a circle of radius |cn|, n times per turn of the pen, starting at the angle of cn. A negative n turns the other way. The spectrum lists those radii; the picture is what they add up to.
A partial sum of a Fourier series is the function smoothed by the Dirichlet kernel, which has negative lobes. More terms make the lobes narrower but no shallower, so next to a jump the overshoot moves closer and stays at about 0.179 for a jump of 2.
A grain on a vibrating plate is thrown about until it lands somewhere that does not move. The nodal lines are where u=0, and near them the distance to the line is about |u|/|∇u|; the plate uses a version of that measure which is exact for a single wave and largest at the antinodes, so the sand clears from there first.
The square's modes φmn and φnm always share a note, by symmetry. A number such as 65=12+82=42+72 gives an accidental coincidence on top of that: four modes, one note, and a family of figures that no single pair of them can make.
A pattern that repeats in two directions can be turned onto itself only by a half, a third, a quarter or a sixth of a turn: the crystallographic restriction. The sum of five waves has ten-fold symmetry because it never repeats. Dan Shechtman found the same ten-fold order in the diffraction pattern of a real alloy in 1982.
What makes the third plate a quasicrystal and not noise is its spectrum: a handful of sharp points on a circle. A random pattern would spread its spectrum over a whole disc; a crystal's points would sit on a lattice.
Two notes a few hertz apart swell and fade at the difference of their frequencies. Two gratings a few per cent apart do the same across space, and the eye, which cannot resolve the slits, sees only the slow difference. The moiré is a beat drawn in light.
Far from an opening, or in the focal plane of a lens, light from every point of the opening arrives at each point of the screen with a phase that depends on direction. Adding those contributions is exactly the Fourier integral, so the pattern on the screen is the transform of the opening (Fraunhofer).
The pattern for light of wavelength λ is the transform evaluated at q=s/λ, where s is the position on the screen. Red light makes the same pattern as blue, only larger, so wherever the pattern is not flat the colours come apart. At the centre every colour is at its brightest and the light stays white.