Spectral

Prologue

1. 0:05A circle turns. On its rim rides a second, on that a third, and so on, each turning at a steady rate. A pen on the last one draws a snowflake.
2. 0:14More circles bring the curve closer to the snowflake, with finer points. Zoom in, though, and it's still only an approximation.
3. 0:21There's another way to write this snowflake down, in full: a list of every circle's speed, radius and starting angle.
4. 0:28That list is called its spectrum.
5. 0:30This film is about how waves and patterns come apart into circles, and why the spectrum is worth looking at.

IOne circle

1. 0:40A point circles at a steady rate; its height over time traces a sine wave. The peak stands one radius above the midline, and each turn makes one rise and fall.
2. 0:50A period is the time the whole drawing takes to repeat. Speed is turns per period: at speed 2, the wave rises and falls twice.
3. 0:58Here, speeds are whole numbers of turns; negative means turning the other way.
4. 1:03Change the starting angle, and the whole wave slides sideways. Three numbers pin down a turning circle: radius, speed and starting angle.

IICircles on circles

1. 1:16Mount a second circle on the first arm's tip. Both angles are measured from the horizontal: each arm turns on its own. The pen's height is the two arms' heights added.
2. 1:26All arms start pointing right; odd speeds, each radius one over its speed.
3. 1:31The more circles, the closer to a square wave: flat top and bottom, a steep jump.
4. 1:36Scramble only the starting angles: no square wave at all. Line them up again, and the square wave is back.
5. 1:43But beside each jump, the curve always overshoots.
6. 1:46More circles never flatten the overshoot; it only narrows, crowding toward the jump. Anywhere away from the jump, enough circles bring the curve close to the square wave.
7. 1:56Other periodic waves can generally be built from circles too. Fourier proposed this, studying heat; later mathematicians worked out exactly when it holds.
8. 2:06Listing every circle's size and starting angle, speed by speed, gives the wave's spectrum.
9. 2:12Shift the wave up, and the only change is a piece that doesn't turn: speed zero.

IIIFinding the circles

1. 2:20Now the reverse question: the pen goes once round, steadily, anticlockwise, from a fixed start. Knowing only its positions, can we recover the circles?
2. 2:30Start simpler: three circles, speeds 1, 2 and 5, draw a curve. Change their order, and the pen's path stays the same.
3. 2:38Now set the whole picture on a turntable turning backwards at speed 2, and watch from outside: the speed-2 arm stands still, and the other speeds appear 2 lower.
4. 2:48From outside, average the pen's position over one period.
5. 2:52The other circles make whole turns, at least one, and average to zero. The still arm remains: its length is the radius, its direction the starting angle.
6. 3:02This needs only the pen's positions, not the circles: try one speed at a time; absent speeds average to zero.
7. 3:09That gives the whole spectrum.
8. 3:11The first two chapters tracked only height. A sine wave is a pen going straight up and down: two half-size circles turning opposite ways, sideways motions cancelling, vertical ones adding.
9. 3:23So, except at speed zero, any up-and-down wave comes apart into matched pairs: equally tall bars on either side.
10. 3:30Turned 60°, the snowflake matches itself. Its spectrum is choosy: only speeds 1, −5, 7, −11 and so on can be nonzero, one more than multiples of 6.
11. 3:43The pen moves steadily, so every sixth of a period it is 60° further round.
12. 3:48In that time, a circle's turn must also line up with 60°; the rest cancel out over the six pieces. Symmetry leaves its fingerprint on the spectrum.

IVStrings and plates

1. 4:01An ideal string, fixed at both ends, vibrates freely in basic patterns called modes: along the string, one, two, three half-waves, and so on.
2. 4:11Watch one point: it vibrates at one, two, three times the lowest frequency, the circles' whole-number speeds. That's where a string instrument's overtones come from.
3. 4:21Pluck the string: its motion is the modes added together, each at its own frequency. Their amounts are this pluck's spectrum.
4. 4:29Pluck elsewhere: same pitch, different spectrum, different tone. Pluck in the middle, where the even modes stay still, and they get nothing at all: symmetry's fingerprint again.
5. 4:39A plate has modes too, each with its own frequency. Sand shaken off the moving parts gathers on lines that stay still: nodal lines.
6. 4:48Real plates are hard to calculate. In a simplified model, two whole numbers label each mode, counting its nodal lines: m down, n across.
7. 4:58Sand shows only what stays still, not how much each place moves.
8. 5:02Turned 90°, mode (2, 5) becomes (5, 2): the square plate is unchanged, and so is the pitch.
9. 5:09Each mixture of the two is a mode too: same pitch, but a different sand pattern and spectrum.
10. 5:15Mix in another pitch, and the pattern keeps changing; the sand can't settle.
11. 5:20In this model, pitch depends only on m² + n². Sixty-five is a sum of two squares in two ways: four modes, one pitch.

VCrystals and quasicrystals

1. 5:33Now spread a sine wave across a flat surface: its crests become parallel stripes, a plane wave.
2. 5:39Its spectrum, like a sine wave's, is a pair of points either side of the centre, on a line perpendicular to the stripes.
3. 5:46Their distance from the centre now counts ripples per unit length, not turns per period: denser stripes, points further out. Flat patterns can generally be taken apart into such stripes.
4. 5:57Two waves of the same spacing, at right angles, make a square grid; its spectrum is four points on a circle.
5. 6:04Three at 60° make a hexagonal grid; its spectrum is six points. The pattern repeats on a lattice, like a crystal.
6. 6:12Five waves of equal strength and spacing, each 36° from the next, all bright at the centre: turn it 36° and it matches itself. That's tenfold symmetry.
7. 6:24Shift it, though, and it never matches exactly; some shifts come close. This is a quasicrystal; this pattern's spectrum is still just ten points on a circle.
8. 6:33Why does it never match? If a pattern repeats when shifted, its rotational symmetry can only be 2-, 3-, 4- or 6-fold.
9. 6:41If it did repeat, there'd be a shortest shift. Tenfold symmetry turns it 36° into another shift, and the step between them is a shift too, but shorter: a contradiction.
10. 6:52So tenfold symmetry and repetition can't coexist.

VILight lays out the spectrum

1. 6:59Parallel light of one colour, once through a small hole, can be taken apart into plane waves in every direction.
2. 7:05A lens brings each direction's light to one point on its focal plane: there lies the small hole's spectrum.
3. 7:12A single hole's spectrum is continuous: a round hole gives a bright central spot with faint rings around it, the Airy pattern.
4. 7:20Light has done the taking apart for us; but all we can see is brightness. The starting angles are lost.
5. 7:26Keep the hole's shape, halve its width, and the pattern doubles in size.
6. 7:30Shrink a shape, and its spectrum spreads out: that holds in general. The steeper a square wave's jump, the higher the speeds it needs.
7. 7:39Use a square hole, and four streaks appear, in two lines: one for each pair of parallel edges.
8. 7:45In a triangle, each edge sends a line through the centre, perpendicular to that edge, a streak each way: three separate lines, six streaks.
9. 7:54A hexagon also gives six; a pentagon, ten.
10. 7:57Evenly spaced slits make a grating: same spacing, more slits, and the bright bands sharpen into separate lines.
11. 8:04So far, it's the holes we've taken apart. Now light itself: it's a vibration too, and colour is how fast it vibrates.
12. 8:12Send white light through: the central line stays white, and each of the others spreads into a spectrum; red, vibrating slowest, bends furthest out.
13. 8:22That spectrum and this film's spectrum mean the same thing: laid out by frequency.
14. 8:27With light of one colour, holes in a lattice give mostly a lattice of bright spots: more holes, sharper spots; closer holes, spots further apart.
15. 8:36Atoms in a crystal aren't holes, but the X-rays they scatter also add up only in certain directions, giving diffraction spots.
16. 8:44The spots, too, record only brightness, not starting angles, so working out how the atoms are arranged takes more effort.
17. 8:51Electrons diffract too. In a rapidly cooled aluminium-manganese alloy, Shechtman saw sharp electron-diffraction spots, with tenfold symmetry.
18. 9:00Sharp spots mean the atoms are highly ordered; but no lattice can have tenfold symmetry. This was a quasicrystal: ordered, yet never repeating.

Coda

1. 9:11With each circle's size and starting angle recorded, shape and spectrum are two sides of one thing: from either, you can compute the other.
2. 9:19Sand and spots of light each show only part of a spectrum; yet that part is enough to reveal the order hidden beneath.
3. 9:26Hence the title: Spectral.