Arithmetic

Prologue

1. 0:05Sunlight on a cup of coffee: across the surface, a bright curve with a sharp point.
2. 0:09Nobody drew it. Each ray bounces off the wall; where the bounced rays crowd together, the light is brightest: that's the curve.
3. 0:18This is a film about number theory. That light will be back.

ITimes tables

1. 0:25Ten points on a circle. Join each number to its double: 1 to 2, 2 to 4.
2. 0:31Five doubled is ten. Ten lands on 0, twelve on 2: past nine, keep counting round, like a clock.
3. 0:38A times table, wrapped around a circle.
4. 0:43More points, same rule. Where the lines crowd together, the light adds up, and a heart appears: a cardioid.
5. 0:50None of the lines is curved. The curve is the edge they all touch, where they crowd thickest.
6. 0:56Times 3: a curve with two sharp tips. Times 4, three.
7. 1:00Between whole multipliers, the curve splits open on one side, and the next tip pushes out of the gap.
8. 1:08All the way to times 10: nine tips, one fewer than the multiplier.
9. 1:13Times 242 gives just two hearts.
10. 1:16Even numbers land where times 2 puts them; odd ones, directly opposite.
11. 1:23162 gives three; 122, four. Small multipliers hide inside big ones.
12. 1:30Put a lamp on the rim. A ray hits the wall; its bounce lands twice as far round, measured from the lamp: the times-two table, and the heart.
13. 1:39Move the lamp away, and the rays turn nearly parallel; the heart changes shape.
14. 1:47In parallel sunlight, it's times three, plus half a turn: the curve from the start, half the times-three table.

IIPascal's triangle

1. 1:57Pascal's triangle: a 1 at the top, 1s down both edges, and inside, each number is the sum of the two above.
2. 2:08Now ask only: odd or even? Light the odd ones. Odd plus odd is even, odd plus even is odd: the two above decide it.
3. 2:17Hundreds of rows down: a triangle with its middle cut out, leaving three, each cut the same way. Sierpiński's triangle.
4. 2:25Now divide by 3, and colour by remainder.
5. 2:27Zoom into a small triangle: it's the whole triangle again, in new colours.
6. 2:32Divide by other primes, 5 or 7, and it happens again: triangles within triangles.
7. 2:43Back to odd and even. Each time the rows double, the new rows are two copies of all above, with a hole between them.
8. 2:50It never stops: this triangle has no bottom.

IIIThe golden angle

1. 2:58Seeds placed one by one, each half a turn from the last: the third lines up with the first.
2. 3:04Each a little further out, they form two straight spokes.
3. 3:11A third of a turn gives three spokes; two fifths, five.
4. 3:15In lowest terms, the denominator counts the spokes.
5. 3:18With a small denominator, the seeds line up before long, leaving wide gaps between the spokes. These fractions creep towards one angle.
6. 3:26About 137.5 degrees: the golden angle. Of all angles, it's the hardest for fractions to match.
7. 3:34No gap ever widens: seen from the centre, the gaps stay close in width, and the disc fills evenly.
8. 3:40Roughly how a sunflower head is laid out.
9. 3:43Seeds 13 apart nearly line up: take 13 golden angles, and you're just short of five full turns.
10. 3:50Further out, 21, then 34: Fibonacci numbers, each the sum of the two before.
11. 3:56At the edge, 55. Those denominators come from this list.
12. 4:01At 137.3 degrees, nearer a fraction than the golden angle, 21 gaps open.
13. 4:07At 137.6, nearer still: 34 gaps.
14. 4:12How near a simple fraction it lies: that decides the pattern.

IVPrime spirals

1. 4:21Place whole numbers like the seeds, each turned one radian more than the last: a little short of a sixth of a turn.
2. 4:28So every sixth number nearly lines up, slipping slightly behind: six curved arms.
3. 4:35Step back: the six arms tangle, then 44 arms form.
4. 4:3944 numbers make just over 7 turns: as with the seeds, the denominator sets the count.
5. 4:45Further still, 710 rays: 710 numbers, 113 turns, almost exactly.
6. 4:52Now only the primes: numbers above 1, divisible only by 1 and themselves.
7. 4:57Apart from 2 and 3, they use just two of the six arms; on the other four, every number is a multiple of 2 or 3.
8. 5:04Of 44 arms, they use 20; of 710, just 280.
9. 5:10Another arrangement: wind the numbers into a square spiral, and light up the primes.
10. 5:15In 1963 the mathematician Stanisław Ulam, doodling through a long talk, found them crowding onto diagonals.
11. 5:23Primes look lawless, yet laid out like this, they show order.
12. 5:26Some diagonals can never hold a prime, like this one: the odd squares. Others are thick with them.

VPrimes from zeros

1. 5:36Count through the primes, climbing a step at each, as high as its logarithm.
2. 5:40Powers of a prime climb the same step. So the steps mark where the primes fall.
3. 5:46The staircase wobbles, but keeps close to a straight line.
4. 5:50Euler found a function, built from the whole numbers, that is also a product over the primes.
5. 5:55Riemann extended it across the plane: the zeta function.
6. 5:59Where the colours circle a dark centre, the function is zero: these are its zeros.
7. 6:08The staircase is exactly a smooth curve, plus one wave for each zero: Riemann saw it, and von Mangoldt proved it.
8. 6:17The higher the zero, the finer its ripples. Add them in, and the gold line begins to hug the steps.
9. 6:29Hundreds of waves trace its outline; with every zero, they are the staircase.
10. 6:34Zeta's zeros say where the primes fall.
11. 6:38The zeros in this strip: do they all sit on the middle line?
12. 6:42Over ten trillion have been checked, and every one does. The Riemann hypothesis says they all do. No one has proved it.

Coda

1. 6:51A heart from chords. Triangles within triangles. A sunflower's spirals. Arms of whole numbers.
2. 6:57Even the primes, written in a set of zeros.
3. 7:00Multiply, add, take remainders, count on: a few rules, laid out, and order shows itself.